Calculation of NMR Relaxation Times for Quadrupolar Nuclei in the Presence of Chemical Exchange
01 Mar 1970-Journal of Chemical Physics (American Institute of Physics)-Vol. 52, Iss: 5, pp 2527-2534
TL;DR: In this paper, a correlation function for a quadrupolar nucleus which may exchange its environment between two kinds of sites in solution has been calculated from an appropriate joint conditional probability for changes in site and in electric field gradient direction with time.
Abstract: The correlation function for a quadrupolar nucleus which may exchange its environment between two kinds of sites in solution has been calculated from an appropriate joint conditional probability for changes in site and in electric field gradient direction with time. In the extreme narrowing limit, the resultant NMR linewidth depends both on the exchange lifetimes and also on the correlation times for rotation at each site; the NMR linewidth may thus be used to obtain information about any of these quantities. The results bear directly on the use of quadrupolar nuclei in the presence of chemical exchange of nuclei between suitable sites to obtain motional information about macromolecules in dilute solution. In particular, the proposed correlation function may account for anaomalously small calculated rotational‐correlation times for large molecules previously obtained from such studies. A precise criterion for applicability of McConnell‐modified Bloch equations is proposed. Other applications of the correl...
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TL;DR: The backbone dynamics of the pleckstrin homology (PH) domain from dynamin were studied by 15N NMR relaxation and steady state heteronuclear 15N [1H] nuclear Overhauser effect measurements, and by molecular dynamics simulations.
232 citations
TL;DR: In this article, the effect of chemical exchange on nuclear magnetic relaxation has been modified to take account of nonexponential relaxation "in the absence of exchange" at one or both of the sites involved in two-site exchange.
195 citations
TL;DR: The circumstances under which the various cases of line splitting and line broadening can occur are indicated, and the consequences of these circumstances for the relaxation times, as determined by pulsed NMR techniques.
Abstract: In the following we shall consider the nature of nuclear magnetic resonance signals of 23Na as observed in tissues and biological materials.’, The explanation of the observed spectra is based on quadrupolar interactions, these being the dominant cause of line broadening. A careful consideration of the theoretical aspects reveals that in many cases a single species of sodium ion will give rise to both a broad and a narrow component of the resonance line. These two components are due to transitions between different spin states of the same species of sodium ion, and hence it is incorrect in these cases to interpret the two components as resulting from a “bound” and a “free” species of sodium ion. The nature of the resonance is further strongly influenced by the extent of molecular order present in the sample, and by the diffusion of sodium ions between regions of different average quadrupolar interaction. The validity of this interpretation is derived from experiments on oriented Liand Na-DNA, muscle, and ion exchange resins. Pulsed nuclear magnetic resonance (NMR) methods have been used to determine the relative magnitude and the spin-spin and spin-lattice relaxation times of the resonance components. The ?Li and 23Na resonance signals of oriented DNA, as observed in collaboration with Dr. A. Rupprecht of the University of Stockholm, have been published b e f ~ r e . ~ The interpretation of a broad and a narrow resonance component in terms of a single species of sodium ion has been put forward by Shporer and Civan? on the basis of sodium NMR spectra of unoriented liquid crystals of sodium linoleate in water; they also observed that the relative amount of “bound” sodium in various tissues, as measured by several authors, is remarkably constant. We shall proceed to indicate the circumstances under which the various cases of line splitting and line broadening can occur, and the consequences of these circumstances for the relaxation times, as determined by pulsed NMR techniques. Finally, we shall discuss some of the experimental pitfalls that lure the experimenter into unjustified conclusions.
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TL;DR: In this article, the authors studied the effect of the thermal motion of the magnetic nuclei upon the spin-spin interaction in a rigid lattice and the line width of the absorption line.
Abstract: The exchange of energy between a system of nuclear spins immersed in a strong magnetic field, and the heat reservoir consisting of the other degrees of freedom (the "lattice") of the substance containing the magnetic nuclei, serves to bring the spin system into equilibrium at a finite temperature. In this condition the system can absorb energy from an applied radiofrequency field. With the absorption of energy, however, the spin temperature tends to rise and the rate of absorption to decrease. Through this "saturation" effect, and in some cases by a more direct method, the spin-lattice relaxation time ${T}_{1}$ can be measured. The interaction among the magnetic nuclei, with which a characteristic time $T_{2}^{}{}_{}{}^{\ensuremath{'}}$ is associated, contributes to the width of the absorption line. Both interactions have been studied in a variety of substances, but with the emphasis on liquids containing hydrogen.Magnetic resonance absorption is observed by means of a radiofrequency bridge; the magnetic field at the sample is modulated at a low frequency. A detailed analysis of the method by which ${T}_{1}$ is derived from saturation experiments is given. Relaxation times observed range from ${10}^{\ensuremath{-}4}$ to ${10}^{2}$ seconds. In liquids ${T}_{1}$ ordinarily decreases with increasing viscosity, in some cases reaching a minimum value after which it increases with further increase in viscosity. The line width meanwhile increases monotonically from an extremely small value toward a value determined by the spin-spin interaction in the rigid lattice. The effect of paramagnetic ions in solution upon the proton relaxation time and line width has been investigated. The relaxation time and line width in ice have been measured at various temperatures.The results can be explained by a theory which takes into account the effect of the thermal motion of the magnetic nuclei upon the spin-spin interaction. The local magnetic field produced at one nucleus by neighboring magnetic nuclei, or even by electronic magnetic moments of paramagnetic ions, is spread out into a spectrum extending to frequencies of the order of $\frac{1}{{\ensuremath{\tau}}_{c}}$, where ${\ensuremath{\tau}}_{c}$ is a correlation time associated with the local Brownian motion and closely related to the characteristic time which occurs in Debye's theory of polar liquids. If the nuclear Larmor frequency $\ensuremath{\omega}$ is much less than $\frac{1}{{\ensuremath{\tau}}_{c}}$, the perturbations caused by the local field nearly average out, ${T}_{1}$ is inversely proportional to ${\ensuremath{\tau}}_{c}$, and the width of the resonance line, in frequency, is about $\frac{1}{{T}_{1}}$. A similar situation is found in hydrogen gas where ${\ensuremath{\tau}}_{c}$ is the time between collisions. In very viscous liquids and in some solids where $\ensuremath{\omega}{\ensuremath{\tau}}_{c}g1$, a quite different behavior is predicted, and observed. Values of ${\ensuremath{\tau}}_{c}$ for ice, inferred from nuclear relaxation measurements, correlate well with dielectric dispersion data.Formulas useful in estimating the detectability of magnetic resonance absorption in various cases are derived in the appendix.
4,973 citations
Book•
01 Jan 1963
TL;DR: In this article, the effect of changing the precession frequency of the magnetic field has been studied using NMR to study rate properties. But the effect is limited to the case of double and double resonance.
Abstract: 1. Elements of Resonance.- 2 Basic Theory.- 3. Magnetic Dipolar Broadening of Rigid Lattices.- 4. Magnetic Interactions of Nuclei with Electrons.- 5. Spin-Lattice Relaxation and Motional Narrowing of Resonance Lines.- 6. Spin Temperature in Magnetism and in Magnetic Resonance.- 7. Double Resonance.- 8. Advanced Concepts in Pulsed Magnetic Resonance.- 9. Multiple Quantum Coherence.- 10. Electric Quadrupole Effects.- 11. Electron Spin Resonance.- 12. Summary.- Problems.- Appendixes.- A. A Theorem About Exponential Operators.- B. Some Further Expressions for the Susceptibility.- D. A Theorem from Perturbation Theory.- E. The High Temperature Approximation.- F. The Effects of Changing the Precession Frequency - Using NMR to Study Rate Phenomena.- G. Diffusion in an Inhomogeneous Magnetic Field.- H. The Equivalence of Three Quantum Mechanics Problems.- I. Powder Patterns.- J. Time-Dependent Hamiltonians.- K. Correction Terms in Average Hamiltonian Theory - The Magnus Expansion.- Selected Bibliography.- References.- Author Index.
4,921 citations
TL;DR: In this paper, the modified Bloch equations for nuclear magnetic resonance are modified to describe the magnetic resonance of a single nuclear species X which is transferred back and forth between two (or more) magnetic environments (A,B) by kinetic molecular processes.
Abstract: The Bloch equations for nuclear magnetic resonance are modified to describe the magnetic resonance of a single nuclear species X which is transferred back and forth between two (or more) magnetic environments (A,B) by kinetic molecular processes. The modified Bloch equations involve the usual assumptions of the Bloch theory and, in addition, require (a) that the X nuclear relaxation times be independent of the molecular exchange rates, and (b) that the X nuclear magnetization in A relax independently of the X magnetization in B, and vice versa. The modified Bloch equations are easily solved in the slow passage case, with arbitrary rf saturation. Earlier relations between reaction rates, and resonance line shapes, which were developed by Gutowsky, McCall, and Slichter, and extended by a number of other investigators, are easily derived using the modified Bloch equations. In the present work the modified equations are used to show how rapid exchange rates can sometimes be measured in solutions where the X r...
1,453 citations
TL;DR: In this paper, an investigation was made of the temperature and frequency dependence of T2 for O17 in aqueous solutions containing Mn2+, Fe2+, Co2+, Ni2+, and Cu2+.
Abstract: An investigation was made of the temperature and frequency dependence of T2 for O17 in aqueous solutions containing Mn2+, Fe2+, Co2+, Ni2+, and Cu2+. This represented an extension of the studies previously performed in this laboratory on these ions. Virtually all of the temperature effects predicted by the modified Bloch equations for a two‐species system were verified experimentally. Rates of exchange of water molecules between the bulk of the solution and the first coordination sphere of the paramagnetic cations were determined for all the ions studied. Activation energies for exchange were measured and electronic T1's and coupling constants were determined in some cases. Evidence was found for a tetrahedral Co2+(H2O)4 species in aqueous solutions near 100°C. The data for cupric ion were interpreted in terms of six coordinated water molecules in a distorted octahedron, with a ratio of ∼105 existing for the axial‐water‐exchange rate over that of the equatorial waters. The rates of exchange were compared ...
1,200 citations