Polynomial Background Research Articles

MCGR: An inverse method for deriving the pair correlation function

Abstract MCGR (Monte Carlo determination of G(r)) is an inverse method for determining the pair correlation function from the structure factor. In this paper we describe the algorithm in detail, including recent developments such as the possible application of coordination constraints, Gaussian smoothing of peak shapes and the ability to subtract a (quadratic) polynomial background. We will illustrate how these have important practical implications for the way that experiments can be performed. Examples of the applications of MCGR to liquids/glasses and crystalline materials will be given.

Phenomenological πN→ππN amplitude and analysis of low energy data on total cross sections and 1--D distributions

We develop the phenomenological amplitude of the $\pi N \to \pi \pi N$ reaction describing the exchanges of $\Delta$ and $N_{*}$ along with the OPE mechanism. The contribution of the latter contains 4 independent low energy parameters (up to $O(k^{4})$ order). The terms of the polynomial background are added to stand for far resonances and for contact terms originating from the off-mass-shell interactions. These terms are introduced with the account of isotopic, crossing, $ C $, $ P $ and $ T $ symmetries of strong interactions. The data consisting of total cross sections in the energy region $ 0.300 \le P_{Lab} \le 500 $~MeV/c and 1D distributions from the bubble-chamber experiments for three reaction channels were undergoing fittings to determine free parameters of the amplitude. The best solutions are characterized by $ \chi^{2}_{DF} = $ 1.16. At the considered energies the isobar exchanges are found to be more important than OPE. The obtained solutions reveal the need in more precise data and/or in polarization measurements because of large correlations of isobar parameters with the OPE ones. The theoretical solutions were used for modeling the Chew-Low extrapolation and the Olsson-Turner threshold approach. It is shown that the noncritical application of the former results in 100% theoretical errors, the extracted values being in fact the random numbers. The results of the Olsson-Turner method are characterised by significant systematic errors coming from unknown details of isobar physics.

Advanced, enhanced HEX program for PIXE

The REX code and subsequent HEX code, originating at Florida State University, have been extensively used for PIXE spectra fitting. In 1989 we produced a renovated HEX package: WITS-HEX, enabling the original Fortran program to be more accessible to the PIXE community. We modernised the user interface by replacing the batch mode of operation with an integrated, menu-driven environment. We added the ability to edit support data files from within the program, provided detailed feedback during the fitting process and enhanced spectral plots using high resolution colour graphics. Our prototype also permitted the inclusion of many more peaks and absorption coefficients into the element library than the original HEX, permitting a more extensive element request list to be used during the fitting operation. We have now completed the second phase of the renewal of HEX. The man-machine interface has been upgraded to conform to the IBM SAA Common User Access (CUA) standard. This eliminated several of the sequential (modal) human-computer dialogues, replacing them with a single parallel system. The support utility used in WITS-HEX to convert the binary format of spectra captured using foreign data acquisition systems has been replaced by code to directly access data in ASCII format. The program is now equipped with context-sensitive help and a tutorial. The polynomial background model has been supplemented by a digital filter model, eliminating the associated instability from the fitting process and other spectral features modelled. The program has been validated by comparing results with those obtained from the former versions: WITS-HEX and HEX. A demonstration version is available on request for evaluation purposes.

Gamow-Teller strength in the 208Pb(p,n)208Bi reaction at 134.3 MeV.

The excitation-energy distribution of transition strength to ${1}^{+}$ states was measured for the $^{208}\mathrm{Pb}$(p,n${)}^{208}$Bi reaction at 134.3 MeV for excitation energies up to 38 MeV. Structures observed in neutron time-of-flight spectra with forward-peaked (\ensuremath{\Delta}L=0) angular distributions were identified as ${1}^{+}$ states, except for the transition to the ${0}^{+}$ isobaric analog state. The ${1}^{+}$ strength in these structures was extracted by normalizing the yield above a fitted polynomial background to the Fermi transition strength localized in the isobaric analog state. The Gamow-Teller strength observed in the ${1}^{+}$ peaks is 56% of the 3(N-Z) sum rule when the strength of the ${\ensuremath{\beta}}^{+}$ transitions is assumed to be zero; 45% of 3(N-Z) is observed in the giant resonance and 10% is observed in structures below the giant resonance. Based on a multipole decomposition analysis, an upper limit on the ${1}^{+}$ strength in the apparent continuum to 38 MeV of excitation energy is estimated to be 37% of 3(N-Z). These results are compared with predictions from a shell model that includes a pairing force and a long-range Gamow-Teller force in both the parent and daughter nuclei.

Low-lying structures in the Gamow-Teller strength functions for the double-beta-decaying nuclei 76Ge, 82Se, 128Te, and 130Te.

Excitation-energy distributions of transition strength to ${1}^{+}$ states excited via the (p,n) reaction at 134.4 MeV on targets of $^{76}\mathrm{Ge}$, $^{82}\mathrm{Se}$, $^{128}\mathrm{Te}$, and $^{130}\mathrm{Te}$ were measured for excitation energies up to 25 MeV. Structures observed in the neutron spectra with forward-peaked (\ensuremath{\Delta}L=0) angular distributions were identified as ${1}^{+}$ states, except for the isobaric analog transitions. The total ${1}^{+}$ strength in these reactions was extracted by normalizing the intensity in the ${1}^{+}$ peaks to the Fermi transition strength observed in the isobaric analog state. The Gamow-Teller strength observed in ${1}^{+}$ peaks above a fitted polynomial background is typically 55% of the sum rule obtained by assuming that the strength of ${\ensuremath{\beta}}^{+}$ transitions is negligible. The portion of this strength found at excitation energies less than that of the Gamow-Teller giant resonance varied from 15% for $^{128}\mathrm{Te}$ to 38% for $^{76}\mathrm{Ge}$. Experimental results are compared with predictions of a shell model that includes a pairing force and a long-range Gamow-Teller force in both parent and daughter nuclei. A comparison of the strength functions of the tellurium isotopes is made; this comparison is relevant in determining whether double-beta decay without neutrino emission (0\ensuremath{\nu} decay) is observed in these isotopes.

The Los Alamos PIXE data reduction software

A fortran computer code has been written to perform standard-free analysis of PIXE spectra. The analysis is broken into three components: relative X-ray intensity calculation, detector calibration, and spectrum deconvolution. Nonlinear portions of the problem are limited to detector calibration. Actual spectrum deconvolution is performed as a linear least squares problem with linear inequality constraints on element concentrations and background. Up to 9 K and 15 L lines of 85 elements are available for spectrum fitting. The current model consists of Gaussian peaks and a polynomial background. Development work is underway to add low energy tails to the peak model and absorption edges to the background.

Analytical approximation of the gamma ray spectra from germanium detectors

Several functional forms for shape function are tested using data obtained by a Ge(Li) detector. A function built from a Gaussian, a step function and a symmetric or asymmetric tailing on a linear or polynomial background is needed. A new shape function is proposed and superior fit are obtained.

An analytic approximation to the peaks obtained from Ge(Li) and Si(Li) detectors used in gamma spectrum analysis

A new approach for the analytical approximation to the peaks obtained from Ge(Li) and Si(Li) detectors is described. The form of the analytical function involves three components viz., Gaussian, distorting and background functions. For the representation of the distorting effects a different analytical form is proposed which is added to the pure Gaussian and polynomial background. The validity of the form proposed is investigated by means of the program CAASA developed for high precision gamma spectroscopy. The improvement achieved by this approach is compared with some results reported and published earlier.

Time-resolved studies on contracting muscle using small-angle X-ray diffraction. I. Design of data collection system

Small-angle X-ray diffraction offers a unique method for the study of muscle structure at a molecular level. Structural changes in muscle during contraction can be studied with a 10 ms time resolution with the use of a powerful X-ray generator and electronic detectors. A data-collection system is described which is based on a linear position-sensitive detector capable of synchronizing data from 32 different phases of a muscle contraction cycle and storing it in different subgroups of a multichannel analyzer memory. On completion of an experiment with a muscle the complete data set is transferred via a fast link to a PDP 11/10 computer for processing and storage. A promising approach to the automated analysis of equatorial data is to assume that the pattern is a sum of a polynomial background and Gaussian peaks. A χ2-minimization procedure, which optimizes the parameters used to describe the pattern, gives excellent agreement between experimental data and the proposed model. Finally a detector system is proposed which would allow collection of data at high rates expected from a camera set on a storage ring.

Crystallinity and crystallite size measurement in polyamide and polyester fibres

X-ray diffraction methods for evaluating crystallinity and crystallite size in fibres generally neglect the possibility of peak overlap, and separate peaks and background by arbitrary procedures. The method described here is based on the resolution of normalized diffraction peaks in terms of combined Gaussian-Cauchy profiles for each peak, together with a polynomial background. Peak area crystallinity is then measured as the total area under the resolved peaks over a defined range. Measurements of apparent crystallite size are obtained from the widths of the resolved peaks. Application of this method to high tenacity nylon-6, nylon-6,6, and PET fibres after annealing, indicates that peak area crystallinity is about 100% for the polyamides and 85% for the polyester. The increase in apparent crystallite size is considered to be due to both an increase in the true mean size and to improvements in local lattice order.

Peak resolution and X-ray crystallinity determination in heat-treated cellulose triacetate

X-ray diffraction traces of cellulose triacetate fibres heat-treated in the range 220 – 290°C were resolved by a computational method which fits Gaussian or Cauchy functions to the peak profiles. A polynomial background is fitted at the same time so that the total area formed by the resolved peaks is a parameter for estimating crystallinity and the area under the background is a parameter for estimating the non-crystalline or disordered component. This measure of actual crystallinity is shown to be dependent upon the function chosen for the peak profile. After appropriate corrections, crystallite widths were obtained and compared with fibril widths measured on electron micrographs. Cauchy resolution data overestimate crystallite size whilst Gaussian resolution data underestimate it. In all cases the crystallites from fibres annealed at 290°C have greatly increased widths compared with crystallites from fibres annealed at 220°C or 250°C.

Effect of spectrum smoothing on peak area determinations

The effect of a smoothing by digital filter on the determination of spectral peak areas has been studied quantitatively by means of the numerical experiments with a computer. Gaussian shape filters were employed. For the method of Covell's partial peak area determination, the least-squares analysis using a standard peak, and the subtraction of polynomial background, statistical and systematic errors are comparatively observed especially in the case of a single small peak on the curved background.

The resolution of multipeak data in fibre science

A computer program based on a procedure discussed by Powell has been developed for the resolution of overlapping peaks in data output from x-ray diffraction, ion-exchange chromatography with spectroscopic detection, and infrared spectroscopy. A polynomial background is also fitted to the data so that recourse to arbitrary graphical methods for separating peaks and background is no longer necessary.

Numerical Approach to the Low-Energyπ−πBootstrap

The $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ amplitude in the low-energy region is parametrized in a crossing-symmetric way as the sum of the $\ensuremath{\rho}$ and ${f}^{0}$ resonance poles plus a polynomial background. The parametrization is flexible and capable of producing amplitudes having quite different features in the energy region below 1 GeV. The parameters are then varied so as to minimize the deviation from elastic unitarity on a set of closely spaced points. In addition, negative-moment finite-energy sum rules are used to connect the low-energy region with assumed Regge asymptotic behavior in the $I=1, 2$ amplitudes. With the mass and width of the ${f}^{0}$, the mass of the $\ensuremath{\rho}$, and the slope of the $\ensuremath{\rho}$ trajectory fixed, an approximate solution satisfying the constraints is found, yielding a $\ensuremath{\rho}$ width of 80\ifmmode\pm\else\textpm\fi{} 30 MeV. The solution displays the usual characteristics of the low-energy $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ amplitudes suggested by other analyses, namely, small scattering lengths and a large $I=0$ $S$-wave phase shift near the mass of the $\ensuremath{\rho}$. This resonantlike behavior is found without introducing an $S$-wave pole in the parametrization, while the small scattering lengths are obtained as results of the numerical bootstrap, although no current-algebra constraints are included. Our proposed solution is also found to satisfy various inequalities proposed by Martin for the ${\ensuremath{\pi}}^{0}\ensuremath{-}{\ensuremath{\pi}}^{0}$ scattering amplitude.