Scattering Power Invariant

Mathematically spoken k is the zero-dimensional projection / 0 of the scattering intensity. After calibration to absolute units 7(s) turns into 7(s) /V - its scattering power is known as Porod s invariant... 

Figure 8.15. The invariant as a function of the composition of a two-phase material. Between 30 and 70 vol.-% the scattering power is almost constant. The regions 0-30 vol.-% and 70 -100 vol.-% exhibit almost linear relations... 

If we know that in our two-phase material the volume fraction of one fraction is between 30 and 70%, then the term v (1 - v) 0.23 is constant to a first approximation (cf. Fig. 8.15). If in this case during the experiment a considerable change of the invariant is observed, it is probably caused by a variation of the contrast67. If, on the other hand, the contrast is known to be constant and nanostructure is evolving from a homogeneous phase, the initial increase of the scattering power is proportional to the change of the materials composition. 

For many years it was believed that hydrogen atoms could not be seen" in the electron density maps produced by X-ray diffraction. The reason for this is that the atomic X-ray scattering power is proportional to the square of the atomic number. This statement was generally, but not invariably, true until the demise of film-recording methods in the mid-1960 s and the advent of the computer-controlled X-ray diffractometers which could provide very accurate X-ray diffraction intensities. 

The invariant Q was defined, in Section 1.5.4, as the quantity that represents the total scattering power of the sample, and it can be evaluated by integrating the observed intensity I(q) over the whole reciprocal space, as indicated by Equation (1.85), or in the case of an isotropic material by... 

Projections are integral operators which map functions onto subspaces of their definition domain. They shall be denoted by a pair of curly parentheses. The best known projection in the field of scattering theory maps the scattering intensity I ( ) onto a zero-dimensional subspace, which is the number Q, known as "invariant or scattering power ... 

Fourier transformation and the scattering power (or invariant) Q is defined... 

Uncoupled Rate Constants. An initial evaluation of polymerization kinetics is presented in Figure (2), constrained by viscosity invariant rate constants K. The slopes of these straight lines give initial estimates of Rgg/Kp according to Equation (14). Figure 3 presents graphically a power law relationship between K g/Kp and viscosity at 21°C and at 16.6 C. More scatter In Yu s data may be attributed to the use of an older GPC instrument of relatively low resolution. The ratio Kgq/Kp is temperature-sensitive a change of the order or five times is observed if the temperature is reduced by 4.4°C and viscosity is kept constant. 

As we have already discussed, a lossless FDN results when the feedback matrix is chosen to be unitary. Smith and Rocchesso have shown that the waveguide interpretation leads to a more general class of lossless scattering matrices [Smith and Rocchesso, 1994], This is due to the fact that each waveguide may have a different characteristic admittance. A scattering matrix is lossless if and only if the active complex power is scattering-invariant, i.e., if and only if... 

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