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Gamma-ray field in air from radioactive sources uniformly distributed in the ground is calculated by numerically solving the photon transport equation. The scattered flux is developed on the Legendre polynomials basis of the spherical harmonics method and a double P1 approximation is applied. The method is implemented in the Octave programming package. The calculation of the gamma-ray field is also done by Monte Carlo simulation using an optimised geometry model of the soil-air medium that is implemented in Geant4 simulation code. The results obtained by the two methods are in good agreement. The computing time for the spherical harmonics deterministic method is significantly less than the Monte Carlo simulation. Dose rate conversion factors in the air are calculated for the natural radioactive series of 238U and 232Th and for 40K nuclide and for soils contaminated by various radionuclides. The results are in agreement with the published theoretical and experimental studies.

Objective: To determine the distribution of the positron range of radionuclide 18F using Monte Carlo Simulation. Method: 18F is widely available for routine clinical use. In this work, we perform a theoretical calculation of the distribution of the 18F positron range using Monte Carlo (MC) simulation. To collect a statistically significant sample of distance values between the beta emitter point and the Line Of Response (LOR), approximately50000 tracks in water were generated and propagated until annihilation. The radial cumulative probability distribution G2 D(δ) as a function of was adjusted and the analytical formulae was obtained. Finding: The maximum, mean and 1D values of the positron range for 18F simulated and propagated in water are calculated and compared with other studies. The application of Monte Carlo simulation for positron range calculation emphasizes the adoption of this calculation for the radionuclide properties and their propagated medium. The calculation results of beta particle energy loss are in good agreement with many studies that dealt with the same matter having the same energy range. The bin size, source shape, simulation code and propagation medium are the main parameters responsible on the slight differences in the positron range calculation results. Application: It is well known from most existing studies that PET resolution blurring is coupled to the positron range and that it is important to take this range into account in the image reconstruction process. In this study, we calculated the positron range of 18F in water, which is not an obvious affect and it is related to the values of Emean and Emax of 18F 0.250 MeV and 0.635MeV, respectively. These values will cause positron particles to propagate in the medium with a range not exceeding 2 mm with an FWHM(px) is 0.16 mm and FWTM(px) is 1.05 mm. Keywords: Geant4 Code, Positron Emission Tomography, Positron Range

We derive the general solution of the semi-relativistic spinless Salpeter equation in the presence of a time-dependent linear potential via the Lewis–Riesenfeld framework and using two forms of the invariant. For comparison, we reobtain the solution in the momentum-space directly by applying a time-momentum transformation to the involved partial differential equation. We also investigate the classical-quantum correspondence for the model in the case of time-periodic force using a momentum-space Gaussian wave-packet.

In this study we investigate the radiological hazard of naturally occurring radioactive material in Tunisian and Algerian phosphorite deposits. Eight samples of phosphorite were collected from the phosphorite mines. The Tunisian and Algerian phosphorites occur in the Late Paleocene and Lower Eocene (Ypresian-Lutetian) in age (Beji Sassi 1984 and Zaier 1999). Activity concentrations in all the samples were measured by alpha spectrometry and gamma spectrometry. Alpha spectrometry analyses show that the specific activity values of 238U, 234U and 235U in the samples of Tunisian phosphorite were 327 ± 7 (321–327), 326 ± 6 (325–331) and 14.50 ± 0.72 (13.90–15.57) Bq kg−1, respectively. Specific activity measured by gamma spectrometry in the samples of the Tunisian and Algerian phosphorite shows a small difference. Specific activity levels of 40K, 226Ra, 232Th, 235U and 238U in the phosphorite samples from Tunisia were, respectively, 71.10 ± 3.80, 391.54 ± 9.39, 60.38 ± 3.74, 12.72 ± 0.54 and 527.42 ± 49.57 Bq kg−1 and Algeria were 15.72 ± 1.73, 989.65 ± 12.52, 12.08 ± 1.20, 47.50 ± 1.52 and 1,148.78 ± 7.30 Bq kg−1, respectively. The measured value of specific activity of 232Th and 40K in the Tunisian phosphorite samples is relatively higher than that found in the samples of Algerian phosphorite. The measured activity of uranium (238U) in the Tunisian phosphorite (527 ± 49) Bq kg−1 is lower than in Algerian phosphorite. The measured activity of 238U in the Tunisian phosphorite samples was (527–1,315 ± 65) 238U Bq kg−1 which is higher than its maximum background value of 110 Bq kg−1 in soils of the various countries of the world (Tufail et al. Radiat Meas 41:443–451, 2006). Different geological origins of phosphorites deposits are the main reason for the large spread in worldwide specific activities. The obtained results of uranium concentrations in phosphorites of different types (Algerian and Tunisian) demonstrate that the uranium concentrations are mainly governed by the phosphatic material. The present study reveals that phosphorite deposits contain natural radioactivity higher than background level.

The radioactivity of 15 kinds of different granites collected in Tunisia was determined by gamma-ray spectrometry using hyper-pure germanium (HPGe) detector. The average activity concentrations for primordial radionuclides 238U, 232Th and 40K were respectively 33.24, 8.01 and 116.98 Bq/kg. The activity concentrations ranged from 3.59 to 87.37 Bq/kg for 238U, from 0.45 to 25.34 Bq/kg for 232Th and from 24.06 to 380.23 Bq/kg for 40K. The measured activity concentrations were used to assess of the radium equivalent activity ranged from 22.2 to 995.8 Bq/kg, the absorbed dose rate in air from 7 to 1209 nGy/h and the internal (0.1 to 2.8) and external (0.1 to 2.7) hazard indices. The data obtained in this study may be useful for natural radioactivity mapping.

We obtain bound-state solutions of the Duffin–Kemmer–Petiau equation in one spatial dimension in the presence of a Lorentz-scalar linear potential. The exact conditions selecting the eigenvalues of the system have been derived and are shown to match their non-relativistic counterparts in the weak-coupling limit.

The path integral method is used to present an exact treatment of the one-dimensional Klein–Gordon oscillator in the context of quantum mechanics with a deformed Heisenberg algebra of the form , leading to the existence of a minimal observable length . We tackle the problem in the momentum space and we use the Schwinger proper-time method to represent Green's function. Calculations are run with the help of the point canonical transformation technique. The bound-state energy spectrum and the associated momentum space eigenfunctions are obtained, and a detailed comparison with the results for the undeformed case is made.