Kubo relaxation function

Joe T and Albrecht A C 1993 Femtosecond time-resolved coherent anti-Stokes Raman spectroscopy of liquid benzene a Kubo relaxation function analysis J. Chem. Phys. 99 3244-51... 

Fig. 10.9 The Kubo relaxation function 4>(f) as given by Eq. (10.69) for an ensemble in which the correlation function decays exponentially with time constant t. In (A), (indicated in arbitrary time units) is varied, while the rms amplitude of the fluctuations (

Fig. 10.11 (A) The real solid line) and imaginary long dashes) parts of the eomplex relaxation function 0(r) given by Eqs. (10.73) and (10.74) with t = 1 arbitrary time unit and t)). (B) Absorptirai solid lines) and emission dashed lines) spectra calculated as the Fourier transforms of, respectively, the complex relaxation function (r) (Eqs. 10.72 and 10.73) and its complex conjugate t). The autocorrelation time constant was 1 arbitrary time unit, and a was 2, 5 or 10 reciprocal time units, as indicated...

Fig. 11.10 Dependence of three-pulse photon-echo signals on delay time (3, as calculated in the impulsive limit with Eq. (11.43) for /] = T (the delay between pulses 1 and 2) = 5 (A) and with Eq. (11.44) for ti = T =—5 (B). The delay between pulses 2 and 3 (/2 = T) was 0, 10 or 100, as indicated. The units of time are arbitrary. All calculations used the Kubo relaxation function (Eq. 10.69) with tc = 40 time units and

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

Rigorous statistical mechanical analysis indicates that the dielectric relaxation function 0(f) of an isotropic system in the linear response regime is equivalent to an autocorrelation function of a microscopic polarization p(f) fluctuating through the molecular motion at equilibrium (Cole, 1967 Kubo, 1957) ... 

This does not imply that spin precession does not take place. But it is not coherent and thus will only show in a loss of signal amplitude as described by GAO- Th appropriate spin relaxation function GAO for stationary fields was first derived by Kubo and Toyabe (1966) for NMR (where it has little practical use) and is known in p,SR as the (static,... 

The term longitudinal field (LF) refers to an external field iqrplied along the initial direction of muon spin (the z axis). LF measurements are most useful in cases where a Kubo-Toyabe-like relaxation function is observed in ZF. The apphed LF competes with the internal field distribution, trying to hold the muon spin in its original direction (thus trying to prevent relaxation). If the longitudinal field Sl fulfills the condition... 

Static relaxation for non-Gaussian and non-Lorentzian field distributions The Kubo-Toyabe relaxation functions assume a Gaussian (dense spin system) or Lorentzian (dilute spin system) distribution for the three cartesian components of the local field 5. Usually full isotropy of field distribution is assumed, but the non-isotropic case has been treated as well (see sect. 3.2.2). 

Fig. 98. Left The Gaussian-broadened Gaussian relaxation fimction (bottom) (explanation see text). ZF pSR asymmetry spectrum in polycrystalline CeCuo2Nio.jSn at 0.08K (top). The dashed line is a fit of the static Gaussian Kubo-Toyabe relaxation function. The solid line is a fit of the static Gaussian-broadened Gaussian function. From Noakes and Kalvius (1997). Right The minimum polarization achieved by the Monte Carlo RCMMV static ZF muon spin relaxation functions, as a function of the reciprocal of the moment-magnitude correlation length, in units of the magnetic-ion nearest-neighbor separation in the model lattice. The horizontal line represents the 1/3 asymptote, above which the minimum polarization cannot rise. The dashed line is a...

Recently, Larkin et al. (2000) have encountered (in a study of the spin-ladder system Sr(Cui xZnj )203, which falls outside this review, but see sect. 8.3.4) muon spin relaxation functions with too shallow and too broad minima of polarization to be reproduced by Kubo-Toyabe functions. Again, longitudinal field data showed the spin system to be static. These authors used an approach (called Kubo golden rule, KGR), which was originally derived by Kubo (see Kubo 1981 and Yamazaki 1997) to describe their findings. For details we refer to the original papers. The KGR method allows the calculation of the muon spin relaxation function for arbitrary field distributions (if they can be described by arithmetic function). In the spin-ladder compound an exponential field distribution reproduced the data. The approach of Noakes and Kalvius (1997) can be reproduced using KGR. 

For X = 0.75 and 0.4, spin glass (SG) type magnetism was found. The pSR relaxation function at low temperatures is of the Kubo-Toyabe type with the typical recovery to ao/3... 

Although Kubo s relaxation function can describe dephasing on time scales that are either shorter or longer than the energy correlation time, it rests on a particular model of the fluctuations and it still assumes that the correlation function (M t)) decays exponentially with time. Sinusoidal components can be added to M t) in Eq. (10.69) to represent coupling to particular vibrational modes of the molecule [29, 32, 33]. But more importantly, Eq. (10.69) also assumes that the mean energy difference between states n and m is independent of time and is the same... 

Figure 10.1 lA shows the real and imaginary parts of the relaxation function

The Fourier transform of the spin Green s function D(k0 = (0([ S k( X -k]) is connected with Kubo s relaxation function by the relation... 

For the moment, assume that the VE picture is correct and inertial solvent motion causes negligible dephasing. Diffusive motion must be the primary cause of coherence decay. In the VE theory, the diffusive motion is the relaxation of stress fluctuations in the solvent by viscous flow. The VE theory calculates both the magnitude Am and lifetime z0J of the resulting vibrational frequency perturbations. A Kubo-like treatment then predicts the coherence decay as a function of the viscosity of the solvent. Figure 19 shows results for typical solvent parameters. At low viscosity, the modulation is in the fast limit, so the decay is slow and nearly exponential. Under these conditions, the dephasing time is inversely proportional to the viscosity, as in previous theories [Equation (19)]. As the viscosity increases, the modulation rate slows. The decay becomes faster and approaches a... 

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