Method and apparatus for post-acquisition correction of nmr spectra

Abstract

There are described analytical and mathematical methods for post-acquisition correction of NMR spectra distorted by static and dynamic magnetic field inhomogeneity caused by imperfections of main superconducting coils and the cold head operation, typical with cryogen-free magnets. For the dynamic inhomogeneity, the method and system apply a variant of the general reference deconvolution method, complemented with a specific mathematical analysis of spectral parameters. For the static inhomogeneity, the system and method apply a delayed Fourier processing. The system and method contribute to the general field of processing NMR spectra and enable a more extensive use of cryogen-free magnets in high-resolution NMR spectroscopy. The method and system used are suitable for rapid scan NMR spectroscopy and in which dynamical distortions of the spectrum caused by the fast frequency sweep are eliminated using a single-resonance reference. Based on the independence of dynamic distortions and NMR characteristics of the material under study, the phase of the complex quadrature Free Induction Decay (FID) signal of a reference single-resonance sample can be used to extract the dynamic field distortion from any spectrum obtained on the same dynamically distorted magnet.

Description

[0001]METHOD AND APPARATUS FOR POST-ACQUISITION CORRECTION OF NMR SPECTRA TECHNICAL FIELD The present invention relates to a method and apparatus for Post-acquisition correction of NMR spectra distorted by dynamic and static field inhomogeneity of cryogen-free magnets. BACKGROUND TO THE INVENTION Nuclear magnetic resonance spectroscopy is a spectroscopic technique to observe the spin state of atomic nuclei, which gives information on the local magnetic field and the interaction with nearest neighbour atoms. It typically involves measuring the absorption of electromagnetic radiation in the radio frequency region from less than 4 to 900 MHz or more. In practice, a sample to be analysed is placed in a magnetic field and an NMR signal is produced by excitation of the nuclei of the sample by applying radio waves at the magnetic resonance frequency. The intramolecular magnetic field around an atom in a molecule changes the resonance frequency from which it is possible to derive information as to the electronic structure of the molecule and its individual functional groups. As the fields are unique or highly characteristic to individual compounds, NMR spectroscopy has become established as the definitive method to identify monomolecular organic compounds in modern organic chemistry practice. NMR spectroscopy usually involves three steps: 1) alignment (polarization) of the magnetic nuclear spins in an applied, constant magnetic field; 2) perturbation of this alignment of the nuclear spins by a weak oscillating magnetic field, usually referred to as a radio-frequency (RF) pulse and 3) detection and analysis of the electromagnetic waves emitted by the nuclei of the perturbated sample. NMR spectroscopy is a powerful tool for analysis of chemical structures on the atomic level. Due to its relatively low sensitivity and high-resolution demand, NMR requires strong and homogeneous magnetic fields. The latter are usually produced by superconducting magnets that operate at liquid helium temperatures. The majority of superconducting magnets are so-called liquid magnets where the main superconducting coils are immersed into a liquid helium reservoir that enables the coil wires to be kept in a superconducting state. As a result of natural evaporation, this requires a regular top-up of liquid helium. As the sources of helium are currently close to exhaustion, this traditional technology becomes more and more expensive. A solution to this problem is the use of so-called dry cryogen-free magnets, in which the main superconducting coils are located in vacuum and kept in the superconducting regime by heat exchange with a cold head cryostat that operates with helium gas pumped in a closed circuit cycle. This modern technology enables one to avoid the expense of helium consumption and is already used quite widely in dual electron paramagnetic resonance (EPR) and dynamic nuclear polarization (DNP) systems. Another strong advantage of dry cryogen-free magnets is that they allow relatively short magnetic field settling, by fast warming up the magnet to equalise magnetic fields outside and inside the superconducting winding. This enables various magnetic fields to be used during a short period of time, such that several magnetic fields can be applied per day. Cryogen-free magnets are also more compact and enable the probe to be inserted from the top of the magnet, making it gravitationally more stable, as well as from the bottom, which is traditional with liquid cryomagnets. An intrinsic feature of cryogen-free magnets is the mechanical vibration caused by the cold head operation. The latter can create dynamic distortions manifested in a time-periodic perturbation of the main magnetic field. This is appreciably pronounced in cases where the main magnet is not completely rigid and its different parts can mechanically move with respect to each other. In this case, it might be expected that the dynamic distortion is proportional to the main field. The cold head operation frequencies are in the order of 1-2 Hz. The effect of these frequencies is not appreciable in low-resolution NMR, typical for solids and some liquids. In these cases, the broadening caused by the cold head operation is well within the typical width of the NMR spectrum created by the anisotropic NMR interactions and relaxation processes. The latter can be in the order of few kHz or more, unless special high- resolution techniques are applied. By contrast, the cold head operation frequencies are comparable to the typical widths of high-resolution NMR spectra, which tends to create significant broadening distortion, especially in liquid state and also in solids under MAS, CP or spin decoupling. The strong dynamic field inhomogeneity can be an appreciable drawback for the use of cryogen-free magnets in high-resolution NMR spectroscopy, where the high magnetic field homogeneity and stability are vital requirements. In practice, the static magnetic field inhomogeneity created by imperfections in the main coils winding as well as in assembling together different parts of the magnet, is never fully avoidable, even after shimming. The latter is a time-consuming iterative procedure that removes the field inhomogeneity only to a certain extent. It is highly desirable to maximally reduce the effects of both dynamic and static magnetic field inhomogeneity in experimental high-resolution NMR spectra. The effect of the dynamic field distortion due to the cold head operation of a cryogen-free magnet has been analysed and experimentally measured and the subject of prior publication in the art. The applicant has previously suggested an active way of suppressing the dynamic magnetic field distortions, based on the application of an extra electric current to the Z0 coil, alternating at a suitable frequency and phase. This procedure has an advantage before the standard (deuterium or fluorine) field lock that is relatively slow and requires the relevant NMR labelling. It requires a repetitive feedback from the signal and an extra acquisition frequency channel. The instrumental suppression however requires extra hardware equipment and is able to suppress only a part of the magnetic field distortion. SUMMARY OF THE PRESENT INVENTION The present invention seeks to provide an improved a method and apparatus for post-acquisition correction of NMR spectra distorted by dynamic and static field inhomogeneity of cryogen-free magnets. According to an aspect of the present invention, there is provided a post-acquisition method of correction of a NMR spectrum distorted by static and dynamic magnetic field inhomogeneity, comprising the step removing dynamic magnetic field distortions that are independent of the NMR properties of the sample by applying a general reference deconvolution of the NMR spectrum through delayed Fourier processing. Preferably, dynamic magnetic field distortions are removed by a variant deconvolution method. Advantageously, the step of applying delayed Fourier processing enables any broadening of the NMR spectrum caused by fast decaying FID signal components to be removed from the NMR spectrum. In the preferred embodiments, the method includes the step of removing dynamic magnetic field inhomogeneity from an NMR spectrum by taking a Fourier transform of the absolute part of the FID signal only, thereby to provide a Lorentz-type single-peak shape without or with reduced dynamic field distortions. Advantageously, the method comprises the step of removing dynamic distortions caused by cold head operation by reference deconvolution, in which the phase of a complex quadrature FID signal of a reference single-resonance sample is used to extract dynamic field distortions from a spectrum obtained on a common dynamically distorted magnet. The method preferably comprises the step of analysing dynamic field inhomogeneity by extracting a phase of the FID signal, applying a Larmor frequency shift and taking a Fourier transform of an integral term of the FID signal. Preferably, the method includes the step of removing dynamic field inhomogeneity from FID contributions created by different nuclear species. The step of removing dynamic field inhomogeneity from FID contributions created by different nuclear species advantageously removes inhomogeneity simultaneously from all chemically shifted parts of the spectrum. The phase term is preferably extracted from: (i) a single-peak reference signal measured on the same distorted magnet, or (ii) from a single peak separated from other peaks in the spectrum. The method preferably comprises applying a reference deconvolution algorithm for removing dynamic inhomogeneity of the magnetic field by: (1) selection of a reference complex FID signal from a single-peak sample, (2) decomposition of a reference signal into amplitude and phase parts and extracting a phase term therefrom containing information on dynamic field inhomogeneity, and (3) cleaning the FID signal of any other sample. There may be provided a step of pre-cutting a time-domain signal before taking a Fourier spectrum thereof, thereby narrowing the spectrum by removing fast decaying components of the FID signal. Advantageously, the method comprises the step of removing any extra baselines between peaks in the spectrum, preferably to zero. In some embodiments, the method comprises the step of replacing the delayed spectrum by a Lorentzian shape that narrows the spectrum. The replacing step may comprise: (i) fitting an initial phase angle and phasing the initial signal by an adjustment factor; (ii) choosing a time and proceeding to a delayed pre-cut signal; (iii) calculating the delayed Fourier spectrum, (iv) removing the baseline. According to another aspect of the present invention, there is provided a system for post-acquisition correction of a NMR spectrum distorted by static and dynamic magnetic field inhomogeneity, comprising a processing unit configured to remove dynamic magnetic field distortions that are independent of the NMR properties of the sample by applying a general reference deconvolution of the NMR spectrum through delayed Fourier processing. Preferably, the processing unit is configured to remove dynamic magnetic field distortions by a variant deconvolution method. Advantageously, the processing unit is configured to remove from the NMR spectrum any spectrum broadening effects caused by fast decaying FID signal components by applying delayed Fourier processing. In the preferred embodiments, the processing unit is configured to remove dynamic magnetic field inhomogeneity from an NMR spectrum by taking a Fourier transform of the absolute part of the FID signal only, thereby to provide a Lorentz-type single-peak shape without or with reduced dynamic field distortions. The processing unit may be configured to remove dynamic distortions caused by cold head operation by reference deconvolution, in which the processing unit uses the phase of a complex quadrature FID signal of a reference single-resonance sample to extract dynamic field distortions from a spectrum obtained on a common dynamically distorted magnet. The system preferably comprises an analysis unit configured to analyse dynamic field inhomogeneity by extracting a phase of the FID signal, applying a Larmor frequency shift and taking a Fourier transform of an integral term of the FID signal. The processing unit is advantageously configured to remove dynamic field inhomogeneity from FID contributions created by different nuclear species. It may be configured to remove dynamic field inhomogeneity from FID contributions created by different nuclear species by removing inhomogeneity simultaneously from all chemically shifted parts of the spectrum. The processing unit may be configured to extract the phase term from: (i) a single-peak reference signal measured on the same distorted magnet, or (ii) from a single peak separated from other peaks in the spectrum. Advantageously, the processing unit is configured to apply a reference deconvolution algorithm to remove dynamic inhomogeneity of the magnetic field by: (1) selection of a reference complex FID signal from a single-peak sample, (2) decomposition of a reference signal into amplitude and phase parts and extracting a phase term therefrom containing information on dynamic field inhomogeneity, and (3) cleaning the FID signal of any other sample. The processing unit may be configured to pre-cut a time-domain signal before taking a Fourier spectrum thereof, thereby narrowing the spectrum by removing fast decaying components of the FID signal. In some embodiments, the processing unit is configured to remove any extra baselines between peaks in the spectrum, for example by cutting off any such baseline to zero. The processing unit may be configured to replace the delayed spectrum by a Lorentzian shape that narrows the spectrum. Advantageously, the processing unit is configured to replace the delay spectrum by: (i) fitting an initial phase angle and phasing the initial signal by an adjustment factor; (ii) choosing a time and proceeding to a delayed pre-cut signal; (iii) calculating the delayed Fourier spectrum, (iv) removing the baseline. According to another aspect of the present invention, there is provided an NMR system operated by a method as specified herein comprising a system as specified herein. Post-acquisition correction of NMR spectra is an important part of NMR spectroscopy that enables refined NMR spectra to be obtained, clean from undesirable out-phasing, broadening and noising. Below are described analytical and mathematical methods for post-acquisition correction of NMR spectra distorted by static and dynamic magnetic field inhomogeneity caused by imperfections of main superconducting coils and the cold head operation, typical with cryogen-free magnets. For the dynamic inhomogeneity, the method and system apply a variant of the general reference deconvolution method, complemented with a specific mathematical analysis of spectral parameters. For the static inhomogeneity, the system and method apply a delayed Fourier processing. The approach is verified by correction processing of high-field experimental liquid-state 1H NMR spectra of water and ethanol as well as solid-state 13C MAS NMR spectra of adamantane, which has been shown to obtain good results for both static and dynamic field distortions. The system and method contribute to the general field of processing NMR spectra and enable a more extensive use of cryogen-free magnets in high-resolution NMR spectroscopy. In addition to the above-mentioned active instrumental ways of suppressing dynamic field distortions, there are disclosed methods of correction processing of acquired experimental NMR spectra against dynamic and static inhomogeneity of the magnetic field, typical for cryogen-free magnets. It is demonstrated below that dynamic distortions caused by the cold head operation can be removed by a technique that falls within the general methodology well-known in a wide context of NMR spectroscopy as reference deconvolution. The disclosed method and system are suitable for rapid scan NMR spectroscopy and in which dynamical distortions of the spectrum caused by the fast frequency sweep are eliminated using a single-resonance reference. Based on the independence of dynamic distortions and NMR characteristics of the material under study, the phase of the complex quadrature Free Induction Decay (FID) signal of a reference single-resonance sample can be used to extract the dynamic field distortion from any spectrum obtained on the same dynamically distorted magnet. The inventors have discovered that static magnetic field inhomogeneity can be significantly reduced by a delayed Fourier processing, which enables the broadening caused by fast decaying FID signal components to be removed from the spectrum. In high- resolution NMR, especially in liquids, such components originate from nuclear species of the sample container and the inhomogeneity of the static field, the external artefacts, not related to the structure of the chemical compound under analysis. While the method of window functions providing acquisition delays has been applied earlier in NMR spectroscopy, the method and system taught herein provide new features and advantages over the art. The efficiency of the new methods can be verified by correction of experimental liquid-state 1H NMR spectra of water and ethanol and solid-state 13C spectra of adamantane at high magnetic field, with the use of a magnet with an appreciable dynamic and static field inhomogeneity. The described methodology shows its high efficiency and proves good perspectives for using cryogen-free magnets in high-resolution NMR spectroscopy. The mathematical methods disclosed herein can have universal application and can be used in a wider context of post-acquisition refinement of NMR spectra. BRIEF DESCRIPTION OF THE DRAWINGS Embodiments of the present invention are described below, by way of example only, with reference to the accompanying drawings, in which: Figure 1 is a series of graphs showing the magnitudes of the distorted (solid lines) and cleaned (dashed lines) Fourier spectra of the model example of Equation (6) for the fixed r = 1 s−1, b = 4 Hz and various values of ν; Figure 2 is a series of graphs showing the magnitudes of the distorted (solid lines) and cleaned (dashed lines) experimental NMR spectra of a liquid pure water sample; and, to the right, the magnitudes of the Fourier spectra of the distorted magnetic field extracted by the disclosed methodology; Figure 3 is a series of graphs showing the magnitudes of the distorted (top) and cleaned (bottom) experimental NMR spectra of an ethanol sample; Figure 4 is a series of graphs showing the real parts of the initially acquired (top) and post-processed (bottom) single-shot phased 1H NMR spectra of liquid ethanol; Figure 5 is a series of graphs showing the real parts of the initially acquired (top) and post-processed (bottom) phased naturally abundance 13C MAS NMR spectra of adamantane powder; and Figure 6 is a schematic diagram of the principal components of a preferred embodiment of apparatus according to the teachings herein. DESCRIPTION OF THE PREFERRED EMBODIMENTS The method and system described below works on removing dynamic magnetic field distortions that are independent of the NMR properties of the sample, by a variant of the reference deconvolution method. This methodology is complemented by analytical calculations of spectral parameters. The dynamics of the macroscopic magnetization of a single nuclear species (spin 1 / 2 or quadrupole in a symmetric environment) is well described by the Bloch equations. Where m = (m ,m ,m)T is the magnetization 3- T xy zvector, B = (Bx,By,Bz) is the 3-vector of the magnetic feld, γ is the gyromagnetic ratio of the species, R is the diagonal matrix that characterizes relaxation to the thermal equilibrium state mth, the symbol × denotes the vector product. In the absence of the radiofrequency irradiation, small transverse contributions Bx,yof the magnetic field tend to be averaged out in the rotating frame of the strong longitudinal component Bz, enabling keeping the field component only along the main z axis Following from Equation (1), the complex transverse magnetization is decoupled and satisfies the equation: A phenomenological transverse relaxation rate ^ ≥ 0 is introduced. The solution to Equation (2) with the initial value m(t = 0) = m0is The main component of the field gives rise to the time-independent Larmor frequency contribution, to which the extra contribution due to the dynamic field inhomogeneity should be added: This obtains in Equation (3) where Ω = γB0is the time-independent Larmor frequency, Δ(t) is the time-dependent contribution due to the dynamic magnetic field inhomogeneity. The complex magnetization m(t) produces the quadrature FID signal acquired after the 90◦ spin flip. Hence, Equations (3), (4) enable the following important observations to be made for experimental FID single-peak complex signals of a liquid sample: 1. The dynamic inhomogeneity of the magnetic field is entirely contained in the phase Φ(t) of the signal. 2. The dynamic field inhomogeneity enters the signal phase in the form of the temporal integral Δ(t) of the field distortion δ(t). 3. The static field inhomogeneity and relaxation effects are entirely contained in the absolute part of the signal in the form of a decaying amplitude a(t) = |m(t) / m0|. 4. The NMR spectrum of the entire FID signal, shifted by the Larmor frequency, is the convolution of the Fourier spectrum of the amplitude a(t) and the Fourier spectrum of the phase term eiΔ(t). Thus, the effective recipe for cleaning the dynamic magnetic field inhomogeneity off the NMR spectrum is to take the Fourier transform Fa(λ) of the absolute part a(t) of the FID signal only. The result will be a Lorentz-type single-peak shape, fully cleaned from the dynamic field distortions. To analyse the dynamic field inhomogeneity, we extract the phase of the FID signal, make the Larmor frequency shift and take the Fourier transform FΔ(λ) of the integral term Δ(t). The Fourier transform Fδ(λ) of the dynamic inhomogeneity δ(t) is then: As a simple illustrative mathematical model, one can consider a dynamic field inhomogeneity given by a single-sinusoidal mode: One can obtain then from Equation (4) which leads to the Fourier expansion of the phase term where Jk(ζ) are the Bessel functions of the first kind. As seen from Equation (5), the single harmonics of the field distortion generates all partials of the fundamental frequency ν in the FID signal. The latter tend to broaden the Fourier spectrum, creating a combination of undesirable extra peaks around the Larmor frequency. The intensities of the distorting harmonics in the spectrum are described by the first kind Bessel functions of the magnitude ζ. For positive k, the Bessel functions Jk(ζ) decay to zero for both small and large values of ζ. The magnitude ζ decays with the increase of the distortion frequency ν, so high-frequency distortions ν ≫ b = γα have a little effect on the signal, also shifting all the distortion harmonics far away from the spectral linewidth at the main Larmor harmonics. The magnitude ζ grows with the decrease of ν, so low-frequency distortions ν ≪ b have a little effect as well. Distortions with intermediate frequencies ν ∼ b can affect the signal significantly. It should be noted also that the intensities Jk(ζ) oscillate around zero with changing ζ, bringing additional quantitative complications to the spectra. Fourier spectra of the model signal of Equation (3) for the described single- sinusoidal dynamic field distortion (shifted by a Larmor frequency) are shown in Figure 1, which are a series of graphs showing the magnitudes of the distorted (solid lines) and cleaned (dashed lines) Fourier spectra of the model example of Equation (6) for the fixed r = 1 s−1, b = 4 Hz and various values of ν. In Figure 1, the physically reasonable values r = 1s−1 , b = γα = 4 Hz are fixed and several illustrative values of ν are chosen. The fixed value of r corresponds to the 1 s effective transverse relaxation time. The fixed value of b corresponds to the distortion amplitude α = 94 nT (10 ppb of the field of 9.4 T) and the 1H species gyromagnetic ratio γ = 42.58 MHz / T. The dashed lines represent the cleaned / undistorted spectrum for comparison. An application of the described cleaning algorithm to experimental NMR spectra is illustrated in Figure 2. In Figure 2, the graphs on the left show the magnitudes of the distorted (solid lines) and cleaned (dashed lines) experimental NMR spectra of the liquid pure water sample at 9.4 T distorted by the cold head operation and an extra generated single-sinusoidal modulation. The graphs on the right show the magnitudes of the Fourier spectra of the distorted magnetic field extracted by the described algorithm. The star marks the fundamental cold head frequency νch = 1.725 Hz. The large peaks correspond to the modulation frequencies ν0: a) ν0 = 10 Hz, (b) ν0 = 40 Hz, (c) ν0 = 100 Hz, (d) ν0 = 180 Hz. With reference to the panel on the right, the group of peaks around the frequency of 20 Hz is attributed to mechanical partials induced by the cold head operation. The peaks around the frequencies of 50 Hz and 150 Hz (that are multiplies of the mains frequency) are attributed to an effect induced in the receiver coil by the electric power mains. In this illustration, a room temperature water FID signal was acquired with a single shot at a 9.4 T magnetic field distorted by cold head induced oscillations. To demonstrate the power of the method, an extra artificially generated single-sinusoidal modulation was applied to the field sweep Z0 coil. The peak at the fundamental cold head frequency νch= 1.725 Hz, as well as the peaks at the artificially generated frequencies ν0, are clearly seen in the field distortion spectra. All distortions are cleaned off by proceeding to the magnitude of the signal (dashed lines), as described herein. The methodology works well for a single nuclear species that does not interact with other nuclear species in the sample. To apply it to generally more complicated spectra, one can analyse the FID signal from a nuclear spin species I undergoing scalar spin-spin interactions with m other nuclear species I(k), k = 1, ... ,m. This situation is typical for liquid state NMR characterised by nuclear J-coupling due to chemical bonds in a molecule. In this case, the spin interactions within the same nuclear species and the non-secular parts of the interactions between different species are eliminated from NMR spectra. The effective spin Hamiltonian becomes (in frequency units) Where ^^,^(^) ^ are the z-components of the relevant angular momentum operators, Ω, γare respectively the Larmor frequency and the gyromagnetic ratio of the nuclear species under study, δ(t) is the time-dependent perturbation of the main magnetic field, Jkare the strengths of the J-couplings. According to the Hamiltonian of Equation (7), the complex transverse magnetisation mI(t) of the nuclear species I that contributes to the FID signal can be calculated as: It can be seen see from Equations (8) and Equation (4) that the same phase term eiΦ(t) that contains the dynamic field inhomogeneity appears as a common factor in all FID contributions created by different nuclear species. Hence, it can be removed simultaneously from all chemically shifted parts of the spectrum. The phase term can be extracted either from another single-peak reference signal measured on the same distorted magnet or from a single peak, well separated from other peaks in the spectrum. Equations (8) show also that the dynamic inhomogeneity cannot be removed or extracted directly from a multiplet, as the information on the J-couplings will be lost together with the field distortion. It follows from the above analysis that the following reference deconvolution algorithm is valid for cleaning off the dynamic inhomogeneity of the magnetic field: (1) choose a reference complex FID signal mref(t) from a single-peak sample, (2) decompose the reference signal into the amplitude and phase parts: and extract the phase term eiΦ(t) that contains the entire information on the dynamic field inhomogeneity, (3) clean the FID signal m(t) of any other sample, multiplying it by the inverse phase term extracted from Equation (9) The result will be a Lorentz-type multi-peak NMR spectrum, clean from the dynamic magnetic field inhomogeneity. As a reference, the signal from a single-peak water sample or a single peak, well separated from other peaks in a more general NMR spectrum, can be used. To illustrate the methodology, experimental NMR spectra of a liquid ethanol sample were processed, obtained with a single shot at a 9.4 T magnetic field distorted by the cold head operation. The simulation results are shown in Figure 3. In the simulation, the single peak of the well chemically separated OH group was chosen as the reference for the described deconvolution algorithm. Referring to Figure 3, the graphs show the magnitudes of the distorted (top) and cleaned (bottom) experimental NMR spectra of the ethanol sample at a 9.4 T magnetic field distorted by the cold head operation. The three main chemically shifted peaks, corresponding to the OH (left), CH2 (middle) and CH3 (right) groups, are plotted separately to show the details of the splitting by the J-coupling and the dynamic eld distortion. The well separated single OH peak (left) was used as the reference for the deconvolution algorithm. Narrowing the static broadening by pre-cutting the FID signal The disclosed methodology is based on a pre-cut of the time-domain signal before taking its Fourier spectrum, in order to narrow the spectrum by removing fast decaying components of the FID signal that broaden the spectrum. Considering first an FID signal: formed of n harmonics with frequencies ωk, each harmonic having a time-dependent complex amplitude ak(t) that characterizes a poly-exponential decay with m different rates rj≥ 0. The presence of the decay rates rjis a consequence of static inhomogeneities of the magnetic field as well as spin-spin relaxation processes. The methodology makes use of an assumption that before the 90◦ spin flip pulse the macroscopic magnetisation is oriented along the static field. At t = 0, just after the 90◦ pulse, the signal flipped onto the transversal plane is phased, so we can be deemed that all the complex amplitudes akjhave the same phase ϕ, The Fourier transform F(λ) of the signal s(t) compensated for the initial phase ϕ is calculated as: As seen from Equation (11), the real part of the Fourier spectrum: is a combination of Lorentzian peaks at the frequencies ωkbroadened by the exponential decay rates rj. The faster the rate, the larger the contributions it makes to the broadening. This undesirable broadening effect can be reduced as follows. Choosing some delay time τ, the early part of the signal is set to zero, that is proceeds to the pre-cut signal: The latter corresponds to multiplication by the step-like window function that is zero for t ∈ [0, τ ) and equals 1 for t ≥ τ. The Fourier transform Fτ (λ) of the pre-cut signal s′ τ compensated for the initial phase ϕ is calculated as: The real part becomes: Comparing Equation (14) with Equation (12), it can be seen that the amplitudes |akj| acquire the extra factors e−rjτβkj(λ). The larger the rate rj and the delayτ, the smaller the relevant amplitudes become in the delayed spectrum. As a result, the spectral peaks become narrower. Thus, the broadening effect of large rates rj can be reduced by proceeding from the initial signal of Equation (10) to the delayed signal of Equation (13). The time delayτ should be chosen to be long enough to remove the undesirable broadening but not too long in order to keep a reasonably large signal-to-noise ratio. Due to the presence of the factors βkj(λ), the delayed spectrum is no longer Lorentzian and an extra baseline bτ(λ) is created between the peaks, This extra baseline is undesirable, it does not contain useful spectral information and should be removed from the spectrum. There are several ways of doing this, two of which are described below by way of example. The first way is to simply cut off the baseline to zero. According to Equations (14), (15), if the delay timeτ is long enough, the contributions of fast decay rates tend to be removed from the spectrum and the contribution of the slowest rate: dominates. This obtains with a good accuracy: If the delay provides a high-resolution spectrum, then: and so According to Equation (17), the baseline bτ(λ) at the spectral peaks λ = ωkis much smaller than the peak intensities: Hence, the cut-off of the baseline simply narrows the spectrum, keeping all the relevant spectral information, both qualitative and quantitative. Another way is to replace the delayed spectrum by a Lorentzian shape that also narrows the spectrum, still keeping all qualitative and quantitative spectral information. As described above, provided the delay timeτ is long enough, the contribution of the slowest rate r only remains in the spectrum. According to Equation (14), a good approximation can be obtained by: whereβk(λ) are given by Equation (16) and αk are some amplitudes. From this it is possible to find the best fit of the delayed spectrum of Equation (14) to the shape of Equation (18) and use this fit to remove the baseline of Equation (15) to obtain the Lorentzian approximation The frequencies ωkcan be taken directly from the delayed spectrum of Equation (14) as positions of the relevant peaks. The only fitting parameters that remain are the slowest rate r and the amplitudesαk. The latter can also be taken from the delayed spectra in the first fitting approximation. For known values of r, ωk, the standard least-square fitting algorithm can be applied based on the linear regression with respect This helps to adjust effectively the fitting parameters. It is believed also that certain modifications of the linear prediction, harmonic inversion and filter diagonalization methods can be applied. For experimental FID signals and the relevant NMR spectra, the narrowing algorithm comprises (or consists of) the following consecutive steps: (1) fit the initial phase ϕ and phase the initial signal s(t) by the factor (2) choose the time delayτ and proceed to the delayed pre-cut signal ^^ ^ ^ by Equation (13), (3) calculate the delayed Fourier spectrum Fτ (λ), (4) remove the baseline, either simply cutting it off or fitting the delayed spectrum to the form of Equation (18) and proceeding then to the Lorentzian by Equation (19). As an illustration of the application of the described algorithm, the same ethanol sample as in Figure 3 was used, with dynamic field distortions cleaned accordingly and the spectrum delayed by τ = 100 ms. The results for two different magnetic fields are plotted in Figure 4. Compared with the magnitude of the spectrum delayed by 20 ms, Figure 3, bottom, it can be seen that the extra Fourier acquisition delay enables much narrower, well phased spectra to be obtained, while keeping all the relevant information on the spectrum and a good signal-to-noise ratio. Referring to Figure 4, the graphs show the real parts of the initially acquired (top) and post-processed (bottom) single shot phased 1H NMR spectra of liquid ethanol at: (a) 1.175 T, and (b) 9.4 T. The well-separated single peak OH chemical group (left) was used as a deconvolution reference and a Fourier time delay of 100 ms was applied, followed by a Lorentzian correction of the baseline. In panel (b) of Figure 4, the three chemical groups (respectively, from left to right) OH, CH2, CH3 are plotted separately to show the details of the cold head distortion and splitting by J-coupling. By way of solid-state verification, the described methodology was applied to proton decoupled natural abundance 13C MAS NMR spectra of adamantane powder. One of the results is plotted in Figure 5. To better illustrate the efficiency of the suggested algorithms, the cold head distortion amplitude was increased four times by applying an alternating current to the Z0 coil at the cold head frequency. The numerical results confirm a good efficiency of the suggested methodology also in the solid-state high-resolution NMR. Referring to Figure 5, the graphs show the real parts of the initially acquired (top) and post-processed (bottom) phased naturally abundance 13C MAS NMR spectra of adamantane powder at a 9.4 T dynamically distorted field and 7.5 kHz rotor speed. The well-separated single-peak of the CH2chemical group (left) was used as a deconvolution reference and a 50 ms Fourier time delay was applied. The Lorentzian correction of the baseline was made. The two chemical groups CH2(left) and CH (right) are plotted separately to show the details of the cold head distortion. It should be appreciated that the experiments described above were carried out with the use of a prototype superconducting magnet, in which an imperfect assembly of its mechanically separate parts resulted in an appreciable dynamic field in-homogeneity induced by the cold head operation. This magnet, which would normally be considered as unfortunate, tuned out to be a good tool for the experimental estimation, suppression and post-acquisition processing of distorted NMR spectra. Even for this imperfect magnet, the methodology disclosed above displayed high efficiency. CONCLUSION Described are numerical mathematical methodologies for post-acquisition correction of NMR spectra distorted by both dynamic and static magnetic field inhomogeneity, typical for cryogen-free magnets. It has demonstrated the high efficiency and good applicability of the approach for experimental NMR spectra. Along with the active way of suppressing the dynamic magnetic field distortions, the disclosed post- acquisition correction further serves towards a more extensive use of cryogen-free magnets in high- resolution NMR spectroscopy. The results presented are in practice universal and can be used in a wide context of correction processing of NMR spectra. The skilled person will appreciate that the methodology disclosed herein will in practice be implemented into a processing system that automatically corrects NMR spectra distorted by dynamic and static field inhomogeneity of cryogen-free magnets. There is also provided a cryogen-free NMR system comprising such a processing system. Referring now to Figure 6, this shows in schematic form an embodiment of apparatus showing the principle components of processing system. The skilled person will appreciate that a number of the elements of the shown apparatus may be configured as hardware or algorithms. The processing system comprises a input 210 for receiving NMR spectra, a processor unit 200 configured to correct the NMR spectra post-acquisition and to output at 232 refined NMR spectra, clean from undesirable out-phasing, broadening and noising. The system provides for post-acquisition correction of NMR spectra distorted by static and dynamic magnetic field inhomogeneity caused by imperfections of main superconducting coils and the cold head operation, typical with cryogen-free magnets. The processing unit 200 comprises modules for carrying out the steps of the methodology, including a module 214 configured to apply a variant of the general reference deconvolution method, complemented with a module 216 configured to carry out specific mathematical analysis of spectral parameters. For the static inhomogeneity, the system is configured to apply a delayed Fourier processing. While the disclosures herein are addressed to cryogenic-free NMR systems it is to be understood that they can be applied to any NMR system. The apparatus and processing unit comprise: 1) a cryogen-free superconducting magnet 202 (or any alternatively suitable magnet) carrying a magnetic field for NMR experiments, with a warm bore or a cryostat isolated variable temperature insert (VTI) bore, 2) NMR probes 204 with chemical materials suitable for the relevant (warm or VTI) magnet bore and quality NMR signal measurements, 3) a NMR spectrometer 206 including a radiofrequency source, transmitter and receiver coils and amplifiers, equipped with an appropriate NMR signal acquisition and pre-processing software, 4) a post-acquisition processing unit 210 which may be a hardware desktop computer comprising a processor and an operation system supporting high-level programming languages (such as Matlab, Python) and the relevant graphical interfaces, enabling a software and programming codes to be created and executed to carry out the post-acquisition processing according to the methodology described. In the specific embodiment shown in Figure 6, the processing unit 210 is configured to remove dynamic magnetic field distortions that are independent of the NMR properties of the sample and for this purpose may include a general reference deconvolution stage 214 configured to apply a general reference deconvolution of the NMR spectrum through delayed Fourier processing. The stage 214 may be a variant devolution stage configured to remove dynamic magnetic field distortions by a variant deconvolution method. There is also preferably provided a spectrum broadening effects removal stage 216 configured to remove from the NMR spectrum any spectrum broadening effects caused by fast decaying FID signal components by applying delayed Fourier processing. There is in this embodiment also provided a dynamic magnetic field inhomogeneity removal stage 218 to remove dynamic magnetic field inhomogeneity from the NMR spectrum by taking a Fourier transform of the absolute part of the FID signal only, thereby to provide a Lorentz-type single-peak shape without or with reduced dynamic field distortions. A dynamic distortions removal stage 222 is configured to remove dynamic distortions caused by cold head operation by reference deconvolution, in which the processing unit uses the phase of a complex quadrature FID signal of a reference single- resonance sample 220 to extract dynamic field distortions from a spectrum obtained on a common dynamically distorted magnet. An analysis unit 224 is configured to analyse dynamic field inhomogeneity by extracting a phase of the FID signal, applying a Larmor frequency shift and taking a Fourier transform of an integral term of the FID signal. Dynamic field inhomogeneity is preferably removed from FID contributions created by different nuclear species. The stage 218 may be configured to remove dynamic field inhomogeneity from FID contributions created by different nuclear species by removing inhomogeneity simultaneously from all chemically shifted parts of the spectrum. The processing unit preferably comprises a phase term extraction stage 226 configured to extract the phase term from: (i) a single-peak reference signal measured on the same distorted magnet, or (ii) from a single peak separated from other peaks in the spectrum. Advantageously, the processing unit is configured to apply a reference deconvolution algorithm to remove dynamic inhomogeneity of the magnetic field by: (1) selection of a reference complex FID signal from a single-peak sample, (2) decomposition of a reference signal into amplitude and phase parts and extracting a phase term therefrom containing information on dynamic field inhomogeneity, and (3) cleaning the FID signal of any other sample. The processing unit may be configured to pre-cut a time-domain signal before taking a Fourier spectrum thereof, thereby narrowing the spectrum by removing fast decaying components of the FID signal. In some embodiments, the processing unit comprises a baseline extraction stage 228 configured to remove any extra baselines between peaks in the spectrum, for example by cutting off any such baseline to zero. It may also include a delayed spectrum replacement stage 230 configured to replace the delayed spectrum by a Lorentzian shape that narrows the spectrum. Advantageously, the stage 230 is configured to replace the delay spectrum by: (i) fitting an initial phase angle and phasing the initial signal by an adjustment factor; (ii) choosing a time and proceeding to a delayed pre-cut signal; (iii) calculating the delayed Fourier spectrum, (iv) removing the baseline. The skilled person will readily appreciate how to implement the apparatus from the teachings herein. The disclosures in British patent application number GB2302818.6, from which this application claims priority, and in the abstract accompanying this application are incorporated herein by reference.

Claims

CLAIMS 1. A post-acquisition method of correction of a NMR spectrum distorted by static and dynamic magnetic field inhomogeneity, comprising the step removing dynamic magnetic field distortions that are independent of the NMR properties of the sample by applying a general reference deconvolution of the NMR spectrum through delayed Fourier processing.

2. A method according to claim 1, wherein dynamic magnetic field distortions are removed by a variant deconvolution method.

3. A method according to any preceding claim, wherein the step of applying delayed Fourier processing enables any broadening of the NMR spectrum caused by fast decaying FID signal components to be removed from the NMR spectrum.

4. A method according to any preceding claim, comprising the step of removing dynamic magnetic field inhomogeneity from an NMR spectrum by taking a Fourier transform of the absolute part of the FID signal only, thereby to provide a Lorentz-type single-peak shape without or with reduced dynamic field distortions.

5. A method according to any preceding claim, comprising the step of removing dynamic distortions caused by cold head operation by reference deconvolution, in which the phase of a complex quadrature FID signal of a reference single-resonance sample is used to extract dynamic field distortions from a spectrum obtained on a common dynamically distorted magnet.

6. A method according to any preceding claim, comprising the step of analysing dynamic field inhomogeneity by extracting a phase of the FID signal, applying a Larmor frequency shift and taking a Fourier transform of an integral term of the FID signal.

7. A method according to any preceding claim, including the step of removing dynamic field inhomogeneity from FID contributions created by different nuclear species.

8. A method according to claim 7, wherein the step of removing dynamic field inhomogeneity from FID contributions created by different nuclear species removes inhomogeneity simultaneously from all chemically shifted parts of the spectrum.

9. A method according to claim 7 or 8, wherein the phase term is extracted from: (i) a single-peak reference signal measured on the same distorted magnet, or (ii) from a single peak separated from other peaks in the spectrum.

10. A method according to any preceding claim, comprising applying a reference deconvolution algorithm for removing dynamic inhomogeneity of the magnetic field by: (1) selection of a reference complex FID signal from a single-peak sample, (2) decomposition of a reference signal into amplitude and phase parts and extracting a phase term therefrom containing information on dynamic field inhomogeneity, and (3) cleaning the FID signal of any other sample.

11. A method according to any preceding claim, comprising the step of pre-cutting a time-domain signal before taking a Fourier spectrum thereof, thereby narrowing the spectrum by removing fast decaying components of the FID signal.

12. A method according to any preceding claim, comprising the step of removing any extra baselines between peaks in the spectrum.

13. A method according to claim 12, comprising the step of cutting off any such baseline to zero.

14. A method according to claim 12 or 13, comprising the step of replacing the delayed spectrum by a Lorentzian shape that narrows the spectrum.

15. A method according to claim 14, wherein the replacing step comprises: (i) fitting an initial phase angle and phasing the initial signal by an adjustment factor; (ii) choosing a time and proceeding to a delayed pre-cut signal; (iii) calculating the delayed Fourier spectrum, (iv) removing the baseline.

16. A system for post-acquisition correction of a NMR spectrum distorted by static and dynamic magnetic field inhomogeneity, comprising a processing unit configured to remove dynamic magnetic field distortions that are independent of the NMR properties of the sample by applying a general reference deconvolution of the NMR spectrum through delayed Fourier processing.

17. A system according to claim 16, wherein the processing unit is configured to remove dynamic magnetic field distortions by a variant deconvolution method.

18. A system according to claim 16 or 17, wherein the processing unit is configured to remove from the NMR spectrum any spectrum broadening effects caused by fast decaying FID signal components by applying delayed Fourier processing.

19. A system according to any one of claims 16 to 18, wherein the processing unit is configured to remove dynamic magnetic field inhomogeneity from an NMR spectrum by taking a Fourier transform of the absolute part of the FID signal only, thereby to provide a Lorentz-type single-peak shape without or with reduced dynamic field distortions.

20. A system according to any one of claims 16 to 19, wherein the processing unit is configured to remove dynamic distortions caused by cold head operation by reference deconvolution, in which the processing unit uses the phase of a complex quadrature FID signal of a reference single-resonance sample to extract dynamic field distortions from a spectrum obtained on a common dynamically distorted magnet.

21. A system according to any one of claims 16 to 19, comprising an analysis unit configured to analyse dynamic field inhomogeneity by extracting a phase of the FID signal, applying a Larmor frequency shift and taking a Fourier transform of an integral term of the FID signal.

22. A system according to any one of claims 16 to 19, wherein the processing unit is configured to remove dynamic field inhomogeneity from FID contributions created by different nuclear species.

23. A system according to claim 22, wherein the processing unit is configured to remove dynamic field inhomogeneity from FID contributions created by different nuclear species by removing inhomogeneity simultaneously from all chemically shifted parts of the spectrum.

24. A system according to claim 22 or 23, wherein the processing unit is configured to extract the phase term from: (i) a single-peak reference signal measured on the same distorted magnet, or (ii) from a single peak separated from other peaks in the spectrum.

25. A system according to any one of claims 16 to 24, wherein the processing unit is configured to apply a reference deconvolution algorithm to remove dynamic inhomogeneity of the magnetic field by: (1) selection of a reference complex FID signal from a single-peak sample,(2) decomposition of a reference signal into amplitude and phase parts and extracting a phase term therefrom containing information on dynamic field inhomogeneity, and (3) cleaning the FID signal of any other sample.

26. A system according to any one of claims 16 to 25, wherein the processing unit is configured to pre-cut a time-domain signal before taking a Fourier spectrum thereof, thereby narrowing the spectrum by removing fast decaying components of the FID signal.

27. A system according to any one of claims 16 to 19, wherein the processing unit is configured to remove any extra baselines between peaks in the spectrum.

28. A system according to claim 27, wherein the processing unit is configured to cut off any such baseline to zero.

29. A system according to claim 27 or 28, wherein the processing unit is configured to replace the delayed spectrum by a Lorentzian shape that narrows the spectrum.

30. A system according to claim 29, wherein the processing unit is configured to replace the delay spectrum by: (i) fitting an initial phase angle and phasing the initial signal by an adjustment factor; (ii) choosing a time and proceeding to a delayed pre-cut signal; (iii) calculating the delayed Fourier spectrum, (iv) removing the baseline.

31. An NMR system operated by a method according to any one of claims 1 to 15 and / or comprising a system according to any one of claims 16 to 30.

32. An NMR system according to claim 31, wherein the system is a cryogen-free NMR system.

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