Spin autocorrelation functions

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

However, the possibility of Id enhancement is limited by the one-dimensional life time of the electron. Introducing this life time effect into the spin autocorrelation function... 

Other special relaxation fiinctions on a phenomenological basis have been given by Crook and Cywinski (1997) and K.M. Kojima et al. (1997). The treatment of dilute spin glasses based on various spin-spin autocorrelation functions (Keren et al. 1996, 2000) has been mentioned in sect. 8.2.1. 

NMR 13C spin-lattice relaxation times are sensitive to the reorientational dynamics of 13C-1H vectors. The motion of the attached proton(s) causes fluctuations in the magnetic field at the 13C nuclei, which results in decay of their magnetization. Although the time scale for the experimentally measured decay of the magnetization of a 13C nucleus in a polymer melt is typically on the order of seconds, the corresponding decay of the 13C-1H vector autocorrelation function is on the order of nanoseconds, and, hence, is amenable to simulation. 

Figure 44. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full line is the quantum result, the dashed-dotted line is the result of the semiclassical spin-coherent state propagator, and the dashed line depicts the result of Suzuki s propagator. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave functions.

The results obtained for the three-mode Model IVb are depicted in Fig. 45. As was found for the semiclassical mapping approach, the spin-coherent state propagators can only reproduce the short-time dynamics for the electronic population. The autocorrelation function, on the other hand, is reproduced at least qualitatively correctly by the semiclassical spin-coherent state propagator. 

For a ferromagnet the order parameter is the magnetization, while for antiferromagnets it is the sublattice magnetization. For spin glasses the magnetization is zero at all temperatures, and an appropriate order parameter was proposed by Edwards and Anderson [78] as the average value of the autocorrelation function... 

There are many experiments which determine only specific frequency components of the power spectra. For example, a measurement of the diffusion coefficient yields the zero frequency component of the power spectrum of the velocity autocorrelation function. Likewise, all other static coefficients are related to autocorrelation functions through the zero frequency component of the corresponding power spectra. On the other hand, measurements or relaxation times of molecular internal degrees of freedom provide information about finite frequency components of power spectra. For example, vibrational and nuclear spin relaxation times yield finite frequency components of power spectra which in the former case is the vibrational resonance frequency,28,29 and in the latter case is the Larmour precessional frequency.8 Experiments which probe a range of frequencies contribute much more to our understanding of the dynamics and structure of the liquid state than those which probe single frequency components. 

Note that G(t) is also the reduced autocorrelation function.10 The above series, however, converges very slowly. In practice it is therefore necessary to use a modified moment expansion that converges more rapidly. In order to obtain an explicit expression for G(t), we consider a single crystal containing only one species of nuclei with spin such that the nuclei are located at the points of a lattice. We further assume that the only interaction present is the nuclear dipole-dipole interaction. It can be shown that10... 

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

This stretched exponential behavior known from spin-glass theory. [16] Each run produces an estimate of r, that depends on the energy minimum visited as the systems freezes (or perhaps on two where the double peak in t is observed).[35] The ensemble average autocorrelation function is then a sum over many terms of the form which yields... 

There is a large class of relaxation mechanisms which operate on molecules in motion in non-metailic samples. All but one, the spin-rotation interaction, depend on the fact that the change in the molecular orientation or translation modulates the field due to that particular interaction and creates a randomly varying field at the site of the nucleus in question. Any such random motion can have associated with it a special form of an autocorrelation function G(t), expressed in terms of a scalar product of the local field h(t) and the same local field at an earlier time h(0), which is a measure of... 

From a practical point of view, all the above expressions for the orientation autocorrelation function lead to very similar numerical results for NMR spin-lattice relaxation time () calculations. 

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