Applications of Continuous Wave Free Precession Sequences in LowField , Time-Domain NMR

This review discusses the theory and applications of the Continuous Wave Free Precession (CWFP) sequence in low-field, time-domain nuclear magnetic resonance (TD-NMR). CWFP is a special case of the Steady State Free Precession (SSFP) regime that is obtained when a train of radiofrequency pulses, separated by a time interval Tp shorter than the effective transverse relaxation time (T2*), is applied to a sample. Unlike regular pulsed experiments, in the CWFP regime, the amplitude is not dependent on T1. Therefore, Tp should be as short as possible (limited by hardware). For Tp < 0.5 ms, thousands of scans can be performed per second, and the signal to noise ratio can be enhanced by more than one order of magnitude. The amplitude of the CWFP signal is dependent on T1/T2; therefore, it can be used in quantitative analyses for samples with a similar relaxation ratio. The time constant to reach the CWFP regime (T*) is also dependent on relaxation times and flip angle (θ). Therefore, T* has been used as a single shot experiment to measure T1 using a low flip angle (5◦) or T2, using θ = 180◦. For measuring T1 and T2 simultaneously in a single experiment, it is necessary to use θ = 90◦, the values of T* and M0, and the magnitude of CWFP signal |Mss|. Therefore, CWFP is an important sequence for TD-NMR, being an alternative to the Carr-Purcell-Meiboom-Gill sequence, which depends only on T2. The use of CWFP for the improvement of the signal to noise ratio in quantitative and qualitative analyses and in relaxation measurements are presented and discussed.


Introduction
Continuous Wave Free Precession (CWFP) is a special regime of the Steady-State Free Precession (SSFP) sequence introduced by Carr in 1958 [1].The original SSFP sequence uses a train of radiofrequency pulses with the same flip angle and phase, and separated by a time interval (T p ) shorter than the transverse relaxation time, T 2 (T p ≤ T 2 ).In the SSFP regime, the nuclear magnetic resonance (NMR) signal is composed of a free induction decay (FID) after each pulse, and an echo signal preceding the following pulse.When Ernst and Anderson introduced pulse and Fourier transform NMR in 1966 [2], they studied the condition that optimizes the signal to noise ratio (SNR) in high-resolution NMR.They observed that the highest SNR was obtained when T p was shorter than the relaxation times.However, they observed that the Fourier transformed signal shows strong phase and amplitude anomalies when T p reaches the SSFP regime (T p ~T2 ).
In 1971, Freeman and Hill [3] showed that the phase and amplitude anomalies were related to the echo component at the end of the FID and they proposed a procedure, called Scrambled Steady State, to suppress these anomalies using a small variation of T p that cancels the echo component.In the same year, Schwenk [4] proposed the Quadriga Fourier Transform (QFT) to suppress these anomalies.The QFT method was based on a small variation in frequency offset.Although these procedures minimize these anomalies, SSFP sequences were not routinely used in high resolution NMR.
We revised these methods and suggested some modifications to increase the applications of SSFP in high-resolution NMR using the Traff apodization function [5] to minimize the truncation problem, or processing the SSFP time domain signal with a parametric method based on Krylov Basis Diagonalization Method (KBDM) [6].KBDM solves the truncation and phase anomalies but not the amplitude problem.Therefore, SSFP sequences cannot be used to efficiently enhance signal to noise ratio (SNR) in high-resolution NMR.SSFP sequences have been successfully used in magnetic resonance imaging [7,8] and in low-field, time-domain NMR [9][10][11].The use of a SSFP sequence in low field time domain (TD-NMR) started in year 2000 [12,13].The applications of SSFP sequences in TD-NMR can be performed under the drastic condition where T p < 1 ms.Under this condition, a truly continuous wave free precession (CWFP) regime is obtained [13].Normally, CWFP sequences use T p < 0.5 ms, as T p has to be shorter than T 2 *, which is in the order of 1 ms in most TD-NMR spectrometers.
In the CWFP regime, FID and echo signals strongly overlap and are dependent on the precession angle (Φ), where the signal is minimal when the interference is destructive, and maximal when the interference is constructive [12,13].Under the CWFP regime, the signal does not decay and its amplitude does not depend on the longitudinal relaxation time (T 1 ) as in conventional pulsed NMR experiments [12,13].All the proprieties of CWFP methods make them an important tool for industrial protocols, as well as in quantitative analysis where CWFP can enhance the SNR compared with conventional CPMG or FID protocols [14][15][16][17][18][19][20][21][22].
In this review, we outline SSFP/CWFP theory and their applications in low field TD-NMR to enhance SNR, in quantitative analyses, in flow analyses, in thermal diffusivity measurements, for fast simultaneous measurements of T 1 and T 2, , and in the fast T 1 method.CWFP methods have been applied in many conventional TD-NMR spectrometers, such as the Minispec mq20, Bruker (0.47 T) [23], Maran Ultra (0.54 T) [24], SLK-100, SpinLock (0.28 T) [25,26], Tecmag spectrometer [27], and Oxford instrumentations [28,29].The complete ExpSpel code used to implement the CWFP pulse sequence in the Minisepc mq20 spectrometer, Bruker (NF electronics) is available in the Supplementary Materials.

SSFP/CWFP Theory
The SSFP sequence uses a train of radiofrequency pulses (rf) with phase (φ), flip angle (θ), and time between pulses (T p ) shorter than the relaxation time T 2 (T p < T 2 ).SSFP sequences have been used to improve the SNR in pulsed NMR spectroscopy since 1958 [1].In 1966, Ernst and Anderson [2] derived the analytical solution for the SSFP regime.They demonstrated that the SSFP signal is composed of FID and echo signals.The FID component (M+) follows the rf pulses, and the echo component (M-) precedes the pulses, as given by the following equations: where with the precession angle Φ = ΩT p , offset frequency Ω = ω re f − ω 0 , and relaxation components E 1 = exp −T p /T 1 , and These equations allow us to calculate the magnitude of the magnetization in the xy plane after the nth rf pulse in the SSFP condition [4,30]: Therefore, the amplitude of the signal has a strong dependence on the ratio of the relaxation times T 1 and T 2 , the flip angle θ, and the precession angle Φ = Ω T p .Equation (2) shows that the amplitude of the magnetization is null when Φ = 2nπ and is the maximum when Φ = (2n + 1)π, where n is an integer.
These nulls and maximums of amplitude occur due to the destructive or constructive interference between the FID and echo components in the SSFP sequences [12].These complex interferences between FID and echo signals depend on the time interval between pulses T p , precession angle Φ, frequency offset Ω, flip angle θ, and phase alternation [23,24].
The transition from regular pulse sequence to SSFP/CWFP regimes was demonstrated by Azeredo et al. in 2000 [12] and is summarized in Figure 1.The NMR signals of Figure 1A-E were obtained by a numerical simulation using a train of 90 • pulses, T 1 = 200 ms, T 2 = 150 ms, T 2 * = 0.5 ms and T p = 5T 1 (Figure 1A); T p = T 2 (Figure 1B); T p = 2T 2 * (Figure 1C); T p = 0.3 ms < T 2 * and Φ = (2n + 1)π (Figure 1D); and Φ = n2π (Figure 1E).In Figure 1A,B, the pulses (gray bar) are shown at the center of the window of ±2 ms to better present the FID and echo signals [12]. Figure 1C-E show other pulses (gray bars) outside the center.Figure 1A depicts the standard pulse sequence with 5T 1 ≥ T p > 3T 2, where the amplitude of the echo is very small (normally not seen in experiments), and most of the magnetization has returned to the Boltzmann equilibrium.Figure 1B shows the typical SSFP signal after the nth pulse with T 2 * < T p = T 2 .The FID signal has a lower amplitude than in Figure 1A, and a small echo signal forms preceding the pulse.In this regime, the FID/echo does not overlap because of the short T 2 *. Figure 1C, with T 2 > T p > T 2 *, shows the SSFP regime with partial FID and echo overlap.A special SSFP regime occurs when T 2 >> T p < T 2 * (Figure 1D,E).Under this condition, due to the short time pulse intervals T p < T 2 *, the FID and echo signals are fully overlapped.Depending on the precession angle Φ, the FID/echo interference can be constructive for Φ = (2n + 1)π with maximum amplitude (Figure 1D), or destructive for Φ = n2π with minimum of amplitude (Figure 1E).Thus, the SSFP signal does not decay, forming the Continuous Wave Free Precession (CWFP) signal.These equations allow us to calculate the magnitude of the magnetization in the xy plane after the nth rf pulse in the SSFP condition [4,30]: Therefore, the amplitude of the signal has a strong dependence on the ratio of the relaxation times T1 and T2, the flip angle θ, and the precession angle Φ = Ω  .Equation (2) shows that the amplitude of the magnetization is null when Φ = 2 and is the maximum when Φ = (2 + 1), where n is an integer.
These nulls and maximums of amplitude occur due to the destructive or constructive interference between the FID and echo components in the SSFP sequences [12].These complex interferences between FID and echo signals depend on the time interval between pulses Tp, precession angle Φ, frequency offset Ω, flip angle θ, and phase alternation [23,24].
The transition from regular pulse sequence to SSFP/CWFP regimes was demonstrated by Azeredo et al. in 2000 [12] and is summarized in Figure 1.The NMR signals of Figures 1A-1E were obtained by a numerical simulation using a train of 90° pulses, T1 = 200 ms, T2 = 150 ms, T2* = 0.5 ms and Tp = 5T1 (Figure 1A); Tp = T2 (Figure 1B); Tp = 2T2* (Figure 1C); Tp = 0.3 ms < T2* and Φ = (2 + 1) (Figure 1D); and Φ = 2 (Figure 1E).In Figures 1A and 1B, the pulses (gray bar) are shown at the center of the window of ± 2 ms to better present the FID and echo signals [12].Figures 1C to 1E show other pulses (gray bars) outside the center.Figure 1A depicts the standard pulse sequence with 5T1 ≥ Tp > 3T2, where the amplitude of the echo is very small (normally not seen in experiments), and most of the magnetization has returned to the Boltzmann equilibrium.Figure 1B shows the typical SSFP signal after the nth pulse with T2* < Tp = T2.The FID signal has a lower amplitude than in Figure 1A, and a small echo signal forms preceding the pulse.In this regime, the FID/echo does not overlap because of the short T2*. Figure 1C, with T2 > Tp > T2*, shows the SSFP regime with partial FID and echo overlap.A special SSFP regime occurs when T2 >> Tp < T2* (Figures 1D and 1E).Under this condition, due to the short time pulse intervals Tp < T2* , the FID and echo signals are fully overlapped.Depending on the precession angle Φ, the FID/echo interference can be constructive for Φ = (2 + 1) with maximum amplitude (Figure 1D), or destructive for Φ = 2 with minimum of amplitude (Figure 1E).Thus, the SSFP signal does not decay, forming the Continuous Wave Free Precession (CWFP) signal.According to Equation (2), the magnitude of the CWFP regimes, using θ = 90 • and Φ = (2n + 1)π, is dependent on the T 1 /T 2 ratio: This equation shows that the magnitude of the CWFP regimes does not depend on the pulse interval T p , i.e., does not depend on T 1 [12,13].Therefore, it is possible to acquire thousands of CWFP signals during one T 1 period, enhancing the SNR by more than one order of magnitude during the same average time used for FID or echo signals in a conventional pulse sequences [12].Given these proprieties, CWFP methods have been used for quantitative analysis in conventional benchtop spectrometers and in line quality control systems, as is discussed in the next section.
This equation shows that the magnitude of the CWFP regimes does not depend on the pulse interval Tp, i.e., does not depend on T1 [12,13].Therefore, it is possible to acquire thousands of CWFP signals during one T1 period, enhancing the SNR by more than one order of magnitude during the same average time used for FID or echo signals in a conventional pulse sequences [12].Given these proprieties, CWFP methods have been used for quantitative analysis in conventional benchtop spectrometers and in line quality control systems, as is discussed in the next section.
For these sequences, the CWFP regime of the magnetization is obtained after a transition time T* [30], given by: These sequences have been used to calculate T1 and T2 relaxation times in a single experiment using 90° pulses.To calculate T1 using T*, it is necessary to use small flip angle (θ1 ~ 5° to 10°) in CP-CWFPx-x.The sequence was named CWFP-T1 [23].One of the major differences between CWFP sequences with and without phase alternation is in the excitation profile (Figure 3). Figure 3A shows the CWFP and CP-CWFP excitation profiles for pulses with same phase (φ1 = φ2).The maximum and minimum amplitudes are obtained when Φ = (2 + 1) and Φ = 2, respectively.Note that the amplitude for resonance experiments is the minimum, and the distance between minimums of signal amplitudes are spaced in frequency by ∆ = 1/ .Figure 3A shows the importance of setting the resonance frequency according to Tp to For these sequences, the CWFP regime of the magnetization is obtained after a transition time T* [30], given by: These sequences have been used to calculate T 1 and T 2 relaxation times in a single experiment using 90 • pulses.To calculate T 1 using T*, it is necessary to use small flip angle (θ 1 ~5• to 10 • ) in CP-CWFPx-x.The sequence was named CWFP-T 1 [23].
One of the major differences between CWFP sequences with and without phase alternation is in the excitation profile (Figure 3). Figure 3A shows the CWFP and CP-CWFP excitation profiles for pulses with same phase (φ 1 = φ 2 ).The maximum and minimum amplitudes are obtained when Φ = (2n + 1)π and Φ = 2nπ, respectively.Note that the amplitude for resonance experiments is the minimum, and the distance between minimums of signal amplitudes are spaced in frequency by ∆ f = 1/T p . Figure 3A shows the importance of setting the resonance frequency according to T p to form signals with maximum amplitude.For example, for T p = 300 µs (solid line), the maximum signal occurs at 2 KHz and shifts to 2.5 KHz for T p = 200 µs (dashed line).Monaretto et al. [24] showed that the CWFP and CP-CWFP sequences with 180 • phase alternation, called CWFPx-x and CP-CWFPx-x, respectively, always show the maximum signal on resonance, regardless of the T p (Figure 3B).As CP-CWFPx-x shows a larger dynamic range for T* decay and is independent of the T 1 /T 2 ratio, it has been considered the best CWFP sequence [24].form signals with maximum amplitude.For example, for Tp = 300 μs (solid line), the maximum signal occurs at 2 KHz and shifts to 2.5 KHz for Tp = 200 μs (dashed line).Monaretto et al. [24] showed that the CWFP and CP-CWFP sequences with 180° phase alternation, called CWFPx-x and CP-CWFPx-x, respectively, always show the maximum signal on resonance, regardless of the Tp (Figure 3B).As CP-CWFPx-x shows a larger dynamic range for T* decay and is independent of the T1/T2 ratio, it has been considered the best CWFP sequence [24].

Applications of CWFP in Steady State Regime
The first application of SSFP/CWFP sequences in low-field TD-NMR was the enhancement of SNR for quantitative analysis [12].Compared to dynamic nuclear polarization and hyperpolarization [31] techniques, CWFP improves SNR without using sophisticated instrumentation and/or special chemical agents [12].Therefore, it can be applied in any standard low-field NMR spectrometer.
The enhancement in SNR using the CWFP sequence occurs because the magnitude of signal is not dependent on T1, but is dependent on the T1/T2 ratio (Equation ( 3) for θ = 90° and Φ = (2 + 1)).Therefore, the pulse interval Tp can be as short as 100 μs.Normally, Tp is used between 300 to 500 μs depending on T2*.A greater enhancement is observed for samples with longer T1.This way, a one order of magnitude SNR gain is easily obtained for a sample with T1 ≈ 1 s using the same average time of FID or echo signals, using conventional pulse sequences with a 5T1 waiting time.
Figure 4 shows an enhancement of more than one order of magnitude in SNR of a single corn seed in 15 s [13].Figure 4A shows FID and echo acquisition, using pulse sequence π/2 -τe -π, where the echo is observed at 2τe, with τe = 3.5 ms.The FID and echo signal data were acquired with 16 scans.For the CWFP experiment, Figure 4B, more than 16,000 scans were accumulated in the same total time of 15 s.

Applications of CWFP in Steady State Regime
The first application of SSFP/CWFP sequences in low-field TD-NMR was the enhancement of SNR for quantitative analysis [12].Compared to dynamic nuclear polarization and hyperpolarization [31] techniques, CWFP improves SNR without using sophisticated instrumentation and/or special chemical agents [12].Therefore, it can be applied in any standard low-field NMR spectrometer.
The enhancement in SNR using the CWFP sequence occurs because the magnitude of signal is not dependent on T 1 , but is dependent on the T 1 /T 2 ratio (Equation (3) for θ = 90 • and Φ = (2n + 1)π).Therefore, the pulse interval T p can be as short as 100 µs.Normally, T p is used between 300 to 500 µs depending on T 2 *.A greater enhancement is observed for samples with longer T 1 .This way, a one order of magnitude SNR gain is easily obtained for a sample with T 1 ≈ 1 s using the same average time of FID or echo signals, using conventional pulse sequences with a 5T 1 waiting time.
Figure 4 shows an enhancement of more than one order of magnitude in SNR of a single corn seed in 15 s [13].Figure 4A shows FID and echo acquisition, using pulse sequence π/2 -τ e -π, where the echo is observed at 2τ e , with τ e = 3.5 ms.The FID and echo signal data were acquired with 16 scans.For the CWFP experiment, Figure 4B, more than 16,000 scans were accumulated in the same total time of 15 s.Another advantage of CWFP in quantitative analyses is the direct correlation between the |Mss| value and the concentration M0 for samples with the same T1/T2 ratio, enabling samples with similar chemical composition to be measured quantitatively.Figure 5 shows the linearity correlation between |Mss| and concentration of vegetable oil in CCl4 (Figure 5A), and hydrogen content in several organic solvents (Figure 5B).CWFP signals are less demanding than standard pulse sequences in flip angle calibration, B1 homogeneity, and magnetic field drift, which are common problems in low-field TD- Another advantage of CWFP in quantitative analyses is the direct correlation between the |M ss | value and the concentration M 0 for samples with the same T 1 /T 2 ratio, enabling samples with similar chemical composition to be measured quantitatively.Figure 5 shows the linearity correlation between |M ss | and concentration of vegetable oil in CCl 4 (Figure 5A), and hydrogen content in several organic solvents (Figure 5B).CWFP signals are less demanding than standard pulse sequences in flip angle calibration, B 1 homogeneity, and magnetic field drift, which are common problems in low-field TD-NMR [13].Another advantage of CWFP in quantitative analyses is the direct correlation between the |Mss| value and the concentration M0 for samples with the same T1/T2 ratio, enabling samples with similar chemical composition to be measured quantitatively.Figure 5 shows the linearity correlation between |Mss| and concentration of vegetable oil in CCl4 (Figure 5A), and hydrogen content in several organic solvents (Figure 5B).CWFP signals are less demanding than standard pulse sequences in flip angle calibration, B1 homogeneity, and magnetic field drift, which are common problems in low-field TD-NMR [13].CWFP was also used for quantitative analyses in flowing samples [11,32].Experimental and theoretical studies showed that CWFP is much more dependent on flow than SSFP [11].CWFP is sensitive to a smaller magnetic field gradient than SSFP, dependent on frequency offset and flow direction.Therefore, tailored sensitivity can be produced by the tuning frequency offset, which is a useful property for fine flow monitoring and flow control [11].Pusiol [33] proposed the use of these characteristics for real-time measurement of proportion and flow-rate of multicomponent complex CWFP was also used for quantitative analyses in flowing samples [11,32].Experimental and theoretical studies showed that CWFP is much more dependent on flow than SSFP [11].CWFP is sensitive to a smaller magnetic field gradient than SSFP, dependent on frequency offset and flow direction.Therefore, tailored sensitivity can be produced by the tuning frequency offset, which is a useful property for fine flow monitoring and flow control [11].Pusiol [33] proposed the use of these characteristics for real-time measurement of proportion and flow-rate of multicomponent complex fluid in high speed.The main advantages are the improvement in the SNR and the steady state signal depends on the fluid velocity, which is virtually insensitive to self-diffusion effects.
Another example of quantitative measurements in flow using the CWFP regime is the determination of oil content in seeds.Figure 6A shows the CWFP signal for 48 oilseeds flowing at 13 cm/s [32].In this figure, each peak is due to a single seed and the peak amplitude is proportional to oil content (Figure 6B).Therefore, the CWFP method has the potential to measure the oil content in more than 20,000 intact seeds per hour.For this measurement, it is necessary that oil from the sample, from the same or different species, have a similar T 1 /T 2 ratio [32].fluid in high speed.The main advantages are the improvement in the SNR and the steady state signal depends on the fluid velocity, which is virtually insensitive to self-diffusion effects.Another example of quantitative measurements in flow using the CWFP regime is the determination of oil content in seeds.Figure 6A shows the CWFP signal for 48 oilseeds flowing at 13 cm/s [32].In this figure, each peak is due to a single seed and the peak amplitude is proportional to oil content (Figure 6B).Therefore, the CWFP method has the potential to measure the oil content in more than 20,000 intact seeds per hour.For this measurement, it is necessary that oil from the sample, from the same or different species, have a similar T1/T2 ratio [32].The CWFP signal was also used to monitor fast polymerization reactions.Figure 7 shows the CWFP signals during photo-polymerization reactions of methacrylate blends using three photo initiators: phenyl-propanedione (PPD-a), monoacylphosphine oxide (MAPO-b), and canphorquinone-c.The CWFP sequence was applied to the sample to reach the steady state regime with a constant |Mss| amplitude.Then, the sample was illuminated with a blue light, and CWFP decay was measured as a function of time.During the polymerization reaction, T2 values decayed, reducing the magnitude of |Mss|.Therefore, Figure 7 shows that CWFP can be used to monitor fast polymerization of these compounds largely used in cavity fillings in tenths of seconds.This figure shows that canphorquinone (photo initiators c), is the most effective initiator used and the The CWFP signal was also used to monitor fast polymerization reactions.Figure 7 shows the CWFP signals during photo-polymerization reactions of methacrylate blends using three photo initiators: phenyl-propanedione (PPD-a), monoacylphosphine oxide (MAPO-b), and canphorquinone-c.The CWFP sequence was applied to the sample to reach the steady state regime with a constant |M ss | amplitude.Then, the sample was illuminated with a blue light, and CWFP decay was measured as a function of time.During the polymerization reaction, T 2 values decayed, reducing the magnitude of |M ss |.Therefore, Figure 7 shows that CWFP can be used to monitor fast polymerization of these compounds largely used in cavity fillings in tenths of seconds.This figure shows that canphorquinone (photo initiators c), is the most effective initiator used and the methacrylate blend was completely polymerized in less than 20 s [34].The same principle of CWFP decay due to a solidification process was used to measure the thermal diffusivity of natural rubber.The experiments were performed with small pieces of rubber.After the CWFP reached the steady state with constant |Mss|, liquid nitrogen was added to the samples [35].The time constant of the CWFP decay was used to calculate the rubber thermal diffusivity and the result was in the same order of magnitude as the thermal diffusivity measured using a standard method [35].
The CWFP sequence with Tp = 300 μs was used in high-resolution NMR to suppress nondeuterated solvent signals.The frequency offset was adjusted to suppress the solvent signal on resonance.The experiments used a CWFP train with a few hundred pulses, producing a null signal for the solvent line on resonance before the acquisition.The method has been used in onedimensional (1D) and two-dimensional (2D) high-resolution NMR experiments [36].

Fast and Simultaneous T1 and T2 Measurements
In 1977, Kronenbitter and Schwenk [30] proposed the use of the transient time T* (Equation ( 4)) of the SSFP signal to measure T1 and T2 using two experiments.In the first experiment, the T1/T2 ratio was determined by measuring the amplitude of the SSFP signals as a function of the flip angle θ.The second experiment involved measuring the transient time T* at the optimum flip angle, yielding the T1 + T2 value.These results were used for the determination of both relaxation times.
Exploring the same dependence on the CWFP regime, Venâncio et al. [9] demonstrated that it is not necessary to use these two experiments to measure T1 and T2 when θ = 90°.They proposed a fast method to measure both relaxation times in a single CWFP experiment using the evolution of the magnitude of the CWFP from the first pulse to the steady state (Figure 8A).In this figure, the initial part of the signal (dark grey region) is characterized by a strong oscillation of the amplitude following odd and even pulses [9].The amplitude of this signal decays with T2*.When the oscillation ends, the signal was named a quasi-stationary state (QSS) (transition between dark and light grays regions).From the QSS point, the signal decays slowly with a time constant T* (light grey region), given by Equation ( 5), to reach the constant steady state amplitude |Mss| (white region).For a CP-CWFP signal, the first oscillating signal (Figure 8B, dark grey) decays to a minimum signal (QSS).From this point, the signal also increases with T* (light grey) and reaches a constant steady state amplitude |Mss| (white region) [27].The same principle of CWFP decay due to a solidification process was used to measure the thermal diffusivity of natural rubber.The experiments were performed with small pieces of rubber.After the CWFP reached the steady state with constant |M ss |, liquid nitrogen was added to the samples [35].The time constant of the CWFP decay was used to calculate the rubber thermal diffusivity and the result was in the same order of magnitude as the thermal diffusivity measured using a standard method [35].
The CWFP sequence with T p = 300 µs was used in high-resolution NMR to suppress non-deuterated solvent signals.The frequency offset was adjusted to suppress the solvent signal on resonance.The experiments used a CWFP train with a few hundred pulses, producing a null signal for the solvent line on resonance before the acquisition.The method has been used in one-dimensional (1D) and two-dimensional (2D) high-resolution NMR experiments [36].

Fast and Simultaneous T 1 and T 2 Measurements
In 1977, Kronenbitter and Schwenk [30] proposed the use of the transient time T* (Equation ( 4)) of the SSFP signal to measure T 1 and T 2 using two experiments.In the first experiment, the T 1 /T 2 ratio was determined by measuring the amplitude of the SSFP signals as a function of the flip angle θ.The second experiment involved measuring the transient time T* at the optimum flip angle, yielding the T 1 + T 2 value.These results were used for the determination of both relaxation times.
Exploring the same dependence on the CWFP regime, Venâncio et al. [9] demonstrated that it is not necessary to use these two experiments to measure T 1 and T 2 when θ = 90 • .They proposed a fast method to measure both relaxation times in a single CWFP experiment using the evolution of the magnitude of the CWFP from the first pulse to the steady state (Figure 8A).In this figure, the initial part of the signal (dark grey region) is characterized by a strong oscillation of the amplitude following odd and even pulses [9].The amplitude of this signal decays with T 2 *.When the oscillation ends, the signal was named a quasi-stationary state (QSS) (transition between dark and light grays regions).From the QSS point, the signal decays slowly with a time constant T* (light grey region), given by Equation ( 5), to reach the constant steady state amplitude |M ss | (white region).For a CP-CWFP signal, the first oscillating signal (Figure 8B, dark grey) decays to a minimum signal (QSS).From this point, the signal also increases with T* (light grey) and reaches a constant steady state amplitude |M ss | (white region) [27].For θ = 90° and Φ = (2 + 1), the transient time T* for the CWFP and CP-CWFP sequences in Equation ( 4) is only dependent on the relaxation times according to Equation (5): A simple rearrangement of Equations ( 3) and ( 5), results in: or the proper relaxations times: Therefore, both relaxation times can be calculated in a single-scan CWFP or CP-CWFP experiment using the magnitude of M0 (intensity after the first pulse), magnitude at steady state |Mss|, and T*, using Equation ( 7).The T* value is obtained by exponential fitting of the T* decay (light grey region in Figure 8).The obtained relaxation time values are equivalent to those obtained by inversion recovery (T1) and CPMG (T2) pulse sequences [37,38].The major source of error in CWFP and CP-CWFP experiments is the |M0| value, which is strongly dependent on the spectrometer dead time.
Normally, the dead time has to be shorter than 10 μs.
The difference between CWFP and CP-CWFP sequences is the dynamic range (DR) of the T* signal that depends on the T1/T2 ratio.Figure 9 provides a comparison of the CWFP (Figure 9A) and CP-CWFP (Figure 9B) transient regimes for samples with T1 ≈ T2 and T1 >> T2.The T1 ≈ T2 and T1 >> T2 signals were acquired using CuSO4 (T1 ≈j T2), and MnSO4 solution (T1 = 4T2), in a 20 MHz spectrometer.The DR for CWFP is the greatest for T1 >> T2 and narrowest for T1 = T2 (Figure 9A).Conversely, the DR of the CP-CWFP signal is the largest for the sample with T1 = T2, and the narrowest for T1 >> T2 (Figure 9B).4) is only dependent on the relaxation times according to Equation (5): A simple rearrangement of Equations ( 3) and ( 5), results in: or the proper relaxations times: Therefore, both relaxation times can be calculated in a single-scan CWFP or CP-CWFP experiment using the magnitude of M 0 (intensity after the first pulse), magnitude at steady state |M ss |, and T*, using Equation (7).The T* value is obtained by exponential fitting of the T* decay (light grey region in Figure 8).The obtained relaxation time values are equivalent to those obtained by inversion recovery (T1) and CPMG (T2) pulse sequences [37,38].The major source of error in CWFP and CP-CWFP experiments is the |M 0 | value, which is strongly dependent on the spectrometer dead time.Normally, the dead time has to be shorter than 10 µs.
The difference between CWFP and CP-CWFP sequences is the dynamic range (DR) of the T* signal that depends on the T 1 /T 2 ratio.Figure 9 provides a comparison of the CWFP (Figure 9A) and CP-CWFP (Figure 9B) transient regimes for samples with T 1 ≈ T 2 and T 1 >> T 2 .The T 1 ≈ T 2 and T 1 >> T 2 signals were acquired using CuSO 4 (T 1 ≈ j T 2 ), and MnSO 4 solution (T 1 = 4T 2 ), in a 20 MHz spectrometer.The DR for CWFP is the greatest for T 1 >> T 2 and narrowest for T 1 = T 2 (Figure 9A).Conversely, the DR of the CP-CWFP signal is the largest for the sample with T 1 = T 2 , and the narrowest for T 1 >> T 2 (Figure 9B).We demonstrated that the CP-CWFPx-x sequence can be used to measure T1 in a single shot experiment using a low flip angle (5° to 10°), even in a spectrometer with a long dead time [23].Using these small flip angles, the T* is strongly dependent on T1 (Equation ( 4)). Figure 10   We demonstrated that several post-acquisition digital filters (low pass, linear and logarithm compression, Savitzky-Golay, and Wavelet filters) can be used to enhance SNR in CWFP-T1 signals without significant deviation of the T1 values [39].Figure 11 shows a SNR enhancement of one order of magnitude on CWFP-T1 signal using second-order, a 41-points window, and a Savistiky-Golay (SG) filter.Therefore, filtered CWFP-T1 can be a fast T1 method, providing an alternative to CPMG for low-field NMR relaxation measurements.The dependence of DR of T* as a function of the relaxation times ratio was solved using the CP-CWFP with phase alternation, CP-CWFPx-x [24].Figure 9C shows the CP-CWFPx-x signal for the sample with T 1 ≈ T 2 and T 1 >> T 2 .This figure shows that the T* dynamic range is independent of T 1 /T 2 ratio.Furthermore, the CP-CWFPx-x pulse sequence can be performed on resonance and therefore is the best CWFP sequence for measuring both relaxation times in a single experiment.
We demonstrated that the CP-CWFPx-x sequence can be used to measure T 1 in a single shot experiment using a low flip angle (5 • to 10 • ), even in a spectrometer with a long dead time [23].Using these small flip angles, the T* is strongly dependent on T 1 (Equation ( 4)). Figure 10   The dependence of DR of T* as a function of the relaxation times ratio was solved using the CP-CWFP with phase alternation, CP-CWFPx-x [24].Figure 9C shows the CP-CWFPx-x signal for the sample with T1 ≈ T2 and T1 >> T2.This figure shows that the T* dynamic range is independent of T1/T2 ratio.Furthermore, the CP-CWFPx-x pulse sequence can be performed on resonance and therefore is the best CWFP sequence for measuring both relaxation times in a single experiment.We demonstrated that the CP-CWFPx-x sequence can be used to measure T1 in a single shot experiment using a low flip angle (5° to 10°), even in a spectrometer with a long dead time [23].Using these small flip angles, the T* is strongly dependent on T1 (Equation ( 4)). Figure 10   We demonstrated that several post-acquisition digital filters (low pass, linear and logarithm compression, Savitzky-Golay, and Wavelet filters) can be used to enhance SNR in CWFP-T1 signals without significant deviation of the T1 values [39].Figure 11 shows a SNR enhancement of one order of magnitude on CWFP-T1 signal using second-order, a 41-points window, and a Savistiky-Golay (SG) filter.Therefore, filtered CWFP-T1 can be a fast T1 method, providing an alternative to CPMG for low-field NMR relaxation measurements.We demonstrated that several post-acquisition digital filters (low pass, linear and logarithm compression, Savitzky-Golay, and Wavelet filters) can be used to enhance SNR in CWFP-T 1 signals without significant deviation of the T 1 values [39].Figure 11 shows a SNR enhancement of one order of magnitude on CWFP-T 1 signal using second-order, a 41-points window, and a Savistiky-Golay (SG) filter.Therefore, filtered CWFP-T 1 can be a fast T 1 method, providing an alternative to CPMG for low-field NMR relaxation measurements.

Agri-Food Products
Transient CWFP/CP-CWFP signals have been used to analyze agriculture and food products as a useful alternative to CPMG.The first application of CWFP transient signals, using 90° pulses, was to measure the fat content in beef samples.The measurement was based on the |Mss|/M0 ratio and the univariate statistic that shows a better correlation (r = 0.9) than T2 (r = −0.25) to beef fat intramuscular content [26].The full transient signals of beef samples were also analyzed using multivariate (chemometric) analysis.The multivariate analyses of CWFP have been used to classify beef samples based on animal sex or breed [26,40], to predict water loss during beef cooking, and fat and moisture contents [41].More recently, we showed that CWFP-T1 can be a simple method to measure fat content in beef [39].Figure 12 shows the relaxation profile of two beef samples with 20% and 62% fat contents.The peak between 300 and 400 ms was assigned to meat and fat, and the signal at 100 ms to fat only.Therefore, samples with higher fat contents show a large 100 ms peak.The ratio of the area of these two peaks has been used to measure fat content in beef with higher precision than standard CWFP sequences [39].
The CWFP signals have been used to study the oilseed quality in static and flowing modes.CWFP and CP-CWFP signals have been used to measure oil viscosity and fatty acid composition using multivariate statistics [42,43].ratio and the univariate statistic that shows a better correlation (r = 0.9) than T 2 (r = −0.25) to beef fat intramuscular content [26].The full transient signals of beef samples were also analyzed using multivariate (chemometric) analysis.The multivariate analyses of CWFP have been used to classify beef samples based on animal sex or breed [26,40], to predict water loss during beef cooking, and fat and moisture contents [41].More recently, we showed that CWFP-T 1 can be a simple method to measure fat content in beef [39].Figure 12 shows the relaxation profile of two beef samples with 20% and 62% fat contents.The peak between 300 and 400 ms was assigned to meat and fat, and the signal at 100 ms to fat only.Therefore, samples with higher fat contents show a large 100 ms peak.The ratio of the area of these two peaks has been used to measure fat content in beef with higher precision than standard CWFP sequences [39].

Polymers
Transient CWFP signals were used to simultaneously measure the T1 and T2 relaxation times during fast polymerization reaction (10 min) of commercial epoxy resin [44].The advantage of CWFP to measure T1 and T2 over conventional methods is that both relaxation times can be measured in a single reaction.The standard T1 and T2 methods (inversion-recovery and CPMG) require the use of two separated reaction samples that are not ideal, as the two reactions did not have the exact same rates.Rodrigues et al. [28,29] applied CWFP sequences to real-time monitoring of the polymerization crosslinking reaction of acrylamide-based hydrogels.The CP-CWFP results showed high sensitivity and time resolution when monitoring important parameters of radical polymerization.Rodrigues et Figure 13a demonstrates that the relaxation profiles show similar variations in the absorbance measured in 730 nm.The relaxation profiles can also be measured in a colorless complex, such as EDTA-Mn 2+ , which is not detected by spectrophotometry [46].Spectrophotometry is a standard method used to study this complex.The CP-CWFPx-x sequence has been used in the complexation of biopolymers chitosan (CS) with paramagnetic ions (Cu 2+ , Fe 3, and Mn 2+ ) as a function of pH [46].The results indicated that CS strongly interacts with these ions and the complex CS-ion can be easily removed from the solution by centrifugation.

16 Figure 4 .
Figure 4. (A) Free induction decay (FID) and spin echo (SE) signals for a corn seed obtained in 15 s with a spin-echo sequence with τe = 3.5 ms, 16 scans.(B) CWFP signal for the same sample and the same acquisition time of 15 s (16,666 acquisition windows of 3Tp), using 90° pulses, Tp = 0.3 ms, and Φ = 5π.Adapted from Publication [13], Copyright (2003), with permission from Elsevier.

Figure 4 .
Figure 4. (A) Free induction decay (FID) and spin echo (SE) signals for a corn seed obtained in 15 s with a spin-echo sequence with τ e = 3.5 ms, 16 scans.(B) CWFP signal for the same sample and the same acquisition time of 15 s (16,666 acquisition windows of 3T p ), using 90 • pulses, T p = 0.3 ms, and Φ = 5π.Adapted from Publication [13], Copyright (2003), with permission from Elsevier.

Figure 4 .
Figure 4. (A) Free induction decay (FID) and spin echo (SE) signals for a corn seed obtained in 15 s with a spin-echo sequence with τe = 3.5 ms, 16 scans.(B) CWFP signal for the same sample and the same acquisition time of 15 s (16,666 acquisition windows of 3Tp), using 90° pulses, Tp = 0.3 ms, and Φ = 5π.Adapted from Publication [13], Copyright (2003), with permission from Elsevier.

Figure 6 .
Figure 6.(A) CWFP signal amplitude corresponding to 48 macadamia nut fragments with oil masses ranging from 0.1 to 0.7 g. (B) Oil mass as a function of CWFP peak amplitude for macadamia nut fragments with oil masses ranging from 0.1 to 0.7 g.A total of 96 peaks were obtained in a time of 14 s.The linear regression determination coefficient was R 2 = 0.95.Adapted from Publication [32], Copyright (2007), with permission from ACS Publications.

Figure 6 .
Figure 6.(A) CWFP signal amplitude corresponding to 48 macadamia nut fragments with oil masses ranging from 0.1 to 0.7 g. (B) Oil mass as a function of CWFP peak amplitude for macadamia nut fragments with oil masses ranging from 0.1 to 0.7 g.A total of 96 peaks were obtained in a time of 14 s.The linear regression determination coefficient was R 2 = 0.95.Adapted from Publication [32], Copyright (2007), with permission from ACS Publications.

16 Figure 8 .
Figure 8. Numerically simulated signals showing the evolution of the (A) CWFP and (B) CP-CWFP signal magnitudes from the first pulse to the stationary state (|Mss|) using T1 = 150 ms, T2 = 50 ms, Tp = 0.3 ms, and T2* = 20 ms.When the oscillation ends, the signal was named a quasi-stationary state (QSS) (transition between dark and light grays regions).The background color indicates the first transient oscillating signal (dark grey), the slowly signal decay with time constant T* (light grey), given by Equation (5), until it reaches the constant steady state amplitude |Mss| (white region).

Figure 8 .
Figure 8. Numerically simulated signals showing the evolution of the (A) CWFP and (B) CP-CWFP signal magnitudes from the first pulse to the stationary state (|M ss |) using T 1 = 150 ms, T 2 = 50 ms, T p = 0.3 ms, and T 2 * = 20 ms.When the oscillation ends, the signal was named a quasi-stationary state (QSS) (transition between dark and light grays regions).The background color indicates the first transient oscillating signal (dark grey), the slowly signal decay with time constant T* (light grey), given by Equation (5), until it reaches the constant steady state amplitude |M ss | (white region).For θ = 90 • and Φ = (2n + 1)π, the transient time T* for the CWFP and CP-CWFP sequences in Equation (4) is only dependent on the relaxation times according to Equation (5):
provides a comparison between the CP-CWFPx-x with low flip angle (CWFP-T1) transition time T* signal, obtained with a single shot experiment, and inversion-recovery (IR) data obtained with 32 experiments.Although the SNR of CWFP-T1 is much lower than obtained with IR, the T1 values were similar.
provides a comparison between the CP-CWFPx-x with low flip angle (CWFP-T 1 ) transition time T* signal, obtained with a single shot experiment, and inversion-recovery (IR) data obtained with 32 experiments.Although the SNR of CWFP-T 1 is much lower than obtained with IR, the T 1 values were similar.
provides a comparison between the CP-CWFPx-x with low flip angle (CWFP-T1) transition time T* signal, obtained with a single shot experiment, and inversion-recovery (IR) data obtained with 32 experiments.Although the SNR of CWFP-T1 is much lower than obtained with IR, the T1 values were similar.

Figure 11 .
Figure 11.Original (gray) and denoised (black) continuous wave free precession (CWFP- ) signals using a second-order Savitzky-Golay filter with 41-data-point windows of a MnSO4 aqueous solution.The signal to noise ratio (SNR) of the original and smoothed data signals are 39 and 374, respectively.

Figure 11 .
Figure 11.Original (gray) and denoised (black) continuous wave free precession (CWFP-T 1 ) signals using a second-order Savitzky-Golay filter with 41-data-point windows of a MnSO 4 aqueous solution.The signal to noise ratio (SNR) of the original and smoothed data signals are 39 and 374, respectively.

4. 2 .
Analysis of Products Using Transient Cwfp Signals 4.2.1.Agri-Food Products Transient CWFP/CP-CWFP signals have been used to analyze agriculture and food products as a useful alternative to CPMG.The first application of CWFP transient signals, using 90 • pulses, was to measure the fat content in beef samples.The measurement was based on the |M ss |/M 0 Appl.Sci.2019, 9, x FOR PEER REVIEW 12 of 16

Figure 12 .
Figure 12.T1 distribution time of obtained by the inverse Laplace transform of a CWFP-T1 signal acquisition of beef samples with different fat contents: (A) 20% and (B) 62%.Adapted from Publication [39], Copyright (2014), with permission from Wiley.

Figure 12 .
Figure 12.T 1 distribution time of obtained by the inverse Laplace transform of a CWFP-T 1 signal acquisition of beef samples with different fat contents: (A) 20% and (B) 62%.Adapted from Publication [39], Copyright (2014), with permission from Wiley.The CWFP signals have been used to study the oilseed quality in static and flowing modes.CWFP and CP-CWFP signals have been used to measure oil viscosity and fatty acid composition using multivariate statistics [42,43].