The limit diffusion mechanism of relaxation for spin systems

The diffusion limit theorem for stochastic differential equations is applied to analyse the dynamical evolutions of spin systems. Bloch equations are derived and the stability of asymptotic evolutions is proved. The theory is applied to nuclear magnetic relaxation of two spins.     

A simulation algorithm based on Bloch equations and product operator matrix: application to dipolar and scalar couplings

A product operator matrix is proposed to describe scalar couplings in liquid NMR. Combination of the product operator matrix and non-linear Bloch equations is employed to describe effects of chemical shift, translational diffusion, dipolar field, radiation damping, and relaxation in multiple spin systems with both scalar and dipolar couplings. A new simulation algorithm based on this approach is used to simulate NMR signals from dipolar field effects in the presence of scalar couplings. Several typical coupled spin systems with both intra-molecular scalar couplings and inter-molecular dipolar couplings are simulated. Monte Carlo methods are incorporated into simulations as well to analyze diffusion process in these complicated spin systems. The simulated results of diffusion and relaxation parameters and 2D NMR spectra are coincident with the experimental measurements, and agree with theoretical predictions as well. The simulation algorithm presented herein therefore provides a convenient means for designing pulse sequences and quantifying experimental results in complex coupled spin systems.     

Gauge theory approach for diffusive and precessional spin dynamics in a two-dimensional electron gas

We develop a gauge theory for diffusive and precessional spin dynamics in a two-dimensional electron gas. Our approach reveals a direct connection between the absence of the equilibrium spin current and a strong anisotropy in the spin relaxation: both effects arise if spin-orbit coupling is reduced to a pure gauge SU(2) field. In this case, the spin-orbit coupling can be removed by a gauge transformation in the form of a local SU(2) spin rotation. The resulting spin dynamics is exactly described in terms of two kinetic coefficients: the spin diffusion and electron mobility. After the inverse transformation, full diffusive and precessional spin density dynamics, including the anisotropic spin relaxation, formation of stable spin structures, and spin precession induced by a macroscopic current are restored. Explicit solutions of the spin evolution equations are found for the initially uniform spin density and for stable, nonuniform structures. Our analysis demonstrates a universal relation between the spin relaxation rate and spin-diffusion coefficient.     

End-to-end correlation for a C-12 hydrocarbon chain

The 19F nuclear spin-lattice relaxation rate constants were measured as a function of magnetic field strength for 1,12-diaminododecane labeled at one end with a nitroxide radical and at the other with a trifluoromethyl group. The magnetic relaxation dispersion profile (MRD) reports the spectral density function appropriate to the end-to-end correlation function for the doubly labeled molecule. After extrapolation to zero concentration to eliminate the intermolecular relaxation contribution to relaxation, the resulting intramolecular MRD profile was compared with several model approaches. The rotational model for the spectral density functions as included in the Solomon-Bloembergen-Morgan equations does not describe the data well. The earlier model of Freed for nuclear spin relaxation induced by a freely diffusing paramagnetic co-solute is not rigorous for this case because the paramagnet is tethered to the observed nuclear spin and only a restricted space in the immediate vicinity of the nuclear spin is accessible for pseudo-translational diffusion of one end of the molecule with respect to the other. A generalization of the Torrey model for magnetic relaxation by translational diffusion developed by Nevzorov and Freed, which includes the effect of restrictions imposed by the finite length of the chain, describes the experiment within experimental errors. A simple modification of the Hwang-Freed model that does not specifically include the dynamical effects of the finite tether also provides a good approximation to the data when the tether chain is sufficiently long.     

Spin diffusion of dipolar energy in NQR

The theory of spin diffusion was extended to the case of nuclear dipolar order in solids containing paramagnetic impurities and nuclei with spin I > 1/2 having nuclear quadrupole moment. We show that spin diffusion process of dipolar order takes place in solids containing paramagnetic impurities. At the start of relaxation process, the direct relaxation regime is realized with non-exponential time dependence. Then the relaxation regime will be changed to diffusion-limited one. Using obtained expressions for the spin lattice relaxation times for these two relaxation regimes, the diffusion coefficient of the dipolar order in nuclear quadrupole resonance can be estimated from experimental data.     

Diffusion,trapping and detrapping of positive muon in Al-0.047% Mg

Positive muon spin relaxation was measured in Al-0.047% Mg quenched from 873 K under zero external magnetic field from 6.3 K to 270 K. The observed spectra were analysed with the calculated muon spin relaxation function which included the static relaxation rates, the trapping rates, the detrapping rates and thet=0 initial trapping fractions. Due to the precise measurements and the realistic expression of spin relaxation function, above four parameters could be determined distinctly. The trapping rates and the distortions around a muon were determined from the values of static relaxation rates at a trapping site. The diffusion features were clearly described by the trapping and detrapping rates. The diffusion process was determined by the temperature dependence of these parameters.     

Nuclear spin-lattice relaxation mechanisms in kaolinite confirmed by magic-angle spinning

Spin-lattice relaxation mechanisms in kaolinite have been reinvestigated by magic-angle spinning (MAS) of the sample. MAS is useful to distinguish between relaxation mechanisms: the direct relaxation rate caused by the dipole-dipole interaction with electron spins is not affected by spinning while the spin diffusion-assisted relaxation rate is. Spin diffusion plays a dominant role in ¹H relaxation. MAS causes only a slight change in the relaxation behavior, because the dipolar coupling between ¹H spins is strong. ²⁹Si relaxes directly through the dipole-dipole interaction with electron spins under spinning conditions higher than 2 kHz. A spin diffusion effect has been clearly observed in the ²⁹Si relaxation of relatively pure samples under static and slow-spinning conditions. ²⁷Al relaxes through three mechanisms: phonon-coupled quadrupole interaction, spin diffusion and dipole-dipole interaction with electron spins. The first mechanism is dominant, while the last is negligibly small. Spin diffusion between ²⁷Al spins is suppressed completely at a spinning rate of 2.5 kHz. We have analyzed the relaxation behavior theoretically and discussed quantitatively. Concentrations of paramagnetic impurities, electron spin-lattice relaxation times and spin diffusion rates have been estimated.     

Spin-lattice relaxation in hyperfine-coupled systems: Applications to interstitial diffusion and molecular dynamics

The solid state diffusion of hydrogen, or of its pseudo-isotope muonium, provides an interesting example of spin-lattice relaxation in a 2-spin, 4-level system. The local field experienced by the interstitial atom fluctuates as it moves, inducing transitions between the coupled electron and nuclear spin states. Rate equations governing the populations of these states may be solved numerically to simulate the different relaxation functions which would be displayed by ESR, ENDOR and μSR spectroscopies and to assist in extracting motional correlation times from the experimental data. Spin relaxation in molecular radicals may be treated similarly, with different selection rules for different mechanisms: this paper treats the spin rotation mechanism and perturbation to anisotropic or isotropic components of the hyperfine interaction, caused by inter or intra-molecular motion. Conventional magnetic resonance monitors the population differences appropriate to particular transitions; only in sufficiently high fields do these distinguish the electronic and nuclear response. Muon spin relaxation is remarkable in separating out the nuclear spin projection whatever the degree of mixing of the spin states,via the asymmetry in the muon radioactive decay. Experimentally it has the advantage that measurements can be made over a wide range of field, from null external field up to thelevel crossing where the relaxation rate exhibits a striking peak.     

Rotating frame proton spin relaxation in viscous glycerol

The observation of a rotating frame Larmor frequency dependence in the proton spin relaxation of glycerol lends support to a recent study of proton spin relaxation in this substance. Comparison of the data to recent theories describing spin relaxation by translational diffusion strongly suggest that this is the dominant relaxation process in glycerol near room temperature. In addition, the measurement of the self-diffusion coefficient of glycerol, at a temperature lower than has been previously reported, indicates that this rotating frame technique may be useful in the study of diffusion in viscous media.     

Spin diffusion and nuclear spin-lattice relaxation in irradiated solids: a multiple-pulse NQR study

We present a detailed theoretical and experimental NQR multiple-pulse spin-locking study of spin-lattice relaxation and spin diffusion processes in the presence of paramagnetic impurities in solids. The relaxation function of the nuclear spin system at the beginning of the relaxation process is given by exp , where T_1ρ is spin-lattice relaxation time in rotating frame and =d/6, d is the sample dimensionality. Then the relaxation proceeds asymptotically to an exponential function of time, which was attributed to the spin-diffusion regime. Using the experimental data obtained from the analysis of those two relaxation regimes in γ-irradiated powdered NaClO₃, spin diffusion coefficient has been determined and the radius of the diffusion barrier has been estimated.     

Correction of spin diffusion during iterative automated NOE assignment

Indirect magnetization transfer increases the observed nuclear Overhauser enhancement (NOE) between two protons in many cases, leading to an underestimation of target distances. Wider distance bounds are necessary to account for this error. However, this leads to a loss of information and may reduce the quality of the structures generated from the inter-proton distances. Although several methods for spin diffusion correction have been published, they are often not employed to derive distance restraints. This prompted us to write a user-friendly and CPU-efficient method to correct for spin diffusion that is fully integrated in our program ambiguous restraints for iterative assignment (ARIA). ARIA thus allows automated iterative NOE assignment and structure calculation with spin diffusion corrected distances. The method relies on numerical integration of the coupled differential equations which govern relaxation by matrix squaring and sparse matrix techniques. We derive a correction factor for the distance restraints from calculated NOE volumes and inter-proton distances. To evaluate the impact of our spin diffusion correction, we tested the new calibration process extensively with data from the Pleckstrin homology (PH) domain of Mus musculus beta-spectrin. By comparing structures refined with and without spin diffusion correction, we show that spin diffusion corrected distance restraints give rise to structures of higher quality (notably fewer NOE violations and a more regular Ramachandran map). Furthermore, spin diffusion correction permits the use of tighter error bounds which improves the distinction between signal and noise in an automated NOE assignment scheme.     

Zur kernmagnetischen Relaxation in Mehrbereichsystemen mit endlicher räumlicher Ausdehnung

General equations, describing the nuclear spin relaxation behaviour in heterogeneous systems, consisting of regions with different relaxation times and diffusion coefficients, are presented. For a special two region system, by means of analogue computing technique the influence of both diffusivity and height of the barriers between the regions on total relaxation is studied. It is shown, that using appropriate expressions for the mean lifetimes of the nuclei in the regions, the relaxation function of total magnetisation may be approximated by the formulae of ZIMMERMAN and BRITTIN . These relations may be used for a rough estimation of diffusivity in the systems under consideration.     

Exchange and dipolar spin-spin interaction and rotational diffusion in saturation transfer EPR spectroscopy

Saturation transfer EPR spectroscopy (STEPR) provides a means for investigating weak spin-spin interaction between spin-labelled molecules because the spectral intensity is proportional to the effective spin-lattice relaxation time,T ₁ ^eff. Rate equations for the spin population defferences yield equivalent results for the dependence ofT ₁ ^eff on the physical (or chemical) and Heisenberg spin exchange rates and show thatT ₁ ^eff depends on the extent of redistribution of saturation throughout the anisotropic spin label powder lineshape. This approach yields a particularly simple formulation for the dependence of the STEPR lineshape on slow rotational diffusion. The effects of spin exchange are readily distinguished from those of slow rotational diffusion because of the insensitivity of the STEPR lineshape in the former case. The characteristic dependence of the STEPR spectral intensity on spin concentration allows determination of the exchange rate and can be used for studying slow translational diffusion, e.g. of spin-labelled proteins. Dipolar relaxation induced by paramagnetic ions gives a linear dependence of the reciprocal spin label STEPR intensity on metal ion concentration. STEPR measurements with spin-labelled lipid molecules in gel phase membranes in the presence of Ni²⁺ ions yield reliable distance information and provide calibrations for use with other systems.     

Spin-orbit coupling, spin relaxation, and spin diffusion in organic solids

We develop a systematic approach of quantifying spin-orbit coupling (SOC) and a rigorous theory of carrier spin relaxation caused by the SOC in disordered organic solids. The SOC mixes up and down spin in the polaron states and can be characterized by an admixture parameter γ2. This mixing effects spin flips as polarons hop from one molecule to another. The spin relaxation time is τ(sf) = R2/(16γ2 D), and the spin diffusion length is L(s) = R/4|γ|, where R is the mean polaron hopping distance and D the carrier diffusion constant. The SOC in tris-(8-hydroxyquinoline) aluminum (Alq3) is particularly strong due to the orthogonal arrangement of the three ligands. The theory quantitatively explains the temperature-dependent spin diffusion in Alq3 from recent muon measurements.     

Bloch equations with diffusion terms for rotational motion in fluids

The phenomenological Bloch equations for magnetic resonance are supplemented by terms for rotational diffusion in fluids. The equations are used to describe damping of magnetization components and the line shape of nonresonance absorption in parallel fields for ions with an anisotropic g factor and an effective spin of 1/2. In both cases, the contributions of the diffusion terms are comparable to the contribution from longitudinal and transverse relaxation times (T₁, T₂).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 42–46, January, 1973.The authors are indebted to B. M. Kozyrev for his interest in the work and discussions of the results.     

Spin relaxation in the presence of electron-electron interactions

The D'yakonov-Perel' spin relaxation induced by the spin-orbit interaction is examined in disordered two-dimensional electron gas. It is shown that, because of the electron-electron interactions, substantially different spin relaxation rates may be observed depending on the technique used to extract them. It is demonstrated that the relaxation rate of a spin population is proportional to the spin-diffusion constant D(s), while the spin-orbit scattering rate controlling the weak-localization corrections is proportional to the diffusion constant D, i.e., the conductivity. The two diffusion constants get strongly renormalized by the electron-electron interactions, but in different ways. As a result, the corresponding relaxation rates are different, with the difference between the two being especially strong near a magnetic instability or near the metal-insulator transition.     

Optical measurement and control of spin diffusion in n-doped GaAs quantum wells

Transient spin gratings are used to study spin diffusion in lightly n-doped GaAs quantum wells at low temperatures. The spin grating is shown to form in the excess electrons from doping, providing spin relaxation and transport properties of the carriers most relevant to spintronic applications. We demonstrate that spin diffusion of the these carriers is accelerated by increasing the density or energy of the optically excited carriers. These results can be used to better understand and even control spin transport in experiments that optically excite spin-polarized carriers.     

On the magnetic field dependence of nuclear spin relaxation (NRS) on surfaces

We have measured the nuclear spin relaxation rate for nuclear spin polarized⁷Li atoms adsorbed on a hot O-W(110) surface and found that it increases as the magnetic field strength approaches zero. The trend of the nuclear spin relaxation rate generally agrees with a logarithmic divergence, a consequence of the correlation function for two-dimensional diffusion. In principle, such experiments yield information on absolute values of diffusion rates for the adsorbed atoms. Supported in part by a Travel Grant from the North Atlantic Treaty Organization.     

Effect of spin diffusion on the spin resonance in local moment systems

The effect of conduction electron spin diffusion on linewidth, lineshift and residue of the spin resonance in local moment systems is investigated. For the calculation of the transverse susceptibility a previously established kinetic equation approach is used. Low temperature Kondo-type anomalies of spin relaxation and diffusion are taken into account in the framework of Suhl's theory. The susceptibility exhibits considerable structure as a function of wavevector. In particular it is shown that the magnetic resonance bottleneck is broken for sufficiently large wavevectors. The effect of relaxation proceeding to local instead of total equilibrium is investigated and shown to introduce only minor modifications with one possible exception. Application of the theory to transmission electron spin resonance (TESR) is also discussed.     

Nuclear spin relaxation of 8Li adsorbed on surfaces

Nuclear spin relaxation experiments with ⁸Li adsorbed on various surfaces provide new information in surface science which is not obtainable othervise. Both dipolar (Korringa) and quadrupolar relaxation due to diffusion are observed. However, in addition, a fast and presently not understood spin relaxation mechanism is present while dosing during the first 0.5 s the surface with polarized ⁸Li. Most strange in this respect is the fact that those ⁸Li atoms which survive depolarization through this mechanism depolarize afterwards with modest spin lattice relaxation rates. The origin of the fast spin lattice relaxation mechanism is presently unknown. This revised version was published online in July 2006 with corrections to the Cover Date.