Axial Point Source Research Articles

New mathematical approach to calculate the geometrical efficiency using different radioactive sources with gamma-ray cylindrical shape detectors

The geometrical efficiency of a source-to-detector configuration is considered to be necessary in the calculation of the full energy peak efficiency, especially for NaI(Tl) and HPGe gamma-ray spectroscopy detectors. The geometrical efficiency depends on the solid angle subtended by the radioactive sources and the detector surfaces. The present work is basically concerned to establish a new mathematical approach for calculating the solid angle and geometrical efficiency, based on conversion of the geometrical solid angle of a non-axial radioactive point source with respect to a circular surface of the detector to a new equivalent geometry. The equivalent geometry consists of an axial radioactive point source with respect to an arbitrary elliptical surface that lies between the radioactive point source and the circular surface of the detector. This expression was extended to include coaxial radioactive circular disk source. The results were compared with a number of published data to explain how significant this work is in the efficiency calibration procedure for the γ-ray detection systems, especially in case of using isotropic radiating γ-ray sources in the form of point and disk shapes.

Empirical Formulae to Calculate Full-Energy Peak Attenuation Coefficients of Well-Type NaI (Tl) Detectors

NaI (Tl) detectors have been widely used during the recent years based on their response to γ-rays, which is significant mainly for the elemental analysis of neutron-induced prompt γ-rays. The detector functions significantly influence the analysis results, especially the detector efficiency. The present study focused on this problem and has been made of the contribution of the full-energy peak attenuation in the calculation of the detector efficiency. An empirical formula has been derived to calculate the full-energy peak attenuation for 3″×3″ NaI (Tl) well-type detector using isotropic radiating axial point sources. The measured efficiency values are used in this method to find the peak attenuation coefficient for this particular source-to-detector geometry. In addition, the obtaining values were used to calculate the detector full-energy peak efficiency, and compared it with the measured ones. The calculated values of the detector efficiency are found to be in a good agreement with experimental data. This gives indication about the ability to extend the calculated efficiency values to build an efficiency curve as a function of γ-ray energy as not many radioactive sources exist for the standardization process.

Investigating the X-ray time lags in PG 1244+026 using an extended corona model

We present an extended corona model based on ray-tracing simulations to investigate X-ray time lags in Active Galactic Nuclei (AGN). This model consists of two axial point sources illuminating an accretion disc that produce the reverberation lags. These lags are due to the time delays between the directly observed and reflection photons and are associated with the light-travel time between the source and the disc, so they allow us to probe the disc-corona geometry. We assume the variations of two X-ray sources are triggered by the same primary variations, but allow the two sources to respond in different ways (i.e. having different source responses). The variations of each source induce a delayed accretion disc response and the total lags consist of a combination of both source and disc responses. We show that the extended corona model can reproduce both the low-frequency hard and high-frequency soft (reverberation) lags. Fitting the model to the timing data of PG~1244+026 reveals the hard and soft X-ray sources at $\sim6r_{\text{g}}$ and $\sim11r_{\text{g}}$, respectively. The upper source produces small amounts of reflection and can be interpreted as a relativistic jet, or outflowing blob, whose emission is beamed away from the disc. This explains the observed lag-energy in which there is no soft lag at energies $<1$~keV as they are diluted by the soft continuum of the upper source. Finally, our models suggest that the fluctuations propagating between the two sources of PG~1244+026 are possible but only at near the speed of light.

Mathematical method to calculate full-energy peak efficiency of detectors based on transfer technique

The full-energy peak efficiency of high-purity germanium well-type detector is extremely important to calculate the absolute activities of natural and artificial radionuclides for samples with low radioactivity. In this work, the efficiency transfer method in an integral form is proposed to calculate the full-energy peak efficiency and to correct the coincidence summing effect for a high-purity germanium well-type detector. This technique is based on the calculation of the ratio of the effective solid angles subtended by the well-type detector with cylindrical sources measured inside detector cavity and an axial point source measured out the detector cavity including the attenuation of the photon by the absorber system. This technique can be easily applied in establishing the efficiency calibration curves of well-type detectors. The calculated values of the efficiency are in good agreement with the experimental calibration data obtained with a mixed γ-ray standard source containing 60Co and 88Y.

A numerical approach to calculate the full-energy peak efficiency of HPGe well-type detectors using the effective solid angle ratio

In the field of gamma-ray spectroscopy with HPGe detectors, applied tomeasurements of activity when the sample to be measured is small and has lowradioactivity, the well-type HPGe detectors are widely used. To determinethe sample activity, the full-energy peak efficiency is needed. In thiswork, the efficiency transfer method (ET) in an integral form is proposed tocalculate the full-energy peak efficiency and to correct the coincidencesumming effect for the HPGe well-type detector. This approach is based onthe calculation of the effective solid angles ratio for a well-type detectorwith a cylindrical source inside and an axial point source outside thedetector cavity, taking into account the attenuation of the gamma-rays. Thecalculated values of the full-energy peak efficiency are found to be in agood agreement with the measured experimental data obtained by using a mixedradionuclide gamma-ray source containing 60Co and 88Y.

Optimal welding parameters with 10 keV point source electron gun

We present the update on the emission characteristics and results from previously reported (Vacuum 77 (2004) 19–26), experimental study of axial point source thermionic electron beam emitter assembly. Emission characteristics for this gun have been presented. Optimal conditions to obtain the requisite beam characteristics for welding of refractory metals at 10 keV energy are reported.

Contribution of pair production process in full-energy peak efficiency calculation of semiconductor detectors for axial point sources with high-energy γ-rays

Study has been made of the contribution of the pair production process in calculation of detector efficiency. Particular attention has been paid to positron annihilation in flight and subsequent scattering of the annihilation quanta in the detector material. A mathematical formula has been derived to directly calculate the liner attenuation coefficient of the full-energy peak (μp) for a Ge (Li) detector using an axial point source. The calculated values of the photopeak efficiency are found to be in a good agreement with published experimental data.

Joint transform correlator: expansion of the aberration function for a compact design

A method to compute the aberrations of any order introduced into an optical system which performs the first Fourier transform in a joint transform correlator is described. The optical system considered is illuminated by a quasimonochromatic axial point source and it optically processes a structure placed between two centred optical systems, A and B. The structure acts as an aperture stop, its plane is perpendicular to the optical axis of A and B and it consists of two rectangular displays (object and sample) placed symmetrically with respect to this axis. Each display can be considered as a superposition of sinusoidal gratings of different spatial frequencies and orientations. The aberrations which correspond to each spatial frequency must be evaluated separately for the object (scene) and for the sample (target) and then the device is neither centred nor symmetric. Here it is shown that the aberration function for one of the displays and a given spatial frequency can be expanded in Zernike's polynomials considering both the usual symmetric terms and the non-symmetric ones and assuming that the aperture stop is the circle which contains the display. Moreover, if the optical system which performs the second Fourier transform in the correlator is free from aberrations, it is shown that a criterion which can be used to ensure that an accurate correlation signal is obtained is that the difference between the wavefront aberrations introduced in both displays is less than a quarter of a wavelength.

Flat disc, radially and axially nonhomogeneous, lens

A plane surfaced lens can be constructed using radially as well as axially nonhomogeneous medium. The rays from an axial point source are incident on the plane front surface perpendicular to the axis. Assuming that the rays inside the lens obey Snell's laws as for an infinite uniform medium, a refractive index law is given such that the rays all become parallel to the axis at the second plane surface of the lens.

Transient far field waveform on the axis of an elastic circular plate excited by a pulsed axial point source

The transient waveform radiated from a thin elastic clamped circular plate set in a baffle is investigated when the plate is excited axisymmetrically by a spherical single triangular sound pulse. An expression for the on-axis transient waveform in the far field is obtained in the time domain. Some numerical calculations are made for a circular plate of duralumin, and then the deformation of the waveform due to the spherical excitation is discussed and compared with that in the case of plane wave excitation.

Measurements and analysis of farfield scattering from a prolate spheroid

The farfield acoustic scattering by a prolate spheroid with axial point sources near the tip of the body was measured. Data were taken for ka between 10 and 160, where a is the semimajor axis of the spheroid. Comparisons were made with numerical results obtained by an integral equation based on the simple-source method, with appropriate coordinate stretching introduced to permit high-frequency solutions with a minimal number of grid points. Theory and experiment agree within experimental error except for the highest frequencies in the shadow region, where very rapid changes in pressure make precise measurements difficult. The results show that for frequencies of aeroacoustic interest, the scattered field is very large and cannot be ignored.

Turbulent dispersion in drag‐reducing fluids

AbstractTurbulent pipe flow was examined by measuring the radial dispersion of dyed fluid introduced at an axial point source within the flow. From the classical theoretical results of G. I. Taylor it was possible thereby to calculate various parameters that characterize the structure of the turbulent flow field. A comparison was made between water, and aqueous solutions of polyethylene oxide (Polyox) at concentrations up to 50 ppm (by weight). Results indicate that Polyox affects the turbulence in the following ways: (1) intensity is reduced, (2) the energy spectrum is shifted toward low frequency, (3) dispersion occurs largely through large scale motion, and intermittency is increased. Indication is also seen that a simple exponential form for the Lagrangian correlation coefficient is superior to other often recommended models.

Diffraction fields of a sequence of equal-radii circular apertures

For an axial point source, a sequence of free circular apertures with equal radii can be constructed to focus the incident wave. A comparison with a similar (ellipsoidal) system is made. Experimental and theoretical results (analysed by the boundary-diffraction-wave theory) agree well.

Penetration Effects in Wide Aperture Collimators

The problem of penetration of γ-rays through wide aperture cylindrical collimators, placed close to a body organ for radioisotope uptake studies, is examined. Values for the penetration factors are numerically computed for axial point sources and compared with those given by the radius-independent formulae, normally used, to establish criteria for the use of these formulae. Penetration factors are also measured experimentally, and within the limitations of the experimental method there is good agreement with the computed values, even for sources close to wide aperture collimators.

The Calculation of Gamma Ray Penetration of the Walls of Cylindrical and Conical Collimating Holes

The ratio f of quanta from a point source of γ radiation which penetrate the side wall of a collimating hole to those which pass through the exit pupil of the hole has been derived by analytical methods for the cases of a cylindrical and a conical hole. In the case of a cylindrical hole values of f are obtained as a function of μH where μ is the linear absorption coefficient of the material of the collimator and H is the distance of the source from its lower plane. It is shown that for a constant value of H, the value of f is constant for all points lying within the projection of the collimating hole. The formulae given apply to any cylindrical hole within the limits of the approximations made in their derivation.The formulae for focusing collimators, where the holes are truncated cones, are derived for axial point sources only, and it is shown that the value of f is smaller in this case than for the equivalent situation with a cylindrical hole.

Optical Properties of Conic Surfaces I Reflecting Cone

The optical properties of a reflecting cone, which have acquired fresh importance in connection with alignment measurements, are treated on the basis of Huygens’ principle. Approximate expressions for the intensity distributions in diffraction images of an axial point source and an off-axis one are derived by the stationary phase method. The results of the calculation are found to be in good agreement with the photographs of images made by an actual reflecting cone.

Efficiencies of Sodium Iodide Crystals

Efficiencies for monoenergetic gamma rays, emanating from an axial point source and incident on sodium iodide crystals of 1.5 in. diameter and of various heights, have been calculated by numerical integration methods. The crystals considered are right circular cylinders of heights 0.125, 0.25, 0.375, 0.50, 0.75, 1.5, 2.0, and 3.0 in. Seventeen photon energies in the range from 0.025 to 10.0 Mev were used. Fourteen distances from source to crystal face in the range 0.1 to 50.0 cm were used. Efficiencies for crystals of diameters other than 1.5 in. may be obtained from these results. Relative photoelectric efficiencies were not calculated.