Fictitious field

In the rotating frame of reference for an on-resonance peak, the B0 field is exactly canceled by a fictitious field created by the rotation of the axes, so that for nuclei that are on-resonance the only field present is the B field during the spin lock (Z eff =B i). If we place the sample magnetization on the y axis of the rotating frame with a 90° hard pulse (phase —jc), the spin lock can be placed on the y axis (phase y). While the spin lock is on, the sample magnetization is locked on the y axis and will not undergo precession, as the only field present is the B field and the sample magnetization is on the same axis as the B field (Fig. 8.37). 

FIGURE 2.8 Formation of the effective rf field Brii in a frame rotating at o> radians/second. Beff is the vector sum of applied rf field B, along x and the residual field along z resulting from B0 and the fictitious field that represents the effect of the rotating frame. 

This expression applies in the rotating frame where tDo = ft>rf so the fictitious field (cf. Fig. 2.2.7) is zero. If the gradient strength is larger than the local fields corresponding to the spin interactions A, the total Hamiltonian is dominated by the Zeeman interaction with field gradient, i.e.,... 

Fig. 9.2.7 Illustration of the fast adiabatic passage through resonance. The magnetization M follows the direction of the effective field The effective field in the rotating frame is the vector sum of the fictitious field Sbc and rf excitation field B. Both fields are applied in orthogonal directions. Because the fictitious field is proportional to the resonance offset S2, the magnitude of the fictitious field and thus the direction of the effective field can be changed by adjusting the resonance offset frequency S2.

Setting the new independent variable v, /tyt, to zero puts the ensemble in correspondence with the actual chemical equilibrium that inspires its construction. Any other choice would correspond to a completely fictitious field that artificially drives the reaction one way or the other. We will not consider this variant in any detail as ensembles for reacting systems are presented more fully in the chapter by Johnson. 

Thus, except for a constant term 2ho geminal space (hQ = 0 for the spin space), it is clear that the additive "one-electron part Hi(l,2) of the total Hamiltonian operator H in either the spin space or the geminal space corresponds to a Zeeman-like interaction of a particle of spin 1 with a field h, either a real magnetic field By with 2h = gPe fictitious field B, with 2h - g ... 

The first attempt to compare 2D- and 3D-slab models within the same calculation scheme was made in [775] for HF LCAO studies of the surface properties of BaTiOs in the cubic perovskite structure. The authors of [775] concluded that results for periodic 3D-slabs are systematically affected by the interactions among repeated images, and possibly the fictitious field imposed by periodic boundary conditions. 

In the rotating frame, the magnetization therefore precesses about an effective field,, that is the vector sum of field H and fictitious field (b/ y. The equations of motion of magnetization m of a system of spins subjected to effective field H in the rotating frame can then be expressed as ... 

When the Bi (probe) field is applied (with angular frequency a>i = yBi) the x component induces precession of My and about the effective field, and correspondingly for the y component. The phase-sensitive detector monitors and My as the u and V mode signals, respectively. In the rotating frame, u is in-phase with B, (in the x direction) and v out-of-phase (in the y direction), and there is a fictitious field coo/y (equivalent to the rotation) which cancels out Bq at resonance. The Bloch equations then become... 

In the absence of an RF field, H = Hq. At resonance, therefore, the fictitious field w/y exactly cancels H, and Heff becomes zero. When the static magnetic field is in the z direction and the RF field Hi is applied along the x direction (in other words. Hi is rotating clockwise in the x-y plane), the total magnetic field H is... 

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Many authors have shown that residual stresses in glass articles can be formally considered as the thermal stresses due to a certain fictitious temperature field. In the general case... 

Theory of the fictitious temperature field allows us to analyze the problems of residual stresses in glass using the mathematical apparatus of thermoelasticity. In this part we formulate the boundary-value problem for determining the internal stresses. We will Lheretore start from the Duhamel-Neuinan relations... 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

In octahedral symmetry, the F term splits into Aig + T2g + Tig crystal-field terms. Suppose we take the case for an octahedral nickel(ii) complex. The ground term is 2g. The total degeneracy of this term is 3 from the spin-multiplicity. Since an A term is orbitally (spatially) non-degenerate, we can assign a fictitious Leff value for this of 0 because 2Leff+l = 1. We might employ Van Vleck s formula now in the form... 

This field of the centrifugal force, unlike the attraction field, is fictitious, and correspondingly, we observe a volume distribution of fictitious sources with a density proportional to co. A summation of the first and second Equations (2.62 and 2.63) gives the system of equations of the gravitational field at regular points... 

First, we imagine that there are fictitious masses at points of the surface S and they are distributed with surface density a in such way that the potential of their field is equal T. 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

The analytic evaluation of the density matrix requires the diagonalization of Hi, which can be easily performed for the two extreme cases, a>r( Qg. Indeed, in the case of a low rf-field, only off-diagonal terms related to the CT are retained in the H Hamiltonian (39), which thus behaves like a fictitious spin-1/2 operator, affecting only the CT coherences /1 io- These coherences are thus selectively excited with the nutation frequency ... 

In this short review, a brief overview of the underlying principles of TDDFT has been presented. The formal aspects for TDDFT in the presence of scalar potentials with periodic time dependence as well as TD electric and magnetic fields with arbitrary time dependence are discussed. This formalism is suitable for treatment of interaction with radiation in atomic and molecular systems. The Kohn-Sham-like TD equations are derived, and it is shown that the basic picture of the original Kohn-Sham theory in terms of a fictitious system of noninteracting particles is retained and a suitable expression for the effective potential is derived. 

This potential is often written in terms of a fictitious exchange-correlation magnetic field Bxc... 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

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