Method of Relaxation Rates Measurement in Proton-Containing Materials

Sergey Ermak|Vladimir Semenov|Roman Lozov

Abstract—A method of the relaxation time measurement in a flowing sample,based on the comparison of the amplitudes of nuclear magnetic resonance(NMR)signals observed from the reference and tested flowing samples was represented.The inductive and optical methods of NMR signal detection in the flowing sample were compared.The levels of NMR signal (10000 Hz−1/2)and the Hanle magnetometers signal-to-noise ratio (3500 Hz−1/2)were calculated and the possibility of the NMR signal detection with the small-size Hanle magnetometer was demonstrated.Additionally,the absence of a magnet-analyzer and the ability to measure both longitudinal and transverse relaxation time with high accuracy was demonstrated.

1.Introduction

There are many various methods for studying the substances characteristics[1]-[9].Among them,the nuclear magnetic resonance(NMR)method is one of the most used.NMR phenomenon is of fundamental importance and provides the ability to measure the longitudinal and transverse relaxation time with high precision,giving important information about the internal structure of the object under study,as in[10].The use of such information covers various industrial sectors,for example,oil processing,food and pharmaceutical industries,where the accuracy of the measured relaxation constants guarantees the quality of the examination of the studied materials,and in NMR tomography directly affects the contrast of the obtained images,as in[11].

The traditional method of measuring relaxation time in a condensed medium is based on the determination of the time interval during which the amplitude of the NMR signal decreases under the action of a series of radio frequency pulses of a certain duration on the sample,as in[12].The accuracy of measurement depends on the observed signal-to-noise ratio,which increases with the increase of the working magnetic field.In the NMR technique with the flowing liquid,the reception of the preliminary polarization of the flowing sample in the cumulative volume placed in the magnetic fieldBPof～10 kilo Gauss(kG)is used to increase the intensity of the observed signal,as in[13].The NMR signal itself is recorded in the magnetic fieldBA,the intensity of which is one or two orders of magnitude less than the value ofBP.At the same time,the longitudinal relaxation timeT1is estimated by

whereVPis the cumulative volume;VTis the volume of the connector pipeline between the cumulative volumeVPand the receiving coil;qis the optimal flow rate of the flowing liquid corresponding to the maximum amplitude of the NMR signalS.

2.Proposed Approaches

The elementary analysis of the observed NMR signal shows that even with a relatively high signal-to-noise ratio(about 100 in the 1 Hz registration band),the relative accuracy ofT1time measurement is tens of percent,which confirms the unsuitability of such a technique in terms of its metrological capabilities.On the other hand,the exact measurement of the transverse relaxation timeT2(for example,by the duration of the NMR signal wiggles underBAfield modulation)is not possible due to the gradients of the working magnetic field in the analysis zone.The above mentioned determines the expediency of finding new approaches to reduce the measurement error in the completed experiments both by modification of traditional methods of detection of the NMR signal in a reception radio frequency coil (inductive method)and a high-sensitivity quantum magnetometer with optical pumping,as in [14].The purpose of such studies is to improve the signal-to-noise ratio in the recording channel,since this parameter ultimately forms the error of the measuring device.In high-resolution NMR spectrometers,the accuracy of measurement of relaxation constants is a fraction of a percent,but such designs require the use of powerful magnets of appropriate size and weight.

In our work,we carried out research of relatively small-sized variants of NMR spectrometers with the flowing liquid,which was used to measure the relaxation constants by comparing the amplitudes of NMR signals observed from the reference and tested flowing samples,as in[15]and[16].Since the time of longitudinal relaxation of the reference sample is known with high accuracy,the observed change in the amplitude of the signals from flowing samples compared to that of the reference in the same experimental conditions can be associated only with a change in the values ofT1andT2.At the same time,the accuracy of their determination depends on the absolute values of the detected signals,which necessitates the development of a number of criteria related to the selection of the standard and the installation of optimal values of the flow rateqand the amplitude of the alternating magnetic field in the receiving radio frequency coil.The specificity of the inductive method is the fact that the timeT2is determined mainly by the gradient of the working magnetic field in the sample placement zone,regardless of the variety of the flowing sample.Therefore,the choice of a standard liquid with a pre-known valueT1is based on the establishment of the range of probable values of this parameter,estimated by(1).

3.Proposed Method of Relaxation Time Measurement

As the calculations have shown,the reference sample should be selected with the time of longitudinal relaxation near the lower(or upper)boundary of this parameter range for tested samples,which guarantees the linearity and high steepness of the corresponding dependence of the NMR signals amplitudeS(T1)for various tested samples.In a more complex case,when the difference in the transverse relaxation time between the reference sample and the tested samples should be taken into account,an independent procedure is needed to determine the transverse relaxation time,for example,by the decay time measurement of the free precession signal and applying this time inS(T1).Since the timeT2of the test sample may not coincide with the time of transverse relaxation of the standard,the relaxation time measurement procedure is based on the comparison of signals from the reference sample and test samples in the coordinate system,the axes of which reflect the values of the relaxation constants and the amplitude of the observed NMR signalS(T1,T2),respectively.When determining the transverse relaxation time,it is necessary to take into account the factor of span broadening of the NMR line,the effect of which on the measurement results can be minimized by reducing the consumption of the flowing sample under study.

3.1.Inductive Detection of NMR Signal

The proposed method was experimentally tested on a laboratory setup(see Fig.1),which included the tube with the flowing liquid,magnet-polarizer with magnetic inductionBP,magnet-analyzer with magnetic inductionBA,nutation coil,modulation coil,autodyne detector with detection coil and sound generator.As the flowing sample passes through the tube inside the magnet-polarizer,the polarization of protons occurs,which is detected in the magnet-analyzer by the magnetic resonance signal using the autodyne detector and the standard method of magnetic field scanning.The nutation coil is used to change the orientation of the nuclear magnetization vector.

Fig.1.Structural diagram of the NMR spectrometer with the flowing sample and inductive signal detection.

Examples of observed NMR signals from water(a)and ethyl alcohol(b)protons are presented in Fig.2.These signals(wiggles),as in[12],have differences in the form of oscillations,depending onT1andT2.

Fig.2.NMR signal from the flowing sample as a function of time from:(a)water and(b)ethyl alcohol.

The sequence of measurement to determine the longitudinal relaxation time of the flowing sample was carried out as follows:

◆ The flow rate of the flowing test sample,the amplitude of the radio-frequency field,as well as the frequency and amplitude of the modulation of the constant magnetic field,at which the amplitude of the NMR absorption signal reaches the maximum value,were measured;

◆ The range of the optimal flow rate within which the amplitude of the NMR absorption signal of the flowing test sample is reduced by a fixed value determined by the signal-to-noise ratio(for example,by 1% at the signal-tonoise ratio of 100,or by 10% at the signal-to-noise ratio of 10)was determined;

◆ The range of values of the longitudinal relaxation time corresponding to the measured range of the optimal flow rate of the flowing test sample was determined;

◆ The flowing test sample was replaced by a flowing reference sample with the known value of the longitudinal relaxation time,the value of which is within the range of values of the longitudinal relaxation time of the test sample;

◆ The flow rate of the flowing reference sample,the amplitude of the radio-frequency field,as well as the frequency and amplitude of the modulation of the constant magnetic field,at which the amplitude of the NMR absorption signal reached its maximum value,were measured;

◆ The flow rate of the flowing reference sample was reduced to a value corresponding to the amplitude of the NMR absorption signal,reduced by half compared with the maximum value;

◆ The calibration dependence of the relative amplitude of the NMR absorption signal from the flowing reference sample on the longitudinal relaxation time in the range of values corresponding to the measured range of the optimal flow rate of the flowing test sample was built;

◆ The flowing reference sample was replaced by the flowing test sample,and the test sample signal amplitude was measured relative to the maximum amplitude of the NMR absorption signal at the same values of the flow rate,the amplitude of the radio-frequency field and the frequency and amplitude of the modulation of the constant magnetic field;

◆ The relative amplitude of the NMR absorption signal from the flowing test sample was compared with the data of the calibration dependence and the deviation of the longitudinal relaxation time from the known longitudinal relaxation time of the flowing reference sample was determined.

In practice,a new method of the relaxation timeT1measurement consists of the following steps.On the first step,S(k),the dependencies of NMR signal on the saturation factorkwere built(see Fig.3).In this case,the saturation factor is equal toγH1T2(γis the gyromagnetic ratio andH1is the intensity of the resonant radio-field (tunable parameter)).The dependencesS(k)were taken for samples with the well-known relaxation timeT1.On the second step the dependence of NMR signal on the relaxation timeT1was built (see Fig.4).The dependenceS(T1) (see Fig.4)makes it possible to determine the relaxation time for varied sugar concentrations with high accuracy.It is important for this method to determine the amplitude of the NMR signals correctly.This problem is discussed in the next part of the paper.

Fig.3.NMR signal from the flowing sample as a function of saturation factor k:1:Water;2,3,and 4:Water solution of sugar.

Fig.4.NMR signal from the flowing sample as a function of relaxation time T1.

Oxygen-rich water with the known longitudinal relaxation time of 0.65 s was used as a reference,and specially prepared decontaminated water obtained as a result of multiple boiling procedures was used as a test sample.In the experiment,in both cases,the transverse relaxation time was determined by the gradient of the working magnetic field of the magnet-analyzer and was 1 ms,while the longitudinal relaxation time of the test sample was equal to 3.25 s.Such a significant difference inT1values is unambiguously explained by the influence of the electron paramagnetic (oxygen)on the relaxation rate in the reference sample,which should be taken into account in the course of measurement,as in[10].

This account assumes the selection of reference samples in which the corresponding procedure of environment decontamination to exclude the factor of similar influence is carried out.In this case,we emphasize the fundamental difference between the considered method of measuring the time of longitudinal relaxation of the liquid from similar methods for stationary samples.In particular,we are talking about the method of a quick passage of a constant magnetic field across the resonance,as in[17],according to which we measure the ratio of the NMR signals corresponding to two successive fast (in comparison with the rate of longitudinal relaxation)passages separated by a period of timeτ.The signal observed at the first pass is proportional to the transverse macroscopic magnetizationM0at the optimal rate of passage and the corresponding intensity of the radiofrequency field,as in

whereχis the static nuclear susceptibility andH0is the constant magnetic field.The signal of the second fast passage through the resonance in the opposite direction is proportional to the longitudinal component of the macroscopic magnetizationMZ:

For the symmetric passage of a constant magnetic field,the NMR signal corresponds to the stationary value of the nuclear magnetizationM,determined by the expression:

The longitudinal relaxation timeT1is determined by two measured amplitudes of the NMR signal with different values ofτ.For the flowing sample,(1)is valid only if the timeτis significantly less than the time spent by the flow of sample in the analyzer volumeθ,which is directly determined by the size of the volume and the rate of the liquid(otherwise,atθ<<τ,the sample will be changed by flowing in the analyzer volume,which leads to a loss of information about the dynamics of macroscopic nuclear magnetization in the period of the first passage through the resonance).As an example,let us take the volume of the analyzer of 1 cm3at a flow rate of 50 cm3/s,(corresponding to the practice of NMR for the maximum of the signal-to-noise ratio).In this case,the timeθis equal to 0.02 s,which means the timeτshould be at least several times less than this value,for example,0.005 s.It is easy to calculate as in(1)that the change of the timeτtwice from 0.005 s to 0.01 s corresponds to the relative change in the amplitude of the signals of only 0.2%,which is significantly lower than the limits of error of measuring the NMR signal amplitude in the flowing sample.

3.2.Optical Detection of NMR Signal

The alternative to the induction method of information retrieval in an NMR spectrometer with a flowing liquid is to use a highly sensitive magnetic sensor with optical pumping in the registration zone,as in[14].This method was first proposed in[18],where the NMR scheme of the spectrometer in combination with the quantum magnetometer on the Hanle effect was considered,as in[19].The practical implementation of such technique was developed much later with the advent of laser pumping sources.As in Fig.5,the laser beam orients the Cs atoms placed in the absorption cell,which detect the rotation of the magnetization vector M.The radio-frequency coils(RF coils)produce the change in the orientation of the magnetization vector M in the external magnetic field(BEXT).Lowfrequency modulated coils(LF)produce the modulation fieldB1.Unlike the resonance magnetometer with optical pumping,in which the measured magnetic field is detected by the frequency of resonance,in the Hanle magnetometers the signal absorption is measured.The amplitude of this signal is proportional to the value of measured magnetic fieldH0,the direction of which is oriented perpendicular to the circularly polarized light beam(see Fig.5).Parallel to the measured fieldH0an auxiliary modulating fieldH1cosωtis applied,the frequency of which satisfies the conditionω>>Γ,whereΓis the relaxation width of the resonance line.In weak fields,when the inequalityγH0<<Γ(γis the gyromagnetic ratio)is fulfilled,a slow precession of the magnetization vector M created by light will occur,the value of which decreases with the velocityΓ,while due to the fulfillment of this inequality,the average resulting magnetization vector will practically coincide with the direction of the pump light.At a non-zero magnetic field the resulting magnetization vector will be smaller in magnitude and directed at some angle to the direction of propagation of the pump light.When the inequalityγH0>>Γis fulfilled,the resulting magnetization vector becomes even smaller and,in the limit of an infinitely large magnetic field,tends to zero.

Fig.5.Structural diagram of the NMR spectrometer with the flowing sample and optical detection.

When the variable fieldH1cosωtis applied in the vicinity of the zero magnetic field,the vector M oscillates along theх-axis with the doubled frequency of 2ω.By increasing the magnetic field to a value comparable to the width of the resonance line,the oscillations of the vector increase in amplitude and occur at frequencyω.The presence of the fieldH1cosωtleads to the fact that the formation of the oscillating magnetization is realized not only in the vicinity of the zero-value of the fieldH0,but also for other values satisfying the resonance condition:

wherenis an integer number.The widths of these resonances are the same for any value ofnand are equal to 2Γγ−1and the amplitude of the Hanle signal of the magnetometer is determined by the product of Bessel functionsJ1(γH1/ω)andJ0(γH1/ω).

The sensitivity of the Hanle magnetometer at the parametric resonance (the minimum possible detectable variation)is equal or more than 10−9Oe,which makes it possible to reliably detect the precession signal of the nuclear moments of atoms,located near the absorption cell of the magnetometer as in the proposed device.A similar signal in the form of a damped precession of the nuclear moments of helium-3 atoms was observed,as in[20],for 11 hours,which exceeds the time of transverse relaxation of helium-3 nuclear moments due to the gradients of the working magnetic field,polarized in the conditions of spin-exchange interactions with optically oriented vapors of alkali metals,as in[21].According to this work,in the case of the optical pumping of potassium vapor and inert gases (helium-3 and neon)mixture,the values of the inhomogeneous effective magnetic field generated by the atoms of electron and nuclear paramagnetic materials was equal to 10−6Oe and 10−3Oe,respectively,at the laser pumping power of 100 mW.

The operation of the Hanle magnetometer requires preliminary calibration,the degree of reliability of which depends on the orientation of the optical axis of the device relative to the vector of the measured field.In the actual operation of such a device,the transverse components of the fieldH0differ from zero,and their magnitude and directions are random variables.The estimates show that for the compensation of transverse field components with the accuracy of 0.01 line width,the accuracy of magnetic field measurement approaches the sensitivity threshold determined by the relaxation width of the lineΓdivided by the signal-to-noise ratio.

One of such ultra-miniature magnetic sensor(volume of several cubic millimeters),as in[22],provides accurate measurement of the transverse relaxation time by using a special volume built into the pipeline with the flowing sample and placed in a high-uniform magnetic field of 250 mG,almost excluding the influence of magnetic gradient on the measurement result.Such a technique,undoubtedly,can be in demand in the designs of NMR spectrometers with the flowing liquid,where the size of the working pipeline is orders of magnitude larger than the miniature analogue,as in[18].In this regard,it is interesting to compare the parameters of the useful signal of the Hanle magnetometer(in particular the signal-to-noise ratio)with the signal observed by the inductive method on the laboratory setup.The typical value of the Hanle parameter of magnetometers is equal to 3500 Hz−1/2,as in[19].For the inductive method applied to the flowing sample,as in[20],the signal-to-noise ratio in a uniform magnetic fieldBAcan be estimated as follows:

wherefis the noise factor taking into account the noise of the equipment,γis the gyromagnetic ratio of protons,ηis the fill factor,Δνis the bandwidth of the input amplifier,Qis the quality factor of the receiving radio frequency coil,βis the coefficient taking into account the reduction of the macroscopic magnetization during transportation of the flowing liquid through the connecting pipeline with the volumeVTfrom the magnet-polarizer with inductionBPto the magnet-analyzer with inductionVA,χis the nuclear magnetic susceptibility,αis the coefficient having an order of magnitude of 10−2taking into account the weight contribution of the volume of the connecting pipeline in the analyzer volume,kis the Boltzmann constant,andTis the temperature.With reference to the laboratory setup(parameters:f≈2;η≈1;Q≈100;β≈0.1;BA≈500 G;BP≈5000 G;Δν≈1 Hz;αVT≈1 cm3;χ≈3×10−10;Т≈300 K),the value is equal to 10000 Hz−1/2,which is in order of magnitude close to this parameter in the Hanle magnetometers.At the same time,an important advantage of using a quantum magnetometer with optical pumping is the absence of the magnet-analyzer and the ability of measuring both the longitudinal and transverse relaxation time with high accuracy comparable to the accuracy of measurement carried out by high-resolution NMR equipment.

The principal difference between the optical pumping modes in the cells with the anti-relaxation coating and with the buffer gas is the different temperature of the absorption cells,which in the case of using the buffer gas can significantly exceed the level of 100 °C.This creates a problem of effective isolation of the Hanle magnetometer sensor from the volume with the flowing sample and complicates the design of the entire device.In this regard,it is preferable to use absorption cells with anti-relaxation coating,the operability of which is provided at lower temperatures.In the case of cesium atoms,the operating temperature of the absorption cell with linear dimensions of 5 cm is close to room temperature and the quality factor of the observed signal is an order of magnitude higher than this parameter in the buffer gas cells.At the same time,the high variation of the sensitivity of magnetometers with optical pumping at the level of 10−14T·Hz−1/2,as in[14],does not guarantee reliable detection of the NMR signal of the flowing sample in the case of ineffective suppression of variations of the external magnetic field.For example,to reduce the relative variation of the geomagnetic field of 10−5in the area of the Hanle magnetometer sensor placement to the above level,it is required to use a magnetic shield with a dynamic shielding coefficient of 5×104.

4.Conclusions

We have demonstrated the new approach to the relaxation time measurement in the flowing sample.This approach based on comparing the amplitudes of NMR signals observed from the reference and tested flowing samples.We have compared the inductive and optical detection of the NMR signal from the flowing sample.The ability of both longitudinal and transverse relaxation time measurement has been demonstrated.The measurement is possible in the absence of a magnet-analyzer.The calculated levels of the NMR signal(10000 Hz−1/2)and Hanle magnetometers signal-to-noise ratio(3500 Hz−1/2)have demonstrated the possibility of the NMR signal detection with the small-size Hanle magnetometer.