Density dependent modelling

Figure 6. MIT density dependent model applied to both up (over metal) and down (between metal) oxide thickness resulting from oxide CMP ...

Fig. 8 Schematic real-space model and normalized electron density profiles < p(z)>/pix> (where p is the bulk electron density of mercury) obtained from the fits of reflectivity data to a density-dependent model for n-octadecanethiol (bold line) and n-dodecanethiol (thin line). The upper and lower figures are aligned with each other. Vertical lines in the model mark the position of the three outermost surface layers of mercury, with the origin of z coinciding with the first mercury layer [89]. 

IV. AN ANALYTICAL, ENERGY- AND LOCAL-DENSITY-DEPENDENT MODEL FOR SELF-ENERGY... 

Fig. 48. Neutron-proK>n density ims radii differences for "Ca deduced using the NR density dependent model (solid dots) and the NRIA model (crosses), from ref. [Ra90]. Correlations are included in both sets of analyses. The NRIA values ate off-set for clarity.

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

Table 6.2 summarizes the low pressure intercept of observed shock-velocity versus particle-velocity relations for a number of powder samples as a function of initial relative density. The characteristic response of an unusually low wavespeed is universally observed, and is in agreement with considerations of Herrmann s P-a model [69H02] for compression of porous solids. Fits to data of porous iron are shown in Fig. 6.4. The first order features of wave-speed are controlled by density, not material. This material-independent, density-dependent behavior is an extremely important feature of highly porous materials. 

In both models the rotational shift of the line 5co is the same either in the static limit, where it is equal to zero, or in the case of extreme narrowing where it reaches its maximum value coq. A slight difference in its dependence on tE is observed in the intermediate region only. The experimentally observed density dependence of the shift shown in Fig. 3.5 is in qualitative agreement with theory. 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

Frame, K.K. and W.S. Hu, "A Model for Density-Dependent Growth of Anchorage Dependent Mammalian Cells", Biotechnol. Bioeng., 32, 1061-1066 (1988). 

Fig. 9.8. Trend of beryllium abundance with metallicity compared to predictions from two models (a) CRS denotes cosmic-ray acceleration in superbubbles rich in iron and oxygen as predicted from theoretical supernova yields (in this case those of Tsujimoto and Shigeyama 1998) and (b) CRI denoting cosmic rays accelerated from the general interstellar medium. The density dependence comes from its influence on the delay in the deposition of the synthesized Be. Virtually identical results were obtained using the yields from Woosley and Weaver (1995). After Ramaty et al. (2000).

To summarize the present situation in Fig. 4 the resulting density dependence of the SE for the approaches discussed above are compared (excluding the 3NF contribution). One sees that the covariant models predict a much larger increase of the SE with the density than the non-relativistic approaches. The lowest-order BHF method predicts a somewhat higher value for 04 than both the VCS and SCGF methods, which lead to very similar results whether that can be ascribed to a consistent treatment of correlations in these methods, or is fortuitous, is not clear. 

Subsequently Furnstahl [20] in a more extensive study pointed out that within the framework of mean field models (both non-relativistic Skyrme as well as relativistic models) there exists an almost linear empirical correlation between theoretical predictions for both 04 and its density dependence, po, and the neutron skin, All lln — Rp, in heavy nuclei. This is illustrated for 208Pb in Fig. 5 (from ref.[20] a similar correlation is found between All and po). Note that whereas the Skyrme results cover a wide range of All values the RMF predictions in general lead to AR > 0.20 fm. 

In principle the density dependence of the SE at higher densities (and further away from N = Z) can be probed by means of heavy-ion reactions using neutron rich radioactive beams. In ref. [35] possible observable effects from the isovector field are considered in terms of the RMF model. Of particular... 

One important condition is constituted by the fact that certainly in symmetric nuclear matter no phase transition is observed below 3po- In fact some theoretical interpretation of the heavy ion experiments performed at the CERN SPS [30] points to a possible phase transition at a critical density pc ss 6po ss 1/fm3. We will in the following take this value for granted and use an extended MIT bag model [31] (requiring a density dependent bag constant ) that is compatible with this condition. 

In the original MIT bag model the bag constant B 55 MeV fm-3 is used, while values B 210 MeV fm-3 are estimated from lattice calculations [34], In this sense B can be considered as a free parameter. We found, however, that a bag model involving a constant (density independent) bag parameter B, combined with our BHF hadronic EOS, will not yield the required phase transition in symmetric matter at pr 6po 1/fm3 [28]. This can only be accomplished by introducing a density dependence of the bag parameter. (The dependence on asymmetry is neglected at the current level of investigation). In practice we use a Gaussian parameterization,... 

Concerning the quark matter EOS, we found that a density dependent bag parameter B p) is necessary in order to be compatible with the CERN-SPS findings on the phase transition from hadronic to quark matter. Joining the corresponding EOS with the baryonic one, maximum masses of about 1.6 M are reached, in line with other recent calculations of neutron star properties employing various phenomenological RMF nuclear EOS together with either effective mass bag model [39] or Nambu-Jona-Lasinio model [40] EOS for quark matter. 

The method is based on the model-free parameterization of the spectral density. Depending on the number of available experimental parameters, up to 7 different models for SD could be considered, listed in Tab. 12.1 [9]. 

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