Density-to-flux transformation

To analyze the data, first perform the vx- and ty-corrections and the time-to-speed transformation to make the velocity volume element the same for all data points, and then normalize each Doppler-selected TOF spectrum according to the averaged ID Doppler profile from several independent scans, I vz) = )T)(, vyS(vy vx 0,vz)dvy. Due to the large exothermicity of this reaction, the problematic density-to-flux transformation is not negligible (despite the large probe laser size used to minimize its effects) and needs to be accounted for (the wy-correction , see Sec. 3.3). By combining all the resulting TOF spectra, the product 3D velocity flux contour... 

The density here refers to the spatial coordinate, i.e. the concentration of the reaction product, and is not to be confused with the D(vx,vy,vz) in previous sections which refers to the center-of-mass velocity space. Laser spectroscopic detection methods in general measure the number of product particles within the detection volume rather than a flux, which is proportional to the reaction rate, emerging from it. Thus, products recoiling at low laboratory velocities will be detected more efficiently than those with higher velocities. The correction for this laboratory velocity-dependent detection efficiency is called a density-to-flux transformation.40 It is a 3D space- and time-resolved problem and is usually treated by a Monte Carlo simulation.41,42... 

Fig. 5. The / ,-< < iirection (i.e. the density-to-flux transformation) function for the current experimental setup.

A second property, closely related to the first, is the abiHty of the heat pipe to effect heat-flux transformation. As long as the total heat flow is ia equiHbrium, the fluid streams connecting the evaporatiag and condensing regions essentially are unaffected by the local power densities ia these two... 

The detection technique can also have an effect upon the angle- and velocity-dependent intensities. Cross sections refer to fluxes of molecules into a given range of velocities and angles. The commonly employed technique of mass spectrometric detection provides a measure of the density in the ionization region. Since density and flux are related by the velocity, we must include a factor of 1/v in making the transformation indicated in equation (B2.3.10) from the CM cross sections to the measured laboratory intensities. 

The operator (E t - r ) transforms the neutron source to the uncollided flux. Weighting this uncoHided flux distribution with the flux importance function gives the ultimate contribution of the neutron source to the total neutron flux in the critical reactor. The source importance, , on the other hand, gives the ultimate contribution of the same source to the total neutron density in that reactor. Thus, the normalization constant Cj is the ratio of the total neutron density to the total neutron flux (i.e., the inverse of the average neutron velocity) in the critical reactor. 

Here the primes denote quantities referring to the moving coordinate system. The above transformation laws (as well as the local equilibrium averages to be obtained) have been given by Green, and may be obtained directly from the explicit form of the densities and fluxes. 

The densities n(v",K") measured by LBF at the crossing point are not strictly proportional to the rate this has been taken into account since early LBF crossed beam works through the use of the density-flux transformation of the product recoiling with laboratory velocity V[ab(v ,K") [21] ... 

Figure 2-68 illustrates the basics of a transformer. First there is a core, usually constructed of a material of high magnetic permeability to achieve a high magnetic flux density. The core has two windings of conductors, a primary coil (designated as... 

In order to focus on the driving force for phase transitions induced by a magnetic field it is advantageous to use the magnetic flux density as an intensive variable. This can be achieved through what is called a Legendre transform [12], A transformed Helmholtz energy is defined as... 

For the purposes of considering diffusion at microelectrodes, it is convenient to introduce two categories of electrodes those to which diffusion occurs in a linear fashion and those to which diffusion occurs in a nonlinear fashion. The former category consists of cylindrical and spherical electrodes. As shown schematically in Figure 12.2A, the lines of flux (i.e., the pathway followed by material diffusing to the electrode) are straight, and the current density is the same at all points on the electrode. Thus, the diffusion problem is one-dimensional (i.e., distance from the electrode surface) and involves solution of the appropriate form of Fick s second law, Equation 12.7 or 12.8, either by Laplace transform methods or by digital simulation (Chap. 20). 

Such fluctuations of the photon flux, emitted from a molecule, have been predicted to be due to cooperative effects (21.22). The theory is based on an idea of Prigogine and coworkers (s. e.g.(22)) who treated the irreversible part of a physical process by transforming the wavefunctions of a dissipative system into another space using a "dynamical" non-unitary representation D = exp(-iVT /fI) with a "star-Hermitian" time operator 3 and V describing the interaction of a relevant local system Hq, e.g. the complex chromophore, and the total system H, i.e. our crystal. In the new representation y>=D Y no additional time dependence is introduced, dD/dt = 0, any expectation value of an operator M=DMD should be unchanged = M> and the total Hamiltonian is transformed by 1T=DHD 1 = Hq to the local system Hamiltonian (21.22). To describe the time development in the new representation, the electron density... 

Equation 3.13 fulfills our basic objective in describing average molecular motion. However, inasmuch as we cannot ordinarily measure directly the average velocity of molecules or ions undergoing transport, it is advantageous to transform 0 into the flux density /, a parameter that is more directly observable. The flux density of a component is the number of moles carried through a unit areain unit time. It is related to that component s mean molecular velocity 0 and concentration c by... 

The ultimate goal of a basic study of separations is to obtain a description of how component concentration pulses (zones or peaks) move around in relationship to one another. The flux density J tells how solute moves across boundaries into and out of regions, but it does not detail the ebb and flow of concentration. To do the latter we must transform J into a form that directly yields concentration changes. The procedure followed below for this is standard in many fields. It is followed, for example, in treatments of heat conduction and diffusion [14,15]. We shall continue to simplify our treatment to one dimension. 

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