Moment gradient

Typically, at least two different values of m1 (besides ml = 0) are used because there are invariably phase shifts that arise from various factors that do not depend on the gradient moments, resulting in a non-zero intercept of c > versus mx. Thus, a velocity image is time consuming because each set of measurements with a value of m1 is an image in itself. 

The higher-order gradient moments encode parameters of motion like velocity v and acceleration a and are related to the Fourier conjugate variables and e of these... 

Fig. 7.2.12 [Capl] Schemes for compensation of gradient moments. Left gradient waveforms without refocusing pulses. Right gradient waveforms for use in Hahn-echo sequences.

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Since atomic nuclei are not perfectly spherical their spin leads to an electric quadnipole moment if I>1 which interacts with the gradient of the electric field due to all surrounding electrons. The Hamiltonian of the nuclear quadnipole interactions can be written as tensorial coupling of the nuclear spin with itself... 

Pulay P, FogarasI G, Pang F and Boggs J E 1979 Systematic ab initio gradient calculation of molecular geometries, force constants and dipole moment derivatives J. Am. Chem. Soc. 101 2550... 

In addition to total energy and gradient, HyperChem can use quantum mechanical methods to calculate several other properties. The properties include the dipole moment, total electron density, total spin density, electrostatic potential, heats of formation, orbital energy levels, vibrational normal modes and frequencies, infrared spectrum intensities, and ultraviolet-visible spectrum frequencies and intensities. The HyperChem log file includes energy, gradient, and dipole values, while HIN files store atomic charge values. 

For a quantum mechanical calculation, the single point calculation leads to a wave function for the molecular system and considerably more information than just the energy and gradient are available. In principle, any expectation value might be computed. You can get plots of the individual orbitals, the total (or spin) electron density and the electrostatic field around the molecule. You can see the orbital energies in the status line when you plot an orbital. Finally, the log file contains additional information including the dipole moment of the molecule. The level of detail may be controlled by the PrintLevel entry in the chem.ini file. 

Finally, if the gas molecule possesses a quadrupole moment Q—examples are CO, COj and Nj—this will interact strongly with the field gradient F to produce a further contribution fQ to the energy. ... 

Moisture gradient is the moisture profile ia a material at a specific moment duriag dryiag, which usually reveals the mechanisms of moisture movement ia the material up to the moment of measurement. 

Moisture gradient refers to the distribution of water in a solid at a given moment in the diying process. 

Why is this relevant to the diffusion of zinc in copper Imagine two adjacent lattice planes in the brass with two slightly different zinc concentrations, as shown in exaggerated form in Fig. 18.5. Let us denote these two planes as A and B. Now for a zinc atom to diffuse from A to B, down the concentration gradient, it has to squeeze between the copper atoms (a simplified statement - but we shall elaborate on it in a moment). This is another way of saying the zinc atom has to overcome an energy barrier... 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

For specimens where gradients in the ms etic moment are of interest, similar arguments apply. Here, however, two separate reflectivity experiments are performed in which the incident neutrons are polarized parallel and perpendicular to the surfiice of the specimen. Combining reflectivity measurements under these two polarization conditions in a manner similar to that for the unpolarized case permits the determination of the variation in the magnetic moments of components parallel and perpendicular to the film surface. This is discussed in detail by Felcher et al. and the interested reader is referred to the literature. 

Once an approximation to the wavefunction of a molecule has been found, it can be used to calculate the probable result of many physical measurements and hence to predict properties such as a molecular hexadecapole moment or the electric field gradient at a quadrupolar nucleus. For many workers in the field, this is the primary objective for performing quantum-mechanical calculations. But from... 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

The dipole polarizability, the field gradient and the quadrupole moment are all examples of tensor properties. A detailed treatment of tensors is outside the scope of the text, but you should be aware of the existence of such entities. 

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often are modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

Here q is the net charge (monopole), p, is the (electric) dipole moment, Q is the quadrupole moment, and F and F are the field and field gradient d /dr), respectively. The dipole moment and electric field are vectors, and the pF term should be interpreted as the dot product (p F = + EyPy + Ez z)- "I e quadrupole moment and field... 

The A matrix involves elements between singly excited states while B is given by matrix elements between doubly excited states and the reference. The P/Q elements are matrix elements of the operator between the reference and a singly excited state. If P = r this is a transition moment, and in the general case it is often denoted a property gradient , in analogy with the case where the operator is the Hamiltonian (eq. (3.67). 

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