• Volume 92, Issue 4

April 2019

• A $\mathcal{PT}$ -symmetric simple harmonic oscillator

We consider a simple harmonic oscillator with non-Hermitian term and study it classically and quantummechanically. We conclude that this version of oscillator, which breaks parity and time reversal, displays all thefeatures possessed by the usual harmonic oscillator. In particular, we calculate its spectrum, adiabatic invarianceand Wigner functions, and show that there is a consistency between the classical and quantum descriptions.

• A family of conservative chaotic systems with cyclic symmetry

In this paper, we propose a family of circulant systems with conservative property. Various dynamical properties of the circulant systems are derived and investigated. Bifurcation plots are derived and presented for a system and the Lyapunov exponents are derived to show the existence of chaotic oscillations, and their sum being zero confirms the conservativeness for certain values of parameters. One of the proposed systems is then implemented in field programmable gate array (FPGA) to show the hardware reliability. We used the hardware–software cosimulation to see the phase portraits of the FPGA implemented system. The discrete integrators required for solving the initial value problem are implemented using the Euler’s method. The register transfer level schematics of the FPGA implemented system and the resources used for the implementations are presented.

• Non-planar electron-acoustic waves with hybrid Cairns–Tsallis distribution

Non-planar electron-acoustic waves having Cairns–Tsallis distributed hot electrons are investigated under multiple temperature electrons model in unmagnetised plasma. In this model, Korteweg–de Vries (KdV) equation is obtained in the cylindrical/spherical coordinates.On the basis of the solutions of KdV equation, variation of solitary wave features (amplitude, velocity and width) with different plasma parameters are analysed. Dispersion and nonlinear coefficients obtained depend on the particle density $\alpha$, non-extensive parameter $q$, electron temperature ratio $\theta$ and non-thermal parameter $\gamma$ . Combined effect of all these plasma parameters significantly changes the properties of the solitary waves in non-planar geometry. It is observed that increasing the number of non-thermal electrons in the medium increases the amplitude, velocity as well as width of the non-planar waves whereas with the increase in temperature, the velocity of waves decreases and this impact is dominant in spherical waves. This two-parameter ($\gamma, q$) distribution model (C–T) is applicable to a wide range of observed plasmas, i.e. auroral region and magnetosphere of the Earth.

• Beyond the conventional collisional absorption of laser light in under-dense plasma: A particle-in-cell simulation study

Collisional absorption of laser light in an under-dense plasma is studied by particle-in-cell (PIC) simulation with Monte Carlo binary Coulomb collisions between charge particles. For a given plasma thickness of a few times the wavelength of 800 nm laser, fractional absorption ($\alpha$) of the laser light due to Coulomb collisions (mainly between electrons and ions) is calculated at different electron temperature $T_{e}$ with a total velocity $v = (v^{2}_{th} + v^{2}_{0}/2)^{1/2}$ dependent Coulomb logarithm ln $\Lambda(v)$, where $v_{th}$ and $v_{0}$ are thermal and ponderomotive velocity of an electron. In the low-temperature regime ($Te \lesssim 15 eV$), it is found that $\alpha$ increases with increasing laser intensity $I_{0}$ up to a maximum corresponding to an intensity $I_{c}$, and then it drops (approximately) obeying the conventional scaling of $\alpha \varpropto I^{−3/2}_{0}$ when $I_{0}$ > $I_{c}$. Such a non-conventional increase of $\alpha$ with $I_{0}$ in the low intensity regime was demonstrated earlier in experiments, and recently explained by classical and quantum models [Phys. Plasmas 21, 13302 (2014); Phys. Rev. E 91, 043102 (2015)]. Here, for the first time, we report this non-conventional collisional laser absorption by PIC simulation, thus bridging the gap between models, simulations, and experimental findings. Moreover, electron energy distributions naturally emanating during the laser interaction(in PIC simulations) are found to be anisotropic and non-Maxwellian in nature, leading to some deviations from the earlier analytical predictions.

• Comparison of substorm onsets during different levels of interplanetary magnetic field $B_{z}$

The magnetospheric response during the substorm events to solar wind driving, as determined by the level and sign of the interplanetary magnetic field (IMF) $B_{z}$, is analysed. Using the superposed epoch analysis, solar wind and geomagnetic conditions under three levels of $B_{z}$ are characterised, i.e. northward or $\rm{BZN} (B_{z}$ > $0 \rm{nT}$, 75 events), very weak or $\rm{BZ}0 (B_{z} \sim 0 \rm{nT}$, 78 events) and southward or $\rm{BZS} (B_{z}$ < $0 \rm{nT}$, 80 events). No northward turning is observed during BZS, while northward turning occurs 20 and 50 min prior to onset, during the BZN and BZ0 classes, respectively. IMF has a strong duskward component and the solar wind speed is also the fastestduring BZN onsets. Auroral activity, as measured by the AL index, takes a longer time to decay to preonset values during BZ0 and BZS onsets compared to the BZN onsets. The level of IMF $B_{z}$ does not seem to influence the oval thickness in the noon sector. The oval is the thinnest during BZ0 events in all the sectors. The rate of auroral widening in the dawn sector is found to be evidently slower than in the dusk sector during the BZS group.

• Synchronisation of integer-order and fractional-order discrete-time chaotic systems

This paper studies the synchronisation of integer- and fractional-order discrete-time chaotic systems with different dimensions. Control laws are proposed for the full-state hybrid projective synchronisation (FSHPS) of a master–slave pair, where the difference equations of the master have an integer order while those of the slave have a fractional order. Moreover, inverse FSHPS laws are proposed for a fractional-order master and an integer-order slave. The Lyapunov stability theory of integer-order maps and the stability theory of linear fractional-order mapsare utilised to establish the asymptotic stability of the zero equilibrium corresponding to the synchronisation error system. Numerical results are presented to verify the findings of the study.

• Minimal length Schrödinger equation via factorisation approach

The fourth-order modified Schrödinger equation due to the generalised uncertainty principle is considered in one dimension with a box problem. The factorisation of fourth-order self-adjoint differential equations is then discussed and thereby the wave functions and energy spectra of themodified Schrödinger equation are derived.

• Determination of molybdenum target parameters for transmission X-ray tube: A Geant4 simulation study

Carbon nanotube-based transmission X-ray tubes are widely used in different applications including mammography and X-ray fluorescence (XRF) experiment. Molybdenum (Mo) is one of the suitable target materials for this type of tube. In this paper, we used a well-known simulator, Geant4, to achieve some of the parameters of the Mo target. The optimum thickness for maximum production of usable X-ray from Mo target was obtained when it was exposed to the electron beam with an energy of 50 keV. In addition, according to the results, hemispheregeometry was recommended for the target at least for XRF application. By increasing target thickness, Mo acts as a filter resulting in limited X-ray energy. For estimating the target’s temperature generation, the amount of deposited energy was also evaluated.

• Comparative study of discharge characteristics and associated film growth for post-cathode and inverted cylindrical magnetron sputtering

In this study, an experimental investigation of a DC cylindrical magnetron discharge for argon gas in post-cathode (i.e. direct) and hollow-cathode (i.e. inverted) configurations was carried out. The discharge properties at different externally applied magnetic fields and operating pressures were measured and compared for both the configurations. The discharge current ($I$)–voltage ($V$) characteristics obey $I \propto V^{n}$, where the value of $n$ is in the range of 3–8. The discharge current increases linearly with the magnetic field in the post-cathode configuration, whereas it saturates at higher magnetic fields in the case of inverted configuration. Measurement of plasma potential indicated a considerable anode fall in the inverted magnetron configuration, whereas a negligible anode fall and strong cathode fall were observed in the case of post-cathode configuration. The plasma density and electron temperature, measured using a double Langmuir probe, were observed to be higher in the inverted magnetron configuration. The plasma density was found to be maximum at around 3–4 cm away from the respective inner electrode in both the configurations. A clear change in surface morphology of copper thin film was observed in the case of inverted magnetron configuration, which might be due to the extra ionisation near the anode owing to the anode fall.

• Electrical features in AlGaN/GaN high electron mobility transistors with recessed gate and undoped region in the barrier

This study considers electrical parameters of AlGaN/GaN high electron mobility transistor (HEMT) with the recessed gate and un-doped region (URG-HEMT) in the barrier layer. We have investigated the main electrical factors such as the lateral electric field, breakdown voltage ($V_{B}$), drain current ($I_{D}$), threshold voltage ($V_{T}$), output conductance ($g_{o}$) and gate capacitance ($C_{g}$). Simulation findings compare these parameters in the single heterostructure (SH-HEMT), recessed gate (RG-HEMT) and the proposed (URG-HEMT) structures. Regarding the simulation outcomes, the maximum lateral field in the URG is less than those in the SH and RG-HEMTs. This improves the breakdown voltage of the suggested device up to 160 V, while the breakdown voltage in the SH and RG transistors is about 90 V. Therefore, breakdown voltage of the reported device is about 80% larger than that of the other transistors. Also, undoped region in the novel transistor reduces the output conductance and gate-to-drain capacitance. But, the recessed gate and undoped regions in the URG structure decrease in 2-DEG electron density and then reduce drain current.

• Theoretical investigation of chemically reactive flow of water-based carbon nanotubes (single-walled and multiple walled) with melting heat transfer

This study reports the chemically reacting flow of carbon nanotubes (CNTs) over a stretchable curved sheet. The flow is initialised due to a stretched surface. A heat source is present. Water is considered as the base liquid. The vital interest of this work is that heat phenomenon is studied via melting heat transfer. Xue relation of nanoliquid is implemented to explain the properties of both single- and multiwall CNTs. Mathematical systems (partial differential equations) for the flow field are obtained. Appropriate transformations are utilised in order to transform partial differential systems into nonlinear ordinary differential systems. Further, these systems are solved numerically. Variations in flow, temperature, concentration, skin friction coefficient and Nusselt number via the involved influential variables are illustrated graphically.

• The impact of surface plasma on the total emission charge from PZST cathode induced by nanosecond electric pulse

Electron emission from antiferroelectric ($\rm{Pb_{0.99}Nb_{0.02})[(Zr_{0.80}Sn_{0.20})_{0.952}Ti_{0.048}]_{0.98}O_{3}}$ cathode has been investigated. The PZST cathode can maintain a metastable ferroelectric phase by the application of a high enough field, thus implying three possibly pulse-loading configurations for electron emission measurements. The fact that emission charge is larger than the non-compensated charge indicates that the surface plasma contributes to the total emission charge. Furthermore, $_{χi} (i = \rm{A, B, C})$, characterising the contribution of surface plasma to the total emission charge, was defined. It was found that the emission charge increases almost linearly with $_{χi}$. Our results are of great importance for a better understanding of electron emission in antiferroelectric/ferroelectric cathodes.

• Long-time dynamics of a vertical-cavity surface-emitting laser under optical feedback

External optical feedback in vertical-cavity surface-emitting laser (VCSEL) is found to influence its output intensity.We studied the effect of the amount of total output polarisation feedback and polarisation-selective feedback on the output intensities of a VCSEL for a low-resolution sampling and for long temporal duration. A 40 μs resolution time-series correlation analysis is performed for different feedback conditions and the characteristicdynamics is investigated.We found a correlated fluctuation in VCSEL output for a moderate amount of total feedbackand polarised feedback. The period of such fluctuations is found to be reduced from total feedback to the polarised feedback of the system.

• A statistical probe into the word frequency and length distributions prevalent in the translations of Bhagavad Gita

A statistical study has been conducted on Bhagavad Gita. Four measures have been derived for the original text in Sanskrit and its translations in Hindi, English and French. First, word frequency distributions for the documents were modelled. Power law was observed with the longest tail in the case of Sanskrit. For other versions, the distributions well replicated the Zipf–Mandelbrot pattern. Second, the Kullback–Leibler (KL) divergence betweenthe documents has been computed with the highest value recorded in all three translations from the Sanskrit text. Next, a Shannon entropy-based measure: vocabulary quotient has been calculated, which estimates the vocabulary richness the texts offer; the highest being in the case of Bhagavad Gita in Sanskrit. Finally, word-length distributions were obtained with the longest word length in Sanskrit. The results attribute to the inflectional nature of Sanskrit.

• Quantum quench dynamics of the one-dimensional Ising model in transverse field

The quantum quench dynamics of the one-dimensional Ising model in transverse field is investigated using the quantum renormalisation group method. The analytic expression of concurrence $C(t)$ is obtained, where the initial state is a superposition state which is constructed from the eigenstates of pure Ising system. The effect of parameter $a$ in the period and range of concurrence are exhibited respectively in the vicinity of the critical point, which show scaling behaviour. When effective magnetic field $g$ is big enough, the maximum limit value is 1.0. However, the minimum value is different, which is also dependent on the evolved time $t$.

• Cascading failure model for the mitigating edge failure of scale-free networks

By studying the classical betweenness and the universal degree methods, we put forward a new model to control the spread of cascading failure on scale-free networks. The new model is based on defining the load of an edge with respect to the betweenness centrality of the two connected nodes. The iterative process of a cascading failure on scale-free networks is analysed by removing one edge.We find that the proposed new model can control the spread of cascading failure more significantly. To make our conclusions more convincing, we have explored the performance of new models in real network by the power grid of the western United States. We further give the following reasonable explanations: First, the reason why the new model shows a more stable performance than the others has been explained. Secondly, we have explored the reason why the new model shows different advantages depending on the load for different networks and lastly, we have studied the exact difference between these two methods and the network structure. This paper might be useful for preventing and mitigating the cascading failure in real life.

• Brief report: Extremely dense general relativistic polytropes of index $n = 1$

Polytropic gas spheres of index 1 and extremely high central densities are analysed with the help of general relativistic field equations. Parameters such as the radius and mass are calculated for different central densities. The limiting values of these quantities are obtained and the physical nature of sound waves in these bodies is verified.

• C-matrix and invariants in chemical kinetics: A mathematical concept

To treat a realistic chemical system, such as a liquid phase dehydrogenation reaction, a chemical scheme, which describes the chemical kinetics in terms of the small number of reaction progress variables is needed. Based onthe matrix algebra, we analyse the key components, elements and reactions in the mechanism, C-matrix. Reduction techniques exploit the time-scale separation into fast and slow modes by computing the dimension reduced model via the elimination of fast mode subjecting them to the slow one. The two-step reversible reaction mechanism is considered for model reduction and to simplify the complexity of reaction mechanisms. They give a meaningful picture, but for maximum clarity, the phase flow of the solution trajectories near the equilibrium point is exploited. The Lyapunov function is applied for the stability analysis. To describe the physical behaviour of the reaction mechanism, graphical results are measured while refinement of the initial approximation is tabulated at the end.

• On the integrability of some two-component Camassa–Holm-type systems

Some two-component Camassa–Holm-type systems are proposed. These systems are shown to be integrable with Lax pairs and bi-Hamiltonian structures. We construct dual hierarchies of these two-component Camassa–Holm-type systems via the tri-Hamiltonian duality method, and derive spectral problems of these dual hierarchies.

• Asymptotic iteration and variational methods for Gaussian potential

In this paper we studied the approximate solutions of radial Schrödinger equation with the attractive Gaussian potential. We used asymptotic iteration method and variational method to obtain energy eigenvalues for any $n$ and $l$ quantum numbers. Our results are in good agreement with the other studies.

• Efficiency calibration of $\gamma$-ray detector for extended sources

Precise identification and quantification of radioisotope in a sample largely depend on the accuracy of the full-energy peak efficiency of the detector. It is generally observed that the efficiency for a given energy of point-like sources is not the same as the extended source. However, number of correction factors such as detector geometry, photon attenuation, coincidence-summing, etc. may reduce such a difference in efficiency regardless of sources, or they should be considered in the measurement of extended samples. In connection to this, the variation of absolute photopeak efficiency of a high-purity germanium (HPGe) detector with the distances from the surface of the detector as well as with the γ-ray energies were investigated using $\gamma$-ray standard point sources. We present a method to determine the loss of efficiencies in radioactivity measurement of the thin extended radioactive disc samples in comparison to point source.

• A modified efficiency centrality to identify influential nodes in weighted networks

It is still a crucial issue to identify influential nodes effectively in the study of complex networks. As for the existing efficiency centrality (EffC), it cannot be applied to a weighted network. In this paper, a modified efficiency centrality ($\rm{EffC}^{m}$) is proposed by extending EffC into weighted networks. The proposed measure trades off the node degree and global structure in a weighted network. The influence of both the sum of the average degree of nodes in the whole network and the average distance of the network is taken into account. Numerical examples are used to illustrate the efficiency of the proposed method.

• # Pramana – Journal of Physics

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