Superposition model

As approaches the crack can also propagate by hydrogen embrittlement processes during the higher load parts of the stress cycle. This forms the basis of various models which have been developed to describe corrosion fatigue, probably the best-known of which are the superposition models due to Wei"". In its most recent version this model takes the form ... 

The effective CF models, intended to include covalence effects via effective charges and shielding parameters [46] (superposition model [47], effective charge model [48], simple overlap model [49, 50]), keep the radial (M-L distance) dependence of the CF parameters as in the simple (point charge) electrostatic model. Dedicated studies have shown, however, that the radial dependence of these parameters deviates strongly from the latter for the whole series of lanthanide ions [51, 52]. 

On the other hand, the estimate of the Breit interaction energy is, in all cases, quite satisfactory in any atomic or ionic superposition model. The reason for this has already been discussed the Breit interaction energy arises due to electron current density in the neighbourhood of the nuclei, which is dominated by the core electrons and is apparently insensitive to the valence electron environment. 

Walther, D., Eisenhaber, F., and Argos, P. (1996). Principles of helix-helix packing in proteins The helical lattice superposition model./. Mol. Biol. 255, 536-553. 

Superposition model 541 5.3.4. Energy transfer processes in semi- ... 

Fig. 7. Variation of the crystal-field parameters and the crystal-field strength of / OCl Pr3+ (R = La, Pr, Gd) under pressure. The dashed fines show a superposition model calculation of die crystal-field strength.

A further possibility to derive local distortions can be based on models like the superposition model, which explicitly relate the crystal-field parameters to structural parameters. This has been done for CaI Cl Sm2+ (Shen and Holzapfel, 1995b) and MFCl Sm2+ (M = Ba, Sr, Ca) (Shen and Holzapfel, 1996). These results will be discussed in the next section. 

The crystal-field parameters introduced in sect. 4.1 still contain all the structural information about the local environment. Therefore, a direct comparison of crystal-field parameters derived from different hosts, even with the same site symmetry, is not reasonable. In addition, the crystal-field parameters cannot be directly related to the distance and angle variations induced by the high-pressure application. Widely used models which extract the structural information from the crystal-field parameters are the angular-overlap (Jprgensen et al., 1963) and superposition model (Bradbury and Newman, 1967). In the case of f elements, the superposition model has been employed widely for the analysis of crystal-field parameters. 

The underlying assumption of the superposition model is that the one-electron crystal field is additive and can be regarded as a superposition of the contributions from individual ions... 

In contrast to the LaCb case, for ROC two different ligands must be taken into account. In this host the Pr3+ ion is surrounded by four O2- and five Cl" ions. The four O2- ions are located in a plane below the central ion at a distance Ro, four Cl- ions in a plane above the central ion at a distance Rcu and one Cl- ion above the Cl- plane at a distance RcY. Within the superposition model, the sum of the contributions from these ligands for the crystal-field parameters Bk can be written as ... 

According to the superposition model the intrinsic parameters found for SrFCl Sm2+ were simply transferred to either CaFCl Sm2+ or BaFCl Sm2+. Then, combining the intrinsic parameters with crystal-field parameters gained from optical high pressure studies on CaFCl Sm2+... 

On the other hand, a more practical approach is to analyze the various parameters with respect to their influence on the anomalous multiplets. According to this procedure Li and Reid (1990) could show that taking into account only the parameters G410Aq, a pronounced improvement for the description of the 2H(2)n/2 multiplet of Nd3+ could be achieved. To study the dependence of correlation effects on the interatomic distances, Jayasankar et al. (1993) analyzed spectroscopic data for LaCbiNd3"1" at pressures up to 10 GPa. Using the superposition model, they could derive the distance dependences of the intrinsic parameters B4 —4... 

In many cases the crystal structure of a rare-earth compound studied under high pressure is a priori known. In such studies the quality of the theoretical link between structure and spectra can be tested. However, a different possibility would be to use the experimentally determined spectral variations in connection with a theoretical approach to derive information about the local structure of the rare-earth ions. Such an attempt has been made in sect. 4.4.2, where the local distortions have been derived either directly from the spectra or by applying the superposition model. Similarly, high pressure studies have been used to get information about the structure in more complicated cases of multiple sites or glasses. In addition, the spectra of rare-earth ions have been used to detect phase transitions that often occur under pressure. Results of such studies will be discussed in the next two sections. 

Newman, D. J., Price, D. C. Runciman, W. A. (1978) Superposition model analysis of the near-infrared spectrum of Fe2+ in pyrope-almandine garnets. Amer. Mineral., 63, 1278-81. 

A further complication arises, because many glass formers exist exhibiting an excess wing in addition to a p-process, and it is by no means clear whether in those cases two intrinsic relaxation processes are present [6,142,152,230, 331,137,321]. In particular, for a relatively weak and well-separated p-process, it appears that the excess wing as well as a-peak still manifest themselves distinctly (i.e., the type A characteristics ), and a simple superposition model may apply for both processes [6,284], On the other hand, a strong p-process leads to a pronounced change of the a-peak, and then it is quite impossible to tell whether or not an additional excess wing is present. Furthermore, one may even have to face the fact that different JG processes may be probed in a different manner by the various methods. This, if true, would allow for a further distinction. 

Newman [31] proposed a model known as the superposition model which simplified the calculations of different contributions to the crystal field parameters and facilitated their analysis. According to the superposition model, the crystal fields are built up from separate... 

The intensities of the absorption bands of coordination compounds have been studied [61,112,82,178]. A superposition model has been applied to the analysis of the spectrum of Er3+ in YGG and LuGG and the best-fit parameters obtained are given in Tables 8.21 and 8.22. 

Intrinsic parameters A4 A (cm-1) obtained using the superposition model for various systems. Values are given corresponding to the mean ligand distance in each host [31],... 

The superposition model has also been applied to experimental crystal field parameters obtained for lanthanides [31] substituted into host lattices of oxides, zircons, anhydrous trihalides, oxysulphides, alkaline earth fluorides and some other cubic crystals. The intrinsic parameters obtained from the analysis are given in Table 8.23. The solution spectrum of Er3+ aquo ion is given in Fig. 8.29. 

Fig.33 Alignment and superposition models of 5-[l-aryl-l,4-dihydro-6-methylpyridazin-4-one-3-yl]-2-arylamino-l,3,4-thiadiazoles and corresponding oxadiazoles in CoMFA. (Reprinted with permission from [194]. Copyright 2002 American Chemical Society)...

All four models were used to fit the pyrolysis of wheat straw, but only the superposition model was used to fit the pyrolysis of washed wheat straw. Ail four models have the flaw that they arc not able to predict the char yield, which then has to be given as input to the models. Prediction of char yield can only be done by more complicated models. 

The pyrolysis of wheat straw could be expected to be described by a superposition model, i.e. described by a number of parallel first order reactions, representing the decomposition of the biomass constituents (cellulose, hemicellulose and lignin) [4,5]. In this work, a superposition model has been used, in which the pyrolysis is assumed to be described by N independent first order reactions (for i=l, 2,..., N) ... 

Fig. 8 The experimental data for the pyrolysis of wheat straw at 10°C/min to 600 C ( 1) and to 300 C for 60 min before 600 C ( 2). Model data fitted to 1 and used to model 2 using the kinetics derived from the superposition model with three reactions (based on mass loss rate data).

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