Effective Solid Angle Research Articles

Mathematical formula to calculate the attenuation coefficients for any complex sample: experimental study

This work introduces a mathematical formula to calculate the attenuation coefficient of a cylindrical sample placed on the top of a well-shaped gamma-ray detector. The formula depends on calculations of the average path length of the photon through a sample and the effective solid angle. The obtained results using the introduced formula are compared with the NIST XCOM database and compared with the experimental work using two different-sized detectors and sources located in two different positions. A remarkable agreement between both was observed, which proves the validity of this formula.

Counterpropagating radiation emitted by an electron in the field of a Gaussian laser pulse

The transition from the nonlinear Thomson scattering (electron initially at rest) to the classical limit of Compton scattering (electron moving towards the laser pulse) is considered. The radiation emitted by an electron in the form of plane waves in the far zone is analyzed, and the intensity distribution over the solid angle is obtained. By constructing radiation diagrams, we find numerically the dependences of the intensity and power of radiation on the initial energy of the electron The backscattering intensity as is find analytically is proportional to six power in initial energy and to the peak power of the laser pulse. It is shown that as the initial energy of the electron increases, the effective solid angle of the scattered radiation decreases monotonically and becomes constant.

Efficient Simulation for Visual Comfort Evaluations

This paper provides a validation of a novel sampling, storage, and evaluation method named raytraverse that can quickly and accurately compute glare and visual comfort metrics including vertical illuminance (Ev), Daylight Glare Probability (DGP), and Unified Glare Probability (UGP). The motivation is to provide a path towards understanding the spatial and temporal distribution of daylight conditions in an architectural space. Current spatial temporal simulation workflows are constrained by the trade-offs between simulation time, accuracy, generality, and storage requirements. Raytraverse provides a bridge between illuminance sensor calculations, which are fast to calculate but provide limited information, and high-resolution image generation, which provide more information but have long simulation times. To make this bridging possible, it relies on a pair of strategies that yields both high accuracy and high information data. First, an iteratively guided sampling approach based on the discrete wavelet transformation greatly reduces the number of view rays submitted to the renderer. Second, rather than collecting returned values in a raster image or dense matrix, a spatial data structure is used to both sparsely store and re-weight the samples according to their effective solid angle, allowing for the direct integration of any view based lighting metric and the synthesis of interpretable high dynamic range images (HDRi). These strategies can be coupled with existing rendering and climate based daylight modeling (CBDM) methods. Through a comparison with high-quality reference simulations and a best practice CBDM method based on Radiance, the raytraverse methods are shown to significantly reduce the simulation time needed to accurately simulate saturation, contrast, and combined visual comfort metrics for a complete set of annual hourly sky conditions from a range of locations within an office floor plan. The stored simulation data can be quickly re-analyzed for different view directions, metrics or images, and sky conditions.

Investigation of Spatial Orientation and Kinetic Energy of Reactive Site Collision between Benzyl Chloride and Piperidine: Novel Insight into the Microwave Nonthermal Effect.

Microwave nonthermal effect in chemical reactions is still an uncertain problem. In this work, we have studied the spatial orientation and kinetic energy of reactive site collision between benzyl chloride and piperidine molecules in substitution reaction under microwave irradiation using the molecular dynamics simulation. Our results showed that microwave polarization can change the spatial orientation of reactive site collision. Collision probability between the Cl atom of the C-Cl group of benzyl chloride and the H atom of the N-H group of piperidine increased by up to 33.5% at an effective spatial solid angle (θ, φ) of (100∼110°, 170∼190°) under microwave irradiation. Also, collision probability between the C atom of the C-Cl group of benzyl chloride and the N atom of the N-H group of piperidine also increased by up to 25.6% at an effective spatial solid angle (θ, φ) of (85∼95°, 170∼190°). Moreover, the kinetic energy of collision under microwave irradiation was also changed, that is, for the collision between the Cl atom of the C-Cl group and the H atom of the N-H group, the fraction of high-energy collision greater than 6.39 × 10-19 J increased by 45.9 times under microwave irradiation, and for the collision between the C atom of the C-Cl group and the N atom of the N-H group, the fraction of high-energy collision greater than 6.39 × 10-19 J also increased by 29.2 times. Through simulation, the reaction rate increased by 34.4∼50.3 times under microwave irradiation, which is close to the experimental increase of 46.3 times. In the end, spatial orientation and kinetic energy of molecular collision changed by microwave polarization are summarized as the microwave postpolarization effect. This effect provides a new insight into the physical mechanism of the microwave nonthermal effect.

A new semi-empirical model for cosmic ray muon flux estimation

Abstract Cosmic ray muons have emerged as a non-conventional high-energy radiation probe to monitor dense and large objects. Muons are the most abundant cosmic radiation on Earth; however, their flux at sea level is approximately 104 min−1m−2, much less than that of induced radiation. In addition, cosmic ray muon flux depends on not only various natural conditions, e.g., zenith angle, altitude, or solar activity, but also the geometric characteristics of detectors. Since the low muon flux typically results in long measurement times, an accurate estimation of measurable muon counts is important to improve the efficiency of muon applications. Here we propose a simple and versatile semi-empirical model to improve the accuracy in muon flux estimation at all zenith angles by incorporating the geometric parameters of detectors, and we name this the ‘effective solid angle model.’ To demonstrate the functionality of our model, it is compared with (i) the cosine-squared model, (ii) the PARMA model, (iii) Monte Carlo simulations, and (iv) experimental measurements. Our results show that the muon count rate estimation capability is significantly improved, resulting in increasing a mean C/E level from 0.7 to 0.95. In addition, by selecting an appropriate intensity correlation, the model can be easily extended to estimate muon flux at various altitudes and also underground.

Analytical computation technique for calculation the effective geometrical solid angle and the efficiency of cubic scintillation crystal with side cylindrical hole

In the gamma-ray spectroscopy field, the radiometric examination for small quantities of natural samples is extremely significant. Therefore, the gamma-ray spectrometry calibration process must be prepared with good precision for several energies, matrices of sources or samples, and source-to-detector shapes. This manuscript considers a new uncomplicated analytical computation technique to calculate the effective geometrical solid angle and the efficiency of cubic scintillation crystal with a side cylindrical hole. The computations can be done by using a simple method, with a few essential limitations, that describes radioactive point sources located inside the side cylindrical hole and a high-efficiency cubic NaI(Tl) detector, come together with a low background as well. The technique stands on a trouble-free solid angle analytical formula for the detection system, using an accurate relation for the detector cavity, united with rough formulas controlling the interactions in the gamma-ray source and the materials introduced in between the source and the gamma-ray spectrometry. This new technique is not restricted to certain sources, because several source shapes can correspond to a homogeneous huge number of point sources and the detector geometry can be represented as a set of border points. The technique simply can be useful to obtain the full-energy peak efficiency in the future, challenging developments for low-energy gamma-ray spectroscopy.

Di-chlorido-(η6-p-cymene)[tris-(4-meth-oxy-phen-yl)phosphane]ruthenium(II).

The title compound, [RuCl2(C10H14)(C21H21O3P)], crystallizes with two complex mol-ecules in the asymmetric unit. The RuII atom has a classical three-legged piano-stool environment being coordinated by a cymene ligand [Ru-centroid = 1.707 (2)/1.704 (2) Å], a tris-(4-meth-oxy-phen-yl)phosphane ligand [Ru-P = 2.3629 (15)/2.3665 (15) Å] and two chloride atoms with the Ru-Cl bonds adopting two distinct values of 2.4068 (16)/2.4167 (16) and 2.4016 (15)/2.4244 (16) Å. The effective cone and solid angles for the phosphane ligands were calculated to be 149.5/150.2° and 25.3/25.6°, respectively. In the crystal, weak C-H⋯Cl/O/π inter-actions are observed. The crystal was refined as a two-component twin.

Expanding the momentum field of view in angle-resolved photoemission systems with hemispherical analyzers.

In photoelectron spectroscopy, the measured electron momentum range is intrinsically related to the excitation photon energy. Low photon energies <10 eV are commonly encountered in laser-based photoemission and lead to a momentum range that is smaller than the Brillouin zones of most materials. This can become a limiting factor when studying condensed matter with laser-based photoemission. An additional restriction is introduced by widely used hemispherical analyzers that record only electrons photoemitted in a solid angle set by the aperture size at the analyzer entrance. Here, we present an upgrade to increase the effective solid angle that is measured with a hemispherical analyzer. We achieve this by accelerating the photoelectrons toward the analyzer with an electric field that is generated by a bias voltage on the sample. Our experimental geometry is comparable to a parallel plate capacitor, and therefore, we approximate the electric field to be uniform along the photoelectron trajectory. With this assumption, we developed an analytic, parameter-free model that relates the measured angles to the electron momenta in the solid and verify its validity by comparing with experimental results on the charge density wave material TbTe3. By providing a larger field of view in momentum space, our approach using a bias potential considerably expands the flexibility of laser-based photoemission setups.

Experimental and simulated methods to characterize the response of a scintillator detector

The NaI(Tl) detector is one of the most used in gamma ray spectrometry because it presents high counting efficiency for gamma rays in a wide energy range. This study presents the modeling of a real 1¼ x ¾" NaI(Tl) scintillation detector + photomultiplier using computational simulations carried out by the MCNP6 code, which considers energy resolution. In the mathematical model, the sensitive volume of the detector was adjusted using 241Am and 137Cs radiation sources by means of the absolute photopeak efficiency. Moreover, the MCNP6 code was used to calculate the effective solid angle of the detector in order to obtain the intrinsic efficiency response function. The mathematical model was experimentally validated using calibrated radioactive sources (241Am, 133Ba, 137Cs and 60Co) and the response functions - efficiency curve and energy resolution - were obtained. In addition, an experimental evaluation of the crystal homogeneity of the detector was made using a137Cs radiation source and a 3D printed device developed without stepper motors. The proposed methodology is able to obtain absolute photopeak efficiency in agreement with experimental data with maximum relative error of 5.64%. The gamma scanning procedure indicated that the crystal of the detector remained homogeneous.

Calibration of cylindrical NaI(Tl) gamma-ray detector intended for truncated conical radioactive source

The computation of the solid angle and the detector efficiency is considering to be one of the most important factors during the measuring process for the radioactivity, especially the cylindrical γ-ray NaI(Tl) detectors nowadays have applications in several fields such as industry, hazardous for health, the gamma-ray radiation detectors grow to be the main essential instruments in radiation protection sector. In the present work, a generic numerical simulation method (NSM) for calculating the efficiency of the γ-ray spectrometry setup is established. The formulas are suitable for any type of source-to-detector shape and can be valuable to determine the full-energy peak and the total efficiencies and P/T ratio of cylindrical γ-ray NaI(Tl) detector setup concerning the truncated conical radioactive source. This methodology is based on estimate the path length of γ-ray radiation inside the detector active medium, inside the source itself, and the self-attenuation correction factors, which typically use to correct the sample attenuation of the original geometry source. The calculations can be completed in general by using extra reasonable and complicate analytical and numerical techniques than the standard models; especially the effective solid angle, and the detector efficiency have to be calculated in case of the truncated conical radioactive source studied condition. Moreover, the (NSM) can be used for the straight calculations of the γ-ray detector efficiency after the computation of improvement that need in the case of γ-γ coincidence summing (CS). The (NSM) confirmation of the development created by the efficiency transfer method has been achieved by comparing the results of the measuring truncated conical radioactive source with certified nuclide activities with the γ-ray NaI(Tl) detector, and a good agreement was obtained after corrections of (CS). The methodology can be unlimited to find the theoretical efficiencies and modifications equivalent to any geometry by essential sufficiently the physical selective considered situation.

Measurements correlated to efficiency studies of NaI(Tl) scintillator with vertical hexagonal radioactive source

Nowadays, many scientists work on the geometry change between source-to-detector arrangement to calibrate the 3”× 3” γ-ray NaI(Tl) detector based on the available sample shape in the laboratory. The calculated full-energy peak efficiency is more sensitive in radiation activity determination and can be estimated build on complex analytical and numerical techniques. In the present study, the full-energy peak and total efficiencies beside the peak-to-total ratio (P/T) for γ-ray NaI(Tl) detector concerning vertical hexagonal ^152Eu radioactive source is calculated for all γ-rays involved in the cascade and estimate the suitable correction for coincidence summing effects (COI) by mean of the numerical simulation method (NSM). The extra factors which cause the γ-ray attenuation taken into consideration as well, such as, source composition materials, “self-attenuation within the source itself”, the detector end cap with the other supporting materials around the detector crystal, the source container material, and the holder used during the measurement process. The theory of this methodology depends on the efficiency transfer technique; calculate the effective solid angle, that be positioned in between the source-to-detector system, besides estimating the path lengths of the γ-rays inside the hexagonal ^152Eu radioactive sources and the active medium inside the detector itself as well. The results gained in the present work gave an agreement between simulation and measured data. The results show that to improve the accuracy of the γ-rays detector efficiency, many technical and methodological aspects of the present method are fitting and possible for applications inside the radiation laboratory, industrial, and medical sectors.

Source-detector effective solid angle calculation using MCNP6 code

Nuclear techniques based on the attenuation of gamma radiation are used in the industry to calculate flow rate, determine fluid density, predict inorganic scale in oil pipelines, evaluate industrial mixers, among other applications. In order to use these nuclear techniques is necessary to perform studies of important parameters of radioactive source and radiation detectors, which are part of the measurement geometry, such as detection efficiency and solid angle. The aim of this study is to calculate the solid angle and the effective solid angle subtended by a NaI(Tl) detector. The effective solid angle considers attenuation in the medium (between source and detector) and other effects of radiation interaction with matter. Mathematical models were developed using the MCNP6 code in order to evaluate the proposed measurement geometry. The source was placed in different positions to the detector to evaluate frontal and lateral solid angle contributions, which is an important parameter to obtain the intrinsic efficiency response function. The simulated model consists of a NaI(Tl) scintillator detector and two point isotropic sources (241Am and 137Cs). The results for the geometry used in this study showed that the difference between solid angle and effective solid angle reached 20.17% for 241Am and 2.58% for 137Cs, which means that it is highly recommended to consider the effective solid angle in the calculations.

Efficiency calibration and coincidence summing correction for a NaI(Tl) spherical detector

Spherical NaI(Tl) detectors are used in gamma-ray spectrometry, where the gamma emissions come from the nuclei with energies in the range from a few keV up to 10 MeV. A spherical detector is aimed to give a good response to photons, which depends on their direction of travel concerning the detector center. Some distortions in the response of a gamma-ray detector with a different geometry can occur because of the non-uniform position of the source from the detector surface. The present work describes the calibration of a NaI(Tl) spherical detector using both an experimental technique and a numerical simulation method (NSM). The NSM is based on an efficiency transfer method (ETM, calculating the effective solid angle, the total efficiency, and the full-energy peak efficiency). Besides, there is a high probability for a source-to-detector distance less than 15 cm to have pulse coincidence summing (CS), which may occur when two successive photons of different energies from the same source are detected within a very short response time. Therefore, γ-γ ray CS factors are calculated numerically for a152Eu radioactive cylindrical source. The CS factors obtained are applied to correct the measured efficiency values for the radioactive volumetric source at different energies. The results show a good agreement between the NSM and the experimental values (after correction with the CS factors).

Evaluation of blackbody radiation shift with 2 × 10−18 uncertainty at room temperature for a transportable 40Ca+ optical clock

Blackbody radiation (BBR) temperature and its uncertainty mainly contribute to the BBR shift uncertainty, which is one of the challenges in most ion optical clocks. Here, an improved BBR temperature evaluation method based on the analysis of the effective solid angle of an ion trapping system using the finite element method is presented. To effectively suppress the BBR temperature uncertainty, the structure of an ion trap was optimized to reduce the effective solid angle of the insulating supports, and a BBR shield was employed to reduce the influence of the surrounding BBR in the newly developed ion trap system. Due to radiofrequency field, it is difficult to evaluate the temperature of BBR shield accurately using contact-temperature-sensors inside the vacuum chamber. We used a well shielded platinum resistance temperature sensor to measure the temperature of the shield from the outside of the vacuum chamber. Through a detailed evaluation, the BBR temperature uncertainty of this newly designed system is 0.17 K, corresponding to a BBR frequency shift uncertainty of 2 × 10−18 for the 40Ca+ optical clock. This improvement of the BBR temperature uncertainty using the active adjustment of the distribution of the effective solid angle of the ion trapping system, without a complicated cryogenic trap, is very suitable for transportable optical clocks operating at room temperature (∼295 K).

Phase-space sensitivity (weight functions) of 3 MeV proton diagnostics

The weight function W(X) is a diagnostic sensitivity to phase-space variables X that relates the measured signal C to the distribution function F(X) through the equation . In the present work, an algorithm to calculate W for a diagnostic that measures 3 MeV protons produced in d(d,p)t fusion reactions between a fast ion and a thermal deuteron is developed. The emitted protons escape the tokamak on curved orbits and are detected. These curved orbits constitute effective diagnostic ‘sightlines’. The presented algorithm accounts for the complications associated with these curved sightlines. An initial calculation of time-reversed proton orbits computes effective solid angles and sightlines for the relevant range of incident proton velocity vectors. These precomputed orbits are then used within the framework of FIDASIM (2020 Plasma Phys. Cont. Fusion 62 105008) to calculate the reactivity averaged over the thermal distribution of the ‘target’ deuterons and the probability that a fast ion of specified energy and pitch has a gyroangle that is consistent with the kinematic equations along each of the sightlines. Comparisons with analytic formulas and with independent calculations for the Mega Amp Spherical Tokamak 3 MeV proton diagnostic verify the algorithm.

Towards a calibration of laboratory setups for grazing incidence and total-reflection X-ray fluorescence analysis

Grazing-Incidence X-ray fluorescence (GIXRF) analysis, which is closely related to total-reflection XRF, is a very powerful technique for the in-depth analysis of many types of technologically relevant samples, e.g. nanoparticle depositions, shallow dopant profiles, thin layered samples or even well-ordered nanostructures. However, the GIXRF based determination of the depth-dependent information about the sample is usually based on a modeling of the experimental data. This requires profound knowledge of the geometrical parameters of the setup employed, especially the incident beam profile as well as the detector aperture parameters. Together they determine the incident angle dependent so-called effective solid angle of detection which must be known in order to model any experimental data set. In this work, we demonstrate how these instrumental parameters, which are typically not known with sufficient accuracy, can be determined using dedicated experiments with a well-known calibration sample. In addition, this paves the way for a full calibration of the setup, as also information about other parameters, e.g. the incident photon flux is gained. Here, we are using a commercially available instrument for this demonstration but the principle can also be applied for other GIXRF setups.

Calibration and use of well-type germanium detectors for low-level gamma-ray spectrometry of sediments using a semi-empirical method

Gamma-ray spectrometry is a widely used technique to quantify the presence of numerous radionuclides in environmental samples. In this work, we describe a methodology for efficiency calibration of four well-type germanium detectors and their use for the determination of low-level activities of gamma emitters in sediment samples. An experimental efficiency calibration for each detector was built with three materials for 17 energies, ranging from 46.5 keV to 1460 keV. For efficiency transfer to different geometries and sample types, we used the effective solid angle approach (ET-Ω method). Final calibrations were calculated for all detectors, several counting geometries, and elemental composition of selected sample types. Calibrations were validated with six reference materials. This methodology allowed to reliably analyze nine gamma emitters (210Pb, 241Am, 234Th, 228Ac, 214Pb, 208Tl, 137Cs, 134Cs and 40K) in sediment samples. Using these calibrations, gamma emitter profiles of sediment cores from contrasting aquatic systems (lake, intertidal, marine and deep-sea areas) provided reliable profiles of 210Pb and artificial radionuclides useful for dating and stratigraphic interpretation. A protocol to implement this methodology is also presented.

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.

A new beamline for energy-dispersive high-resolution PIXE analysis using polycapillary optics

A new beamline for high energy resolution PIXE measurements is presented. This new setup includes options for both in-air and vacuum measurements. For the high energy resolution transition-edge sensor array, a polycapillary lens is used for detecting low-energy X-rays down to 0.5 keV and to increase the effective solid angle. X-ray transmission of the polycapillary lens was characterized using two calibration standards. The gain obtained by adding a polycapillary lens was 1.6–2.3 at energies between 2.1 keV and 4.5 keV. From 1.04 to 1.74 keV the gain is increased to 2.1–3.0, and at energies 4.9–8.0 keV the gain is between 1.6 and 0.65. The measured gain agreed well with theoretical calculations. The lightest detectable element in the new setup is oxygen at 0.525 keV.

A generalized mathematical model for efficiency calibration of gamma detectors: Application to practical cases

Efficiency calibration, i. e. determination of detection efficiency, ?p, is a crucial issue in gamma spectrometry (quantification of gamma spectroscopic measurements) with semiconductor and scintillation detectors. Comparing three possible ways to addressing the problem ? relative, absolute and semi empirical ? advantages of the latter are emphasized. Among semi empirical models, efficiency transfer using effective solid angles, ??, is sorted out and briefly elaborated. This approach reduces the problem of efficiency calibration to the determination of ??. It proved reliable and has been broadly used in practice, mainly in the form of the long existing ANGLE software. Progressing further, a generalized mathematical formula for calcu- lations is developed ? first of the kind ? offering an opportunity for advanced applications of gamma spectrometry. The formula enables unlimited flexibility in application, as it conveniently separates the source data from the detector data during the integration procedures ?? calculations). Its practicality is demonstrated for a number of typically encountered counting arrangements, as well as for some exotic ones. The relevant formulae are used in PC calculations and numerical testing is further performed so as to check the validity of the mathematical method and the computer code. Care was taken of the optimization of complex numerical procedures employed (involving fivefold numerical integration), so as to keep computation times as low as possible (in order of minutes or even seconds on ordinary PC). Results obtained are affirmative for both the method and the code. The model will be gradually incorporated into ANGLE software, thus making it readily available for routine use by gamma spectrometry community.