Strength function

The theory is concerned with the problem of determining the probability of failure of a part which is subjected to a loading stress, L, and which has a strength, S. It is assumed that both L and S are random variables with known PDFs, represented by f S) and f L) (Disney et al., 1968). The probability of failure, and hence the reliability, can then be estimated as the area of interference between these stress and strength functions (Murty and Naikan, 1997). 

We present an illustration based on the reaction Cu+ and Co(NH3)5Cl2+. The rate constants for this reaction were determined over the ionic strength range 9.4 X 10 4-3.00 M. They are given in Table 9-6. Figure 9-2 depicts the variation of log k with the ionic strength function. 

The paper is organized as follows. In Section 2, derivation of the the SRPA formalism is done. Relations of SRPA with other alternative approaches are commented. In Sec. 3, the method to calculate SRPA strength function (counterpart of the linear response theory) is outlined. In Section 4, the particular SRPA versions for the electronic Kohn-Sham and nuclear Skyrme functionals are specified and the origin and role of time-odd currents in functionals are scrutinized. In Sec. 5, the practical SRPA realization is discussed. Some examples demonstrating accuracy of the method in atomic clusters and nuclei are presented. The summary is done in Sec. 6. In Appendix A, densities and currents for Skyrme functional are listed. In Appendix B, the optimal ways to calculate SRPA basic values are discussed. 

In study of response of a system to external fields, we are usually interested in the average strength function instead of the responses of particular RPA states. For example, giant resonances in heavy nuclei are formed by thousands of RPA states whose contributions in any case cannot be distinguished experimentally. In this case, it is reasonable to consider the averaged response described by the strength function. Besides, the calculation of the strength function is usually much easier. 

For electric external fields of multipolarity EXjj, the strength function can be defined as... 

It is worth noting that, unlike the standard definition of the strength function with using 5 uj — Uv), we exploit here the Lorentz weight. It is very convenient to simulate various smoothing effects. 

The explicite expression for (49) can be obtained by using the Cauchy residue theorem. For this aim, the strength function is recasted as a sum of V residues for the poles z = Since the sum of all the residues (covering all the poles) is zero, the residues with z = Eojp (whose calculation is time consuming) can be replaced by the sum of residies with z = u) E f(A/2) and z = Esph whose calculation is much less expensive (see details of the derivation in [8]). 

The first term in (53) is contribution of the residual two-body interaction while the second term is the unperturbed (purely Iph) strength function. Further, F z) is determinant of the RPA symmetric matrix (35) of the rank 2K, where K is the number of the initial operators Qk- The symmetric matrix B of the rank 2K + 1) is defined as... 

Figure 1. Photoabsorption cross section for the dipole plasmon in axially deformed sodium clusters, normalized to the number of valence electrons N - The parameters of quadrupole and hexadecapole deformations are given in boxes. The experimental data [39] (triangles) are compared with SRPA results given as bars for RPA states and as the strength function (49) smoothed by the Lorentz weight with A = 0.25 eV. Contribntions to the strength function from p =0 and 1 dipole modes (the latter has twice larger strength) are exhibited by dashed curves. The bars are given in eVA. ...

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

First one has to assume that the poles kn lie very densely on the real axis and that the vn vary smoothly from one kn to the next. Then it makes sense to define a strength function g(x) by setting, as in (III.3.3),... 

Whether or not such a tc exists depends on the nature of the bath. It is certainly necessary that the bath has a continuous, or at least very dense, spectrum with eigenvalues that contribute equally to the interaction in order to have a smooth strength function g(k). Also this function must be virtually constant over the range over which the interaction is felt, i.e., the line width, as seen in 2. It would be desirable to have more concrete criteria, but it is hard to formulate them. 

The dipole response in real time gives access to the response in frequency domain by Fourier transfrom D (a)), from which one can extract the strength function S(n>) = cA b yf and the power spectrum P( ) = I)(a/) 2. The strength function is the more suited quantity in the linear regime, where it can be related to the photoabsorption cross section [31], while the power spectrum better applies for spectral analysis in the non linear regime [24],... 

Many of the applications, particularly the more challenging ones, of fission-product decay data have involved the fission products as a group. Because of their large number (>700) and the fact that almost no data have been available for many of them, it has always been necessary for such applications to utilize calculated values for the unmeasured quantities. To do this, a number of different approaches have been taken to describe the p-decay properties (p-strength functions) of these nuclides [TAK72,... 

MAN82, KLA83]. The most comprehensive of these approaches has been that of Klapdor and his co-workers [KLA83] who, using a specific model for the p-strength function, provide predictions for a wide variety of fission-product-related phenomena in astrophysics and nuclear physics. 

Conventional decay-schemes studies do not seem appropriate, because of the complexity of the decay schemes, the errors to which they are subject (cf. the discussion of 8 Br above) and the large amount of time needed to carry them out. Direct measurement of the p-strength functions themselves, utilizing total-absorption y spectrometry and, where relevant, delayed-neutron-gamma coincidence techniques, promises to provide a means of producing the necessary information in a reasonable time. 

Role of Discrete Levels in Deriving Absolute Dipole Strength Functions... 

Figure 8. 175Lu(n,y) excitation function, calculated using our original dipole strength function sys-tematics, compared with experimental data from [MAC78,BEE80]. 

Figure 10 Recently derived absolute dipole strength functions for 176Lu (solid curves), compared with those predicted by our original sys-tematics (dashed curves). The arrows indicate the value assigned to the one free parameter, Ex. In our recent derivation, Ex = 11 MeV, and in our original systematics, Ex = 5 MeV.

Figure 11. El strength functions for 89Y and Zr, calculated using our revised systematics, compared with measurements from [AXE70, SZE79].

Examples of large-basis shell-model calculations of Gamow-Teller 6-decay properties of specific interest in the astrophysical s-and r- processes are presented. Numerical results are given for i) the GT-matrix elements for the excited state decays of the unstable s-process nucleus "Tc and ii) the GT-strength function for the neutron-rich nucleus 130Cd, which lies on the r-process path. The results are discussed in conjunction with the astrophysics problems. 

We have again utilized the Lanczos algorithm to compute the B strength function for the l3°Cd(0+) l30In(1 + ) Gamow-Teller transitions. Low-... 

Fig. 2. Calculated GT-strength function for the 130Cd ground-state (ON decays to the low-lying 1+ (T=16) states in 130In as a sequence of the number of Lanczos iterations. The abscissa is the excitation energy in 130In. For convenience of drawing, the minimum width is chosen to be 100 keV.

KRU81]). The B strength function for nuclei along the decay back paths [coupled with neutron separation energies (Sn), fission barrier heights (Bf) and B"decay Q-values (Qg)] determines the amount of B delayed fission and neutron emission that occurs during the cascade back to the B stability line. 

The /3-decay strength function is proportional to the square of the matrix elements of /3-operator between final and initial states. So is B and is defined by ... 

Random Phase Approximation Calculations of Gamow-Teller /3-Strength Functions in the A = 80-100 Region with Woods-Saxon Wave Functions... 

We discuss some features of a model for calculation of p-strength functions, in particular some recent improvements. An essential feature of the model is that it takes the microscopic structure of the nucleus into account. The initial version of the model used Nilsson model wave functions as the starting point for determining the wave functions of the mother and daughter nuclei, and added a pairing interaction treated in the BCS approximation and a residual GT interaction treated in the RPA-approximation. We have developed a version of the code that uses Woods-Saxon wave functions as input. We have also improved the treatment of the odd-A Av=0 transitions, so that the singularities that occured in the old theory are now avoided. 

The calculation of the p-strength function involves evaluating the matrix element of the p transition operator between the initial wave function t i. of... 

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