Carr-Purcell Echo Research Articles

Nuclear pair electron spin echo envelope modulation

The interaction of electron spins with homonuclear spin pairs in their vicinity is one of the dominating mechanisms of electron spin echo decay at low temperature and low concentration. Here, we study this mechanism using established concepts of electron spin echo envelope modulation (ESEEM). We obtain analytical expressions for the Hahn echo, the refocused echo, the stimulated echo, and Carr–Purcell pulse trains with small numbers of π pulses. Hahn echo decay is well approximated by the product of nuclear pair ESEEM functions. The same approximation can explain dependence of stimulated echo decay on the first interpulse delay and provides reasonable time scale estimates for decay of Carr–Purcell echos after an odd number of π pulses. Carr–Purcell echoes after an even number of π pulses are rather sensitive to correlations within larger nuclear spin clusters. Approximations improve for both odd and even numbers of π pulses by factorising the nuclear spin bath into disjoint clusters, provided that modulation due to pairs of spins belonging to different clusters is considered in addition to cluster modulation. The analytical ESEEM expressions for the Hahn echo and the Carr–Purcell echo after two π pulses have the same mathematical form as the filter functions of these sequences of spin noise spectroscopy. This coincidence provides a computationally very efficient way of predicting Hahn echo decay induced by homonuclear spin pairs. The analytical pair product approximation predicts the previously observed (Bahrenberg et al., 2021) increase of the refocused echo amplitude when one refocusing time is incremented and other one is fixed but longer. In contrast, the spin-noise concept fails to predict this effect.

Theory of damped quantum rotation in nuclear magnetic resonance spectra. II. Numerical simulations for the benzene rotor

In Part I of this series of papers, the damped quantum rotation (DQR) theory, formulated originally for hindered threefold molecular rotors in solids, was generalized to the N-fold case. The stochastic dynamics of such objects, evidenced in NMR line shapes, was shown to be more complicated than in the standard model of classical jumps between the wells of the N-fold torsional potential. Actually, it comprises certain quantum rate (i.e., coherence-damping) processes subject to the requirements of the Pauli principle. The jump picture is recovered only when the quantum rates fit specific patterns. In this work, one of the ways of approaching such a classical limit is identified for the benzene rotor. This is inferred from a quantum mechanical model whose validity was earlier confirmed for a methyl group. Based on that model, theoretical calculations for the benzene ring dynamics in a clathrate crystal, 1-(9-anthryloxy)anthraquinone/benzene-d6, confronted with the pertinent literature data, point to possible deviations from the classical limit. However, the predicted DQR effects are too small to be observed in solid echo 2H NMR spectra of the C6D6 isotopomer. The chances of detecting the effects are improved when Carr-Purcell echo 1H spectra of a single crystal of the isotopomer including C6H6 as a guest are considered. The substantial differences in the sensitivity to the DQR effects of the spectra of protonated and deuterated benzene are concerned with different magnitudes of the intramolecular dipolar spin couplings. The dynamic isotope effect (C6D6 vs C6H6), which is small in this case, is only of secondary importance. Legitimacy of the use of the jump model in 2H NMR line shape studies of benzene-d6 is fully confirmed by the present considerations. However, the physical significance of the dynamic parameters extracted from such studies is shown from a new perspective.

Carr–Purcell echo spectra in the studies of lineshape effects. Nonclassical hindered rotation of methyl groups in 1,2,3,4-tetrachloro-9,10-dimethyltriptycene

In the standard NMR spectra, the lineshape patterns produced by a molecular rate process are often poorly structured. When alternative theoretical models of such a process are to be compared, even quantitative lineshape fits may then give inconclusive results. A detailed description is presented of an approach involving fits of the competing models to series of Carr–Purcell echo spectra. Its high discriminative power has already been exploited in a number of cases of practical significance. An explanation is given why it can be superior to methods based on the standard spectra. Its applicability in practice is now illustrated on example of the methyl proton spectra in 1,2,3,4-tetrachloro-9,10-dimethyltriptycene TCDMT. It is shown that, in the echo spectra, the recently discovered effect of nonclassical stochastic reorientation of the methyl group can be identified clearly while it is practically nondiscernible in the standard spectra of TCDMT. This is the first detection of the effect at temperatures above 200 K. It is also shown that in computer-assisted interpretation of exchange-broadened echo spectra, the usual description of the stimulating radiofrequency pulses in terms of rotation operators ought to be replaced by a more realistic pulse model.

Protonic conductivity in KH2PO4 family studied by NMR

Protonic conductivity in the KH2PO4 family of hydrogen-bonded insulators has been studied via the proton NMR lineshape and the decay of the NMR transverse nuclear spin magnetization of 87Rb nuclei in a spatially inhomogeneous electric field gradient. A large difference in the time scales of the decays of Hahn spin echo and Carr–Purcell echo train was observed. The Hahn echo decay curves show an exponential decay with the exponent proportional to the cube of time that is characteristic for diffusive motions. The proton NMR spectrum demonstrates the presence of mobile protons which produce a motionally narrowed absorption line above 220 K. The same effects were observed in several pure ferroelectric and antiferroelectric members of the KH2PO4 family as well as in their mixtures – the proton glasses.

What Are the Conditions for Exponential Time-Cubed Echo Decays?

Diffusion of precessing spins through a constant field gradient is well-known to produce two distinctive features: an exp(−bt3) decay of the echo amplitude in response to two pulses and a much slower decay of the Carr–Purcell echo train. These features will appear whenever the spin frequency is described by a continuous random-walk. The present work shows that this may also occur in the presence of motions with long correlation times τc—continuous Gaussian frequency noise with an exponential autocorrelation has the correct properties over time durations smaller than τc. Thus, time-cubed echo decays will occur in situations other than physical diffusion. The decay rate of the Carr–Purcell echo train is shown to vary with the pulse spacing τ whenever the correlation time τc is long; the slower Carr–Purcell decay compared to the two-pulse echo decay is not unique to diffusion. Simulations are presented that display time-cubed decays. The simulations confirm two important criteria: the echo time must be less than τc and the frequency noise must consist of nearly continuous variations, as opposed to step-like changes. These criteria define the range of physical parameters for which time-cubed decays will be observable.

Protonic inter-H-bond motion and ionic conductivity in hydrogen-bonded proton glasses

Low-frequency dynamics of the hydrogen-bonded proton glass ${\mathrm{Rb}}_{1\ensuremath{-}x}{(\mathrm{N}\mathrm{H}}_{4}{)}_{x}{\mathrm{H}}_{2}{\mathrm{AsO}}_{4}$ and its deuterated form has been investigated by measuring the decay of the NMR transverse nuclear-spin magnetization of ${}^{87}\mathrm{Rb}$ nuclei in a spatially inhomogeneous electric-field gradient, ${}^{1}\mathrm{H}$ line shape and ${}^{2}\mathrm{H}$ two-dimensional exchange NMR techniques. A large difference in the time scales of the decay of Hahn echo and Carr-Purcell echo train was observed between room temperature and 10 K. The Hahn echo decay curves showed an exponential decay with the exponent proportional to the cube of time that is characteristic for random stochastic atomic motions. The proton NMR spectrum demonstrates the presence of mobile protons which produce a motionally narrowed absorption line above 220 K. Two-dimensional exchange NMR experiment on the deuterated glass demonstrated the existence of slow thermally activated deuteron hopping between different hydrogen bonds, the process that represents the ionic conductivity. The same effects were observed also in several pure compounds of the ${\mathrm{KH}}_{2}{\mathrm{PO}}_{4}$ family. The observed low-frequency dynamics is not a peculiarity of the proton glass phase but represents the protonic inter-H-bond motion and ionic conductivity that is a common feature of the ${\mathrm{KH}}_{2}{\mathrm{PO}}_{4}$ family.

Magnetic Resonance Imaging of Diffusion in the Presence of Background Gradients and Imaging of Background Gradients

Abstract The measurement of the self-diffusion coefficient D by an NMR technique that uses an applied gradient G A can be corrupted by systems that have a background magnetic field gradient G 0 and also by imaging gradients G 1 , when used in an imaging mode. In a nonimaging mode, the corrupting cross term G A · G 0 can be eliminated in the diffusion measurement by use of an alternating-pulse-field-gradient (APFG) sequence that allows an accurate and uncorrupted measurement of D . A Carr-Purcell echo train enables the measurement of the expectation value, 〈 D G 2 0 〉; assuming D and G 0 to be uncorrelated will allow 〈 G 2 0 〉 to be determined. An image of D or of 〈 D G 2 0 〉 may be obtained without the corrupting G A · G 0 and G A · G I terms by appending a standard imaging sequence to an APFG sequence or a Carr-Purcell sequence, respectively; assuming D and G 0 to be uncorrelated will allow (〈 G 2 0 〉 to be determined within each pixel. Measurements of D and 〈 G 2 0 〉 and their images are made in apple flesh in which minute air bubbles are shown to produce the large 〈 G 2 0 〉. Their values in an 81 g Golden Delicious apple at a measuring frequency of 100 MHz were D = 1.42 × 10 −5 cm 2 /s and [formula] = 8.9 G/cm.

Deuterium quadrupole and carr-purcell echo relaxation in chemically exchanging systems. theory and experiments for the two-site case

A theoretical analysis of the quadrupole echo and Carr-Purcell echo experiments for pairs of I = 1 quadrupole nuclei undergoing mutual exchange is given. Exact numerical calculations of the echo intensities as a function of the time interval between the pulses and the kinetic and magnetic parameters are presented. The results are compared with deuterium NMR measurements performed on two systems involving cyclohexane-d 12 under conditions at which the molecules undergo ring inversion. These include a liquid crystalline solution of C 6D 12 aligned in a magnetic field and a powder sample of the thiourea-C 6 D 12 inclusion compound. Three experimental methods were applied to study these systems: (i) complete analysis of the ID spectrum lineshape, (ii) quadrupole echo intensity measurements as a function of the time interval between pulses, and (iii) Carr-Purcell echo relaxation measurements as a function of the pulse repetition rates. Within the experimental accuracy all three methods gave identical results. It is argued that under certain conditions pulse experiments may be advantageous or even necessary for a complete quantitative analysis of dynamic effects in deuterium NMR of solids.

Spin-echo attenuation by diffusion in nonuniform field gradients

The diffusion-dependent decay of spin-echo signals from samples distributed over non-uniform field gradients is analyzed. This decay deviates from the linear semilogarithmic dependencies on t 3 or t obtained, respectively, in Hahn spin-echo or Carr-Purcell echo trains from spins in constant field gradients. It is demonstrated here that when the average diffusion distance is small compared to the spatial variation of the gradient, the spin-echo decay can be modeled by a simple superposition of signals from a large number of microdomains, each characterized by a different gradient. This is illustrated by two examples, a water sample in a deliberately created quadratic variation of the external field, and a saturated water/ sand mixture with large internal field gradients.

A new approach to automatic shimming

A fast, algebraic technique for automatic magnet shimming is presented. It is based on field maps computed by phase measurement within Carr-Purcell echo sequences. The correction computation itself is carried out by an algorithm which compensates field inhomogeneities within the sample by a linear combination of all shim control functions. The procedure is noniterative, thus avoiding instability problems and improving speed performance. Two algorithms are discussed: (a) a Chebychev approach which minimizes the maximum field deviation (minimax) and (b) a least-squares algorithm.

Chemical exchange and nmr T2 relaxation—The multisite case

The behavior of a Carr-Purcell echo train in a chemical system with more than two exchanging sites is investigated. This paper extends and supplements previous work examining the theory of the spin-echo experiment. By using the condition of microscopic reversibility of the chemical system, it can be shown that the echo maxima, apart from the first few, decay exponentially. The limiting cases of high and low pulse rates are discussed. At high pulse rates the influence of relative chemical shifts between the various sites is eliminated. In this case the echo train agrees with that expected from the Bloch-McConnell equation when chemical shifts are neglected. Closed expressions for the case of exchange between two sites are given.