Calculation Of Solid Angle Research Articles

Developments in the calculation of average solid angle subtended by detectors to sources at any position and orientation, by a Monte Carlo method

This software development uses a Monte Carlo method to calculate the average solid angle subtended by detectors to sources, ranging in shape, at any position and orientation to each other. The average solid angle is required in nuclear science and engineering to calculate the fraction of radiation emitted from a source that is incident on the detector. The software takes advantage of developments in pseudo random number generators, parallel processing and 64 bit computer architecture to achieve accuracies of 0.01%, and feasibly 0.001% in reasonable time with higher core number workstations. The practical limitations of Monte Carlo modelling are discussed. The software also allows visualisation of the detector–source geometry and the simulated emissions.

Efficiency of a cubic NaI(Tl) detector with rectangular cavity using standard radioactive point sources placed at non-axial position

Low cost scintillation detectors as compared with HPGe detectors are considered to be one of most important radiation detection tools. Therefore, these detectors can be manufactured in different shapes and work at room temperature without any cooling systems, which added an extra advantage to it. This work presents a study of a cubic detector with a rectangular cavity in different experimental setup geometries, using standard point-like gamma-ray sources, where the efficiency of the detector in these geometries was the target to be studied. According to this aim, the data from the experimental measurements was used to determine the detector efficiency. An analytical calculation of the detector efficiency was done by using a new mathematical expression, this mathematical expression depends on the efficiency transfer technique and effective solid angle calculations. To support the mathematical model, the source-to-detector arrangement was simulated by Geant4 Monte Carlo code. All the compared efficiency results were found to be promising and trusted based on the calculated deviation percentages.

Steric and electronic evaluations of P(o-tol)R2, where R is phenyl or cyclohexyl: crystal structures of SeP(o-tol)R2.

The crystal structures of SeP(o-tol)R2, where o-tol is ortho-tolyl (2-methylphenyl) and R is Ph (phenyl), namely (2-methylphenyl)diphenylphosphane selenide, C19H17PSe, or Cy (cyclohexyl), namely dicyclohexyl(2-methylphenyl)phosphane selenide, C19H29PSe, were determined to aid in the evaluation of the steric and electronic behaviour of these analogous phosphane compounds. The compounds crystallized in similar monoclinic crystal systems, but are differentiated in their unit cells by a doubling of the number of independent molecules for R= Cy (Z'= 2) and the choice of glide plane by convention. The preferred orientation for the o-tolyl substituent obtained from the X-ray structural analysis is gauche for R= Ph and anti for R= Cy (using the Se-P-Cipso-Cortho torsion angles as reference). Density functional theory (DFT) calculations showed both conformations to be equally probable and indicate that the preferred solid-state conformer is probably due to the minimization of repulsion energies, resulting in a packing arrangement primarily featuring weak C-H...Se interactions and additional C-H...π interactions in the R= Ph structure. A detailed electronic and steric analysis was conducted on both phosphanes using Se-P bond lengths, multinuclear NMR 1JSe-P coupling constants, theoretical topological evaluation and crystallographic and solid-angle calculations, and compared to selected literature examples. The results indicate that the use of the o-tolyl substituent increases both the electron-donating capability and the steric size, but is also dependent on whether the o-tolyl group adopts a gauche or anti conformation. The single-crystal geometrical data are unable to detect electronic differences between these two structures due to the somewhat large displacement parameters observed for the Se atom in the R= Cy structure.

Analytical calculation of the solid angle subtended by an arbitrarily positioned ellipsoid to a point source

We present a geometric method for computing an ellipse that subtends the same solid-angle domain as an arbitrarily positioned ellipsoid. With this method we can extend existing analytical solid-angle calculations of ellipses to ellipsoids. Our idea consists of applying a linear transformation on the ellipsoid such that it is transformed into a sphere from which a disk that covers the same solid-angle domain can be computed. We demonstrate that by applying the inverse linear transformation on this disk we obtain an ellipse that subtends the same solid-angle domain as the ellipsoid. We provide a MATLAB implementation of our algorithm and we validate it numerically.

Design of an alpha-particle counting system at a defined solid angle at Turkish atomic energy authority-Sarayköy nuclear research and training center (TAEK-SANAEM)

Abstract The design details of an alpha-particle counting set-up at a defined solid angle (ACS-DSA) constructed in Radionuclide Metrology Department at TAEK-SANAEM for use in the primary standardization of radioactive solutions and determination of nuclear decay data of alpha-particle emitters is presented. The counting system is designed such that the solid angle is very well-defined and directly traceable to the national standards. The design involves mechanical construction of different parts like the source chamber, various coaxial flanges, and circular diaphragms in front of the passivated implanted planar silicon (PIPS®) detector, distance tubes, a digital caliper and a sliding piston to allow for different measurement configurations. All geometric configurations are easily changeable and characterisable with high accuracy which facilitates the solid angle calculation. A mixed alpha source was counted to check performance of assembled ACS-DSA system and good energy resolution and low peak tailing in the alpha energy spectrum was observed for small diaphragm apertures and far source-to-detector geometries.

The uncertainty of counting at a defined solid angle

Specific uncertainty components of counting at a defined solid angle are discussed. It is potentially an extremely accurate technique for primary standardisation of activity of alpha emitters and low-energy x-ray emitters. Owing to its reproducibility, it is very well suited for half-life measurements. Considered sources of uncertainty are 1) source–detector geometry, 2) solid-angle calculation, 3) energy loss and self-absorption, 4) scattering, 5) detection efficiency. Other sources of uncertainty, such as source weighing, counting, dead time and decay data are common to other standardisation methods. Statistical uncertainty propagation formulas are presented for the solid angle subtended by a circular detector to radioactive sources. Computer simulations were performed to investigate aspects of particle scattering.

Development of a software package for solid-angle calculations using the Monte Carlo method

Solid-angle calculations play an important role in the absolute calibration of radioactivity measurement systems and in the determination of the activity of radioactive sources, which are often complicated. In the present paper, a software package is developed to provide a convenient tool for solid-angle calculations in nuclear physics. The proposed software calculates solid angles using the Monte Carlo method, in which a new type of variance reduction technique was integrated. The package, developed under the environment of Microsoft Foundation Classes (MFC) in Microsoft Visual C++, has a graphical user interface, in which, the visualization function is integrated in conjunction with OpenGL. One advantage of the proposed software package is that it can calculate the solid angle subtended by a detector with different geometric shapes (e.g., cylinder, square prism, regular triangular prism or regular hexagonal prism) to a point, circular or cylindrical source without any difficulty. The results obtained from the proposed software package were compared with those obtained from previous studies and calculated using Geant4. It shows that the proposed software package can produce accurate solid-angle values with a greater computation speed than Geant4.

Electron energy loss spectrometers: An advanced operation mode for the lens system and the quantitative calculation of solid angle and transmission

A new operation mode of the lens system in electron energy loss spectrometers is proposed which brings the cardinal plane of the lenses closer to the sample. Experiments as well as electron optical calculations show an intensity gain by a factor of 3–4. The required voltages on the lens elements are calculated as function of the impact energy at the sample. Furthermore, the lens system is characterized with respect to the solid angle probed by the spectrometer and the transmission which enables the evaluation of spectral intensities in terms of the absolute scattering probability for elastic and inelastic scattering. The results apply to most existing electron energy loss spectrometers as they stem from the same design and use identical lens systems.

High efficiency microfluidic beta detector for pharmacokinetic studies in small animals

Abstract New radiotracers are continuously being developed to improve diagnostic efficiency using Single Photon Emission Computed Tomography (SPECT) or Positron Emission Tomography (PET). The characterization of their pharmacokinetics requires blood radioactivity monitoring over time during the scan and is very challenging in small animals because of the low volume of blood available. In this work, a prototype microfluidic blood counter made of a microchannel atop a silicon substrate containing PIN photodiodes is proposed to improve beta detection efficiency in a small volume by eliminating unnecessary interfaces between fluid and detector. A flat rectangular-shaped epoxy channel, 36 μm×1.26 mm cross section and 31.5 mm in length, was microfabricated over a die containing an array of 2×2 mm 2 PIN photodiodes, leaving only a few micrometers of epoxy floor layer between the fluid and the photodiode sensitive surface. This geometry leads to a quasi 2D source, optimizing geometrical detection efficiency that was estimated at 41% using solid angle calculation. CV – IV measurements were made at each fabrication step to confirm that the microchannel components had no significant effects on the diodes’ electrical characteristics. The chip was wire-bonded to a PCB and connected to charge sensitive preamplifier and amplifier modules for pulse shaping. Energy spectra recorded for different isotopes showed continuous beta distribution for PET isotopes and monoenergetic conversion electron peaks for 99m Tc. Absolute sensitivity was determined for the most popular PET and SPECT radioisotopes and ranged from 26% to 33% for PET tracers ( 18 F, 13 N, 11 C, 68 Ga) and more than 2% for 99m Tc. Input functions were successfully simulated with 18 F, confirming the setup’s suitability for pharmacokinetic modeling of PET and SPECT radiotracers in animal experiments. By using standard materials and procedures, the fabrication process is well suited to on-chip microfluidic functionality, allowing full characterization of new radiotracers.

ANGLE v2.1—New version of the computer code for semiconductor detector gamma-efficiency calculations

New version of the commercially available ANGLE software for semiconductor detector gamma-efficiency calculations is presented. ANGLE allows for accurate determination of the activities of gamma spectroscopic samples for which no “replicate” standard exists, in terms of geometry and matrix. A semi-empirical (“efficiency transfer”) approach is applied, based on the effective solid angle calculations. Advantages of both absolute (Monte Carlo) and relative (calibrated-source-based) methods are combined—while minimizing potential for systematic errors in the former and reducing practical limitations of the latter. ANGLE is broadly applicable, accounting for most of counting arrangements in gamma-spectrometry practice (in respect to detector types and configuration, source shapes and volumes, matrix composition, source-to-detector distance, etc.). Besides the years of practical utilization in many gamma-spectrometry laboratories, accuracy of the software is successfully tested in a recent IAEA-organized intercomparison exercise—ANGLE scored 0.65% average deviation from the exercise mean for E γ >20keV energies.

Detector Solid Angle Formulas for Use in X-Ray Energy Dispersive Spectrometry

With the advent of silicon drift X-ray detectors, a range of new geometries has become possible in electron optical columns. Because of their compact size, these detectors can potentially achieve high geometrical collection efficiencies; however, using traditional approximations detector solid angle calculations rapidly break down and at times can yield nonphysical values. In this article we present generalized formulas that can be used to calculate the variation in detection solid angle for contemporary Si(Li) as well as new silicon drift configurations.

An algorithm for the solid angle calculation applied in alpha-particle counting

An algorithm is presented to calculate the solid angle for a flat source facing an axially symmetric detector set-up with one or more coaxial diaphragms. The calculation is based on numerical integration and its validity was checked by comparing with the results of a Monte Carlo program. Formulas are given for point sources as well as circular and annular sources. The latter is applied in a method to take into account the possible inhomogeneity, non-circular shape, and/or eccentricity of the radioactive source with respect to the symmetry axis of the detector.

Tailoring solid angle calculations to the actual radioactivity distribution of planar sources

In the frame of accurate radioactivity measurements by means of alpha-particle counting at defined low solid angles, a procedure is presented that takes into account the source's actual radioactivity distribution in the solid angle calculation. The method pertains to an axially symmetric set-up in which a thin flat source is placed parallel to the diaphragm(s) and the detector. From a digital autoradiograph, the source's radial activity distribution is assessed. The source is virtually subdivided into concentric rings and the corresponding solid angle values are weighted with the relative contribution of the annular regions-of-interest to the total radioactivity of the source.

Calculation of the average solid angle subtended by a detector to source in a parallel plane by a Monte Carlo method.

A short computer program is described for a PC which uses a Monte Carlo method to calculate the average solid angle subtended by a rectangular or circular detector window to a coaxial or non-coaxial rectangular, circular or point source. The advantage of the Monte Carlo method is that it allows the calculation of average solid angle for source-detector geometries that are difficult to analyse by analytical methods. The values of solid angle are calculated to accuracies of typically better than 0.1%. The calculated values from the Monte Carlo method agree closely with those produced by polygon approximation and numerical integration by Gardner and Verghese, and others.

Small Molecule Recognition: Solid Angles Surface Representation and Molecular Shape Complementarity

Here we examine the recognition of small molecules by their protein and DNA receptors. We focus on two questions: First, how well does the solid angle molecular surface representation perform in fitting together the surfaces of small ligands, such as drugs and cofactors to their corresponding receptors; And second, in particular, to what extent does the shape complementarity play a role in the matching (recognition) process of such small molecules. Both questions have been investigated in protein-protein binding: "Critical Points" based on solid angle calculations have been shown to perform well in the matching of large protein molecules. They are robust, may be few in numbers, and capture satisfactorily the molecular shape. Shape complementarity has been shown to be a critical factor in protein-protein recognition, but has not been examined in drug-receptor recognition. To probe these questions, here we dock 185 receptor-small ligand molecule pairs. We find that such a representation performs adequately for the smaller ligands too, and that shape complementarity is also observed. These issues are important, given the large databases of drugs that routinely have to be scanned to find candidate, lead compounds. We have been able to carry out such large scale docking experiments owing to our efficient, computer-vision based docking algorithms. Its fast CPU matching times, on the order of minutes on a PC, allows such large scale docking experiments.

Improved algorithm for accurate computation of molecular solid angles

A new algorithm involving the calculation of the solid angle of a molecule about a point as a measure of steric size has been developed. The algorithm calculates the total solid angle in a stepwise fashion, summing all regions of individual atom solid angles and overlapped atoms, taking into account all orders of possible overlap of multiple atoms. The results for several molecular fragments have been compared to previous solid and cone angle calculations and improved correlations were observed. © 1996 by John Wiley & Sons, Inc.

Solid angles III. The role of conformers in solid angle calculations

The values of the solid angles Ω for a range of commonly encountered ligands in organometallic chemistry (phosphines, phosphites, amines, arsines and cyclopentadienyl rings) have been determined. The solid angles were derived from a single energy conformer in a geometry constrained in a prototypical metal environment, i.e. with the ligand attached to Cr(CO) 5. This choice permits comparison with recent ligand repulsive energies E R reported by Brown. Correlations between Ω, ER and θ for the range of ligands studied typically gave correlation coefficients greater than 0.8. The values of the weighted average solid angle Ω , using an extension of the methodology by Brown and Mosbo, have also been determined. The minimum conformer solid angle values are correlated with the weighted average solid angle ( r = 0.96), suggesting that the minimum-energy conformation is a good approximate measure of steric size.

A personal computer based program for HPGe detector absolute-peak-efficiency calculation and calibration

A personal computer based program is developed for HPGe detector effective solid angle calculation for various source-detector arrangements. This program uses a simplified detector model to avoid tedious computational algorithms. The source geometry can be point, disk or cylinder. The applicability of using this program for gamma-ray absolute-peak-efficiency calibration has been successfully validated with various combinations of source geometry, source-detector distance and gamma-ray energy.

Calculations of an effective solid angle including self-absorption correction applied to gamma-ray spectrometry analysis of natural samples

A new method is presented for determination of uranium and thorium contents in solid samples by gamma-ray spectrometry. The analytical procedure involved determination of uranium via the 63.288 keV gamma emission of its daughter 234Th and thorium on the basis of 238.578 keV 212Pb peak. The radiochemical equilibrium was assumed. Geometry effects and exact self-absorption were taken into account by measurements of sample linear absorption coefficients and calculation of effective solid angle. IAEA standards were used for determination of detector efficiency. The method provides reliable analytical measurements even for samples much different from standard reference materials in density and gamma absorption.

Monte Carlo calculation of the average solid angle subtended by a parallelepiped detector from a distributed source

A Monte Carlo method utilizing total variance reduction is applied to the calculation of the solid angle subtended by a parallelepiped detector from an arbitrarily positioned point. Although there is an analytical expression for the solid angle subtended by a rectangle from an arbitrary point, the method should prove valuable when a mean solid angle from a distributed source is required. In addition, the algorithm provides an efficient way to sample the directions of gamma rays which intercept a parallelepiped detector, required in the Monte Carlo calculations of the efficiency of square detectors.