Points field

The Champ-Sons model has been developed to quantitatively predict the field radiated by water- or solid wedge- eoupled transdueers into solids. It is required to deal with interfaces of complex geometry, arbitrary transducers and arbitrary excitation pulses. It aims at computing the time-dependent waveform of various acoustical quantities (displacement, velocity, traction, velocity potential) radiated at a (possibly large) number of field-points inside a solid medium. 

It is well established that GO approximation leads to aceurate results if both the source and the observation points are not close to the interface. In practice, this means that both points must be distant from the interface of at least one wavelength. This condition is always fulfilled for the source point. For the field-point, the accuracy of the solution is ensured if the above condition is completed, as shown by comparing exact and approximate results [5]. 

The Champ-Sons model is a most effieient tool allowing quantitative predictions of the field radiated by arbitrary transducers and possibly complex interfaces. It allows one to easily define the complete set of transducer characteristics (shape of the piezoelectric element, planar or focused lens, contact or immersion, single or multi-element), the excitation pulse (possibly an experimentally measured signal), to define the characteristics of the testing configuration (geometry of the piece, transducer position relatively to the piece, characteristics of both the coupling medium and the piece), and finally to define the calculation to run (field-points position, acoustical quantity considered). 

The field points must then be fitted to predict the activity. There are generally far more field points than known compound activities to be fitted. The least-squares algorithms used in QSAR studies do not function for such an underdetermined system. A partial least squares (PLS) algorithm is used for this type of fitting. This method starts with matrices of field data and activity data. These matrices are then used to derive two new matrices containing a description of the system and the residual noise in the data. Earlier studies used a similar technique, called principal component analysis (PCA). PLS is generally considered to be superior. 

A charge density p(r) generates an electrostatic potential 0(R) at the field point R according to the formula... 

Let us restrict our attention for the moment to the case where the number of patterns n << N, and look at the system from a mean-field point of view (see section 7.1.6). 

Clearly this process can rapidly become tedious, especially when one thinks about tracing hundreds or thousands of rays from many different points on the object (field points). Fortunately many computer programs have been written that allow us to harness the power of personal computers to do this. Nonetheless, most commercial computer programs use the exact same technique as that which we have described. 

For a single electron, the velocity operator in eq.(7) corresponds to the current operator j(Ro,B) = —ev, and hence to the following operator representing the magnetic field induced at a field point R... 

Focussing on terms linear in the applied field B, the induced magnetic field at the field point R obtains as the expectation value of B "(R, Ro,B) with respect to the first order wave function corresponding to eq.(6), yielding... 

In an NMR context, i.e. when the field point coincides with the position of a magnetic nucleus, the implications of the separation of the response fields in eqs. (18-20) into parts generated by the various components of the shielding tensor, are appreciated by first noting that the resonance frequency for the nucleus can be written as [28]... 

In previous presentations [16-19,28], the LORG equations are formulated in a nucleus centered coordinate system. Explicit reference to a field point R, can be introduced following eq.(13), and the resulting LORG equations for the i, j th element of the shielding tensor become... 

Numerical results for the shielding field of the benzene molecule are collected in Table 1 for the center of the molecule (labelled COM), and for points along a quarter circle of radius 2.47 A from the -ajcis to the x-axis, see Figure 3 for specification of axes. The radius of the circle corresponds to the distance from the ring center to a proton but, as defined, the points lie in the entirely nucleus-free xz-plane. Except for COM, the entries in the table are labelled by the angle between the z-axes and the direction to the field point. The table includes the isotropic part of the shielding, and the principal... 

To relate the pi electron ring current diagram in Figure 5 to the response surfaces of Figure 2, we note that the diamagnetic pi electron response for a field applied perpendicularly to the molecular plane at the tt/3 field point... 

Ta foil Nuclear forward scattering of synchrotron radiation (NFS) at Ta resonance in Ta foil without and with applied magnetic field, point out advantages over conventional Ta Mossbauer spectroscopy... 

The second important spin-angular operation is the 90° rotation where the polarization is transformed from the z to the x direction or vice versa. A Mezei coil in the x,z plane is adjusted such that the resultant field points exactly in the direction of the bisection of the angle between x and z. A 180° rotation around this axis transforms the z component of polarization to the x direction. At the same time, the sign of the y component is inverted (Fig. lc). 

Major categories of process Raman applications include reaction monitoring, in-process quality checks, and mobile or field point measurements. Quality control laboratory applications often are converted to a continuous process monitoring approach, or could simply be viewed as part of a larger production process. 

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