Particle Spectra Research Articles

Simulation study of a simultaneous beta–gamma-ray detection using a 3-layer phoswich detector and Monte Carlo methods

Combination of two or three dissimilar scintillator materials as a radiation detector has found major role in environmental radiation monitoring. In this paper, a three-layer Phoswich detector including BC-400, YAG, and CsI was designed to efficiently discriminate gamma-ray in the beta events up to 3.2 MeV using a simple rise-time discrimination method. MCNPX Monte Carlo code was used to obtain interaction probability of beta and gamma-rays as well as optimum thicknesses of the layers in the designing process. The optical transport of the system was simulated by GEANT4. In this regard, the pulses from simultaneous beta-gamma emitter sources were detected and discriminated based on pulse's rise-time so that the minimum number of gamma-ray contaminating events was observed in the beta spectrum. The results showed that using the proposed configuration and the method, output pulses with a rise-time shorter than 9 ns have been successfully detected as a beta particle while those with rising time longer than 15 ns have been identified as gamma-ray events. Overall results revealed that using the proposed system, an individual spectrum of beta particles or gamma-rays can be recorded from a simultaneous beta-gamma emitter source that minimizes contribution of the other radiation.

Evolution of massive black hole binaries in collisionally relaxed nuclear star clusters – Impact of mass segregation

ABSTRACT Massive Black Hole (MBH) binaries are considered to be one of the most important sources of Gravitational Waves (GW) that can be detected by GW detectors like LISA. However, there are a lot of uncertainties in the dynamics of MBH binaries in the stages leading up to the GW-emission phase. It has been recently suggested that Nuclear Star Clusters (NSCs) could provide a viable route to overcome the final parsec problem for MBH binaries at the centre of galaxies. NSCs are collisional systems where the dynamics would be altered by the presence of a mass spectrum. In this study, we use a suite of high-resolution N-body simulations with over 1 million particles to understand how collisional relaxation under the presence of a mass spectrum of NSC particles affects the dynamics of the MBH binary under the merger of two NSCs. We consider MBH binaries with different mass ratios and additional non-relaxed models. We find that mass-segregation driven by collisional relaxation can lead to accelerated hardening in lower mass ratio binaries but has the opposite effect in higher mass ratio binaries. Crucially, the relaxed models also demonstrate much lower eccentricities at binary formation and negligible growth during hardening stages leading to longer merger time-scales. The results are robust and highlight the importance of collisional relaxation on changing the dynamics of the binary. Our models are state-of-the-art, use zero softening, and high enough particle numbers to model NSCs realistically.

Quiver Symmetries and Wall-Crossing Invariance

We study the BPS particle spectrum of five-dimensional superconformal field theories on $$\mathbb {R}^4\times S^1$$ with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi–Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi–Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich–Soibelman wall-crossing invariant. We solve these equations iteratively: the pattern arising from the solution is naturally extended to an exact conjectural expression, that we provide for the local Hirzebruch $$\mathbb {F}_0$$ , and local del Pezzo $$dP_3$$ and $$dP_5$$ geometries. Remarkably, the BPS spectrum consists of two copies of suitable 4d $$\mathcal {N}=2$$ spectra, augmented by Kaluza-Klein towers.

Particle Production in Strong Electromagnetic Fields and Local Approximations

We investigate the phenomenon of electron–positron pair production in intense external backgrounds within the strong-field regime. We perform nonperturbative calculations by solving the quantum kinetic equations, and obtain the momentum distributions of particles created and the total number of pairs. In particular, we analyze the validity of the locally constant field approximation (LCFA), which represents a powerful method for treating inhomogeneous external backgrounds. We consider a combination of two consecutive time-dependent Sauter pulses and thoroughly examine the effects of quantum interference and the role of the Pauli exclusion principle. It is shown that the latter can be approximately incorporated within the LCFA when computing the momentum distributions, while the closed-form LCFA expression for the total particle yield completely disregards Pauli blocking. It is demonstrated that in the presence of multiple turning points of classical electron trajectories, one observes interference patterns in the particle spectra, and the LCFA may significantly overestimate the number of pairs. To further elaborate this issue, we perform the analogous calculations in the case of scalar QED. It is shown that the quantum statistics effects enhance the number of bosons produced.

TAUOLA update for decay channels with e+e− pairs in the final state

With the arrival of high luminosity B-factories like the Belle II experiment, τ decay measurements have become more precise than ever, allowing rarer processes to be explored, and finer details of τ decays to be studied. These are important to understand the spectrum of intermediate particles produced in τ decays. Therefore Monte Carlo generators, like the TAUOLA program, have to facilitate precision analysis as well as confront new models that constantly emerge with the availability of high statistics experimental data. New decay channels and models may lead to large variation of matrix elements size within the available phase space, as in the case when e+e− pairs are present in final state. It requires appropriate presampler of the phase space generator and a proper documentation to help users introduce their own models. While releasing a new update, it is important for the TAUOLA Monte Carlo library to maintain the general structure of the previous versions to preserve backward compatibility. The aim is to minimize changes from the user perspective. This paper presents a demonstration of new models implementation facilitated by the current update. Program summaryProgram Title: tauola-bbbCPC Library link to program files:https://doi.org/10.17632/3hyw3f7v32.1Licensing provisions: GPLv2Programming language:FORTRAN/C++Journal reference of previous version: Comput. Phys. Comm., 232:220–236, 2018Does the new version supersede the previous version?: NoReasons for the new version: Meeting the needs of present day experiments.Summary of revisions: Phase space presamplers enriched with the possibility to generate narrow peaks of e+e− pairs, if present in matrix elements of 4- and 5-body decay channels and due to electromagnetic or New Physics interactions. The examples of physical models are implemented for: τ−→π−e−e+ντ, τ−→ν¯μμ−e−e+ντ and τ−→ν¯ee−e−e+ντ channels, which are used for the collected new algorithm validations.Nature of problem: Present day experiments, in particular Belle II, have the capability to measure extremely rare, not yet measured τ decays, as well as precise measurements of known decay channels. They require a Monte Carlo generator capable of generating all desired decays. They also need the possibility to modify and test new models for such decays. At the same time the default initialization of the event generator should contain decent modeling of known τ decays.Solution method: The new version supplements previous publications on TAUOLA. Phase-space presamplers and new decay channels with additional light lepton pairs are introduced.Additional comments including restrictions and unusual features: TAR BALL of the future versions of the program may become available through:http://wasm.web.cern.ch/wasm/Welcome.html; its sub-pages http://tauolapp.web.cern.ch/tauolapp/ or http://wasm.web.cern.ch/wasm/f77.html

Ultra high energy cosmic rays from past activity of Andromeda galaxy

ABSTRACT It is shown that the relativistic jets associated with the growth and past activity of the supermassive black hole in the Andromeda galaxy could be the main source of cosmic rays with energies above 1015 eV. Most of the cosmic ray energy is related to a bow shock of the jet that produces multi PeV cosmic rays with light composition. The highest energy cosmic rays with heavy composition are produced in the jet itself. The spectra of energetic particles produced in Andromeda galaxy and propagated to the Earth are calculated and compared with observations.

Multifield Coupling of Local Surface Plasmon Resonance‐Assisted Au–TiO2 Photocatalysis Considering Bimodal Resonance and Superposition Effect

This study concerns the improvement of the synergetic catalytic efficiency of Au–TiO2 nanorods considering the local surface plasmon resonance (LSPR) effect of different Au nanoparticles (NPs). The absorption spectrum and selective absorption efficiency for visible light, the local electric field, and the generated thermal effect are specifically analyzed based on the multifield decoupling of LSPR‐assisted Au–TiO2 catalysts. The simulation results show that the extinction spectra of both spherical and ellipsoidal Au particles are consistent with the experimental data. Interestingly, the latter is characterized by significant bimodal resonance modes. Comparatively, the simulation results show that the longitudinal mode, which is sensitive to the aspect ratio, is more favorable for the improvement of the photocatalytic activities. It is found that the resonance peaks are highly controllable, and are linear to the particle size and aspect ratio. Meanwhile, the electric field mode of TiO2 is significantly increased under the resonance wavelength. It is worth mentioning that the superposition effect makes a non‐negligible impact on the actual catalysts, leading to a relative shift of resonance wavelength. The consideration of the hot spots caused by the superposition effect influence the photocatalytic results significantly, providing values in diminishing the inadaptability of the theory in near‐touching regions of plasma particles.

Phenomenological implications of the new Littlest Higgs model with T-parity

We investigate the parameter space of the new Littlest Higgs model with T-parity (NLHT) recently introduced to cure some pathologies of the original LHT. The model requires extra fermion content and additional pseudo-Goldstone bosons. While the heavy top quark sector is similar, there are both T-odd and T-even heavy quarks and leptons with masses proportional to just two sets of Yukawa matrices in flavor space, one more than in the LHT. The new scalars are a singlet and real triplet, T-odd, with masses controlled by gauge and Yukawa couplings, independent of the spontaneous symmetry breaking scale f, and hence potentially light. Imposing that no mass exceeds the cutoff scale, applying current lower bounds on vector-like quarks and assuming a simplified model with mass degenerate heavy fermions compatible with the heavy photon as dark matter constituent, we find that f gets constrained within the interval between 2 and 3 TeV, the common Yukawa coupling of heavy leptons gets fixed and the Yukawa coupling of heavy quarks becomes greatly correlated to the top quark Yukawa couplings. The particle spectrum is then bounded from below and above, with the (lightest) heavy photon at about 0.5 TeV, not far from the heavy leptons, the new scalars below 1 TeV, the usual complex scalar triplet close to the heavy weak bosons at about 1.5 to 2.5 TeV, and the heavy quarks and top quark partners between 2 and 5 TeV. The new scalars decay predominantly to a standard and a T-odd lepton and have a width comparable to that of the Higgs.

Inverse design of core-shell particles with discrete material classes using neural networks

The design of scatterers on demand is a challenging task that requires the investigation and development of novel and flexible approaches. In this paper, we propose a machine learning-assisted optimization framework to design multi-layered core-shell particles that provide a scattering response on demand. Artificial neural networks can learn to predict the scattering spectrum of core-shell particles with high accuracy and can act as fully differentiable surrogate models for a gradient-based design approach. To enable the fabrication of the particles, we consider existing materials and introduce a novel two-step optimization to treat continuous geometric parameters and discrete feasible materials simultaneously. Moreover, we overcome the non-uniqueness of the problem and expand the design space to particles of varying numbers of shells, i.e., different number of optimization parameters, with a classification network. Our method is 1–2 orders of magnitudes faster than conventional approaches in both forward prediction and inverse design and is potentially scalable to even larger and more complex scatterers.

Upon the horizon’s verge: Thermal particle creation between and approaching horizons

Quantum particle creation from spacetime horizons, or accelerating boundaries in the dynamical Casimir effect, can have an equilibrium, or thermal, distribution. Using an accelerating boundary in flat spacetime (moving mirror), we investigate the production of thermal energy flux despite non-equilibrium accelerations, the evolution between equilibrium states, and the ‘interference’ between horizons. In particular, this allows us to give a complete solution to the particle spectrum of the accelerated boundary correspondence with Schwarzschild–de Sitter spacetime.

BCS solutions and effective quark energies of the QCD Hamiltonian in the Coulomb gauge

The search of theoretical approximations adequate to the description of the non-perturbative regime of QCD, that is the low-energy portion of the hadron spectrum, requires the adoption of notions more frequently applied to other, more conventional, quantum many body systems, like the atomic nucleus, solid state systems, etc. The identification and the use of effective degrees of freedom is one of the notions whose limitations we shall explore in this work. To start with the choice of the basis to be adopted to represent effective fermionic degrees of freedom is a crucial first step towards the goal of constructing an effective Hamiltonian from which the low energy portion of the hadronic spectrum may emerge as its solution. Here, we propose the use of the harmonic oscillator basis and emphasize its advantages compared to the basis of free momentum states. The starting Hamiltonian is the effective Coulomb plus linear potential, which we have used in previous calculations. We proceed by pre-diagonalizing the Hamiltonian to build a single particle spectrum, namely: the spectrum of effective quark degrees of freedom, starting from arbitrary chosen quark masses, which are later on re-normalized. Then, quark-pair correlations are described in the context of the well-known Bogoliubov transformations. The resulting quasiparticle states are then used to construct meson-like states. The dependence of the results upon the parameters which enter in the calculations is explored in detail.

Random Statistical Analysis of Transverse Momentum Spectra of Strange Particles and Dependence of Related Parameters on Centrality in High Energy Collisions at the LHC

We have studied the transverse momentum ( p T ) spectra of the final-state strange particles, including K ± , ϕ , Ξ , and Ω , produced in high energy lead–lead (Pb–Pb), proton–lead ( p –Pb), xenon–xenon (Xe–Xe) collisions at the Large Hadron Collider (LHC). Taking into account the contribution of multiquark composition, whose probability density distribution is described by the modified Tsallis–Pareto-type function; we simulate the p T spectra of the final-state strange particles by a Monte Carlo method, which is shown to be in good agreement with the experimental data in most the cases. The kinetic freeze-out parameters are obtained. The present method provides a new tool for studying the spectra of various particles produced in high energy collisions, reflecting more realistically the collision process, which is of great significance to study the formation and properties of the produced particles.

Hazma meets HERWIG4DM: precision gamma-ray, neutrino, and positron spectra for light dark matter

We present a new open-source package, Hazma 2, that computes accurate spectra relevant for indirect dark matter searches for photon, neutrino, and positron production from vector-mediated dark matter annihilation and for spin-one dark matter decay. The tool bridges across the regimes of validity of two state of the art codes: Hazma 1, which provides an accurate description below hadronic resonances up to center-of-mass energies around 250 MeV, and Herwig4DM, which is based on vector meson dominance and measured form factors, and accurate well into the few GeV range. The applicability of the combined code extends to approximately 1.5 GeV, above which the number of final state hadrons off of which we individually compute the photon, neutrino, and positron yield grows exceedingly rapidly. We provide example branching ratios, particle spectra and conservative observational constraints from existing gamma-ray data for the well-motivated cases of decaying dark photon dark matter and vector-mediated fermionic dark matter annihilation. Finally, we compare our results to other existing codes at the boundaries of their respective ranges of applicability. Hazma 2 is freely available on GitHub at https://github.com/LoganAMorrison/Hazma.

Recoil and charged particle energy spectra from the natSi(n,x) reaction and the Si semiconductor detector response to the 14 MeV neutrons

The energy spectra of light and heavy charged particles generated by the 14 MeV neutrons in silicon nuclei and in the Si semiconductor detector (SSD) were analysed. The existing measurements were used to validate the corresponding energy differential cross sections from major evaluations ENDF/B-VIII.0 and JEFF-3.3. The required data were extracted by the various codes: NJOY21, SPKA and MCNP6. It was shown that both evaluations underestimate the high energy part of the 28Si(n,x)p- and α-particle spectra by ≤50%, whereas at smaller emission energies no experimental data even exist. The calculation of the atom displacement damage in SSD reveals that the secondary protons produce additionally ≈15% of damage, whereas the contribution from secondary alphas is impossible to estimate since the Si(α,x) recoil data are not available.The single experiment, carried out by K. Yageta and co-workers in 1988, reported the energy spectra in the Si semiconductor down to the rather low energy of ≈0.3 MeV, at which the production of reaction recoils dominates over the light ions. The Monte Carlo simulation of this experiment was performed by MCNP6. The generation and transport of all relevant ions were taken in account. The energy depositions from protons or alphas were summed with their associated recoils after correction for non-ionization losses. The neutron reactions in the dead layer of the SSD were shown to increase the counts in active layer by (1–20)% depending on sort of ions. The impact of the field funneling effect was studied. Consideration of all these underlying processes has eventually allowed a reasonable reproduction of this experiment.

A Parameterization of Interstitial Aerosol Extinction and Its Application to Marine Cloud Brightening

Abstract Marine cloud brightening (MCB) is a geoengineering approach to counteract climate change by the deliberate seeding of sea salt aerosol particles that, once they activated to cloud droplets, directly increase cloud reflectance and hence global albedo. However, a large fraction of the seeded aerosol may remain interstitial, i.e., unactivated particles among cloud droplets. Because the consideration of interstitial aerosol optical properties usually requires computationally expensive simulations of the entire particle spectrum and direct Mie calculations, we develop a simple parameterization to be used with computationally efficient bulk and even bin cloud microphysical schemes that do not treat the unactivated aerosol explicitly. Using parcel and large-eddy simulations with highly detailed Lagrangian cloud microphysics and direct Mie calculations as a reference, we show that the parameterization captures the variability in the interstitial aerosol extinction successfully. By applying the parameterization to typical MCB cases, we find that the consideration of interstitial aerosol extinction is important for the assessment of MCB in shallow clouds with weak updrafts, in which only a small fraction of aerosol particles is activated to cloud droplets. Significance Statement The optical properties of clouds are not only determined by the number and size of cloud droplets. Unactivated aerosol particles, so-called interstitial aerosol, can contribute substantially to the optical thickness of shallow clouds with weak updrafts in aerosol-laden conditions. The consideration of interstitial aerosol optical thickness has been computationally challenging, but the new parameterization presented here allows for an efficient representation in various types of cloud models. The parameterization is shown to be an important addition for the assessment of marine cloud brightening (MCB), a potential geoengineering technique to counteract global warming by increasing the cloud albedo through the deliberate seeding of aerosol.

A numerical study of the interplay between Fermi acceleration mechanisms in radio lobes of FR-II radio galaxies

Context. Radio-loud active galactic nuclei (AGNs) are thought to possess various sites of particle acceleration, which gives rise to the observed nonthermal spectra. Stochastic turbulent acceleration (STA) and diffusive shock acceleration (DSA) are commonly cited as potential sources of high-energy particles in weakly magnetized environments. Together, these acceleration processes and various radiative losses determine the emission characteristics of these extragalactic radio sources. Aims. The purpose of this research is to investigate the dynamical interplay between the STA and DSA in the radio lobes of FR-II radio galaxies, as well as the manner in which these acceleration mechanisms, along with a variety of radiative losses, collectively shape the emission features seen in these extragalactic sources. Methods. A phenomenologically motivated model of STA is considered and subsequently employed on a magnetohydrodynamically simulated radio lobe through a novel hybrid Eulerian-Lagrangian framework. Results. STA gives rise to a curved particle spectrum that is morphologically different from the usual shock-accelerated spectrum. As a consequence of this structural difference in the underlying particle energy spectrum, various multiwavelength features arise in the spectral energy distribution of the radio lobe. Additionally, we observe enhanced diffuse X-ray emission from radio lobes for cases where STA is taken into account in addition to DSA.

Implementation and validation of realistic (n, x) reaction yields in Geant4 utilizing a detailed evaluated nuclear reaction library below 20 MeV

Neutron-induced reactions with charged particle emission play an important role in a variety of research fields ranging from fundamental nuclear physics and nuclear astrophysics, to applications of nuclear technologies, to energy production, and material science. Recently, the capability to study reactions with radioactive targets has become important to significantly advance research in explosive nucleosynthesis and nuclear applications. To achieve the relevant research goals and study (n, x) reactions over a broad neutron beam energy range, the Low Energy Neutron-induced charged-particle (Z) chamber (LENZ) at Los Alamos Neutron Science Center (LANSCE) was developed, along with varied ancillary instrumentation, to enable the aforementioned research program. For the (n, x) reactions of interest at low energies, a precise simulation of the discrete spectrum of emitted charged particles is essential. In addition, since LANSCE is a user facility, a simulation of the LENZ setup easily accessible by users has a high value. With these goals in mind, we have developed a detailed simulation using the Geant4 toolkit. In this work, we present the implementation and the validation of the simulation using experimental data from recent campaigns with the LENZ apparatus. Specifically, we benchmark the simulation against a similar MCNP-based application, and determine the realistic range of applicability for the utilized probability biasing technique. We describe the implementation of an evaluated library with angular distribution and partial cross-section data, and we perform a validation of the simulation based on comparisons of simulated spectra with experimental ones for a number of targets used in previous experimental campaigns. Lastly, we discuss the limitations, caveats, and assets of the simulation code and techniques used.

Effects of electrode fabrication on electrical properties of CdMgTe room temperature radiation detectors

Cd1-xMgxTe (CdMgTe) is an ideal alternative material for room temperature radiation detectors. In this paper, the effects of different electrode structures on the performance of CdMgTe detectors were studied. The electrodes were prepared by vacuum evaporation and chemical precipitation method using Au and Al materials. The results indicated that CdMgTe detectors with Au/CdMgTe/Al electrode structure prepared by vacuum evaporation method had the best electrical properties. For the crystals grown by Cd-excess and Te-rich conditions, the resistivity were 2.89 × 1011 Ω cm and 3.8 × 1011 Ω cm, the leakage current (under 100 V bias voltage) were 4 nA and 3 nA, and the Hall mobility were 2.73 × 103 cm2 V−1 s−1 and 3.69 × 102 cm2 V−1 s−1, respectively. The conductive type of the crystals were weak n-type. 241Am @ 5.48 MeV α particle spectra of the detectors showed that the optimal energy resolution were 8.66% and 8.62%, and the carrier mobility-lifetime products (μτ)e were 1.47 × 10−3 cm2/V and 1.03 × 10−3 cm2/V, respectively. Moreover, the carrier mobility obtained by time-of-flight (TOF) test were 15.01 cm2/(V·s) and 77.79 cm2/(V·s), respectively.

Half-integer Wannier diagram and Brown-Zak fermions of graphene on hexagonal boron nitride

The moir\'e potential of graphene on hexagonal boron nitride (hBN) generates a supercell sufficiently large as to thread a full magnetic flux quantum $\Phi_0$ for experimentally accessible magnetic field strengths. Close to rational fractions of $\Phi_0$, $p/q \cdot\Phi_0$, magnetotranslation invariance is restored giving rise to Brown-Zak fermions featuring the same dispersion relation as in the absence of the field. Employing a highly efficient numerical approach we have performed the first realistic simulation of the magnetoconductance for a 250 nm wide graphene ribbon on hexagonal boron nitride using a full ab-initio derived parametrization including strain. The resulting Hofstadter butterfly is analyzed in terms of a novel Wannier diagram for Landau spectra of Dirac particles that includes the lifting of the spin and valley degeneracy by the magnetic field and the moir\'e potential. This complex diagram can account for many experimentally observed features on a single-particle level, such as spin and valley degeneracy lifting and a non-periodicidy in $\Phi_0$.

Analyses of the collective properties of hadronic matter in Au-Au collisions at 54.4 GeV

We investigated the strange hadrons transverse momentum (${p}_{T}$) spectra in Au-Au collision at $\sqrt{{s}_{NN}}=54.4\text{ }\text{ }\mathrm{GeV}$ in the framework of modified Hagedorn function with embedded flow. It is found that the model can describe the particle spectra well. We extracted the kinetic freeze-out temperature ${T}_{0}$, transverse flow velocity ${\ensuremath{\beta}}_{T}$, kinetic freeze-out volume $V$, mean transverse momentum $⟨{p}_{T}⟩$, the entropy parameter $n$, and the multiplicity parameter ${N}_{0}$. We reported that all these parameters increase towards the central collisions. The larger kinetic freeze-out temperature, transverse flow velocity, kinetic freeze-out volume, and the entropy parameter ($n$) in central collisions compared to peripheral collisions show the early decoupling of the particles in central collisions. In addition, all the above parameters are mass dependent. The kinetic freeze-out temperature (${T}_{0}$), the entropy parameter $n$, and mean transverse momentum ($⟨{p}_{T}⟩$) are larger for massive particles, while the transverse flow velocity (${\ensuremath{\beta}}_{T}$), kinetic freeze-out volume ($V$), and the multiplicity parameter (${N}_{0}$) show the opposite behavior. Larger ${T}_{0}$, $n$, and smaller ${\ensuremath{\beta}}_{T}$ as well as $V$ of the heavier particles indicate the early freeze-out of the heavier particles, while larger $⟨{p}_{T}⟩$ for the heavier particles evince that the effect of radial flow is stronger in heavier particles. The separate set of parameters for each particle shows the multiple kinetic freeze-out scenario, where the mass-dependent kinetic freeze-out volume shows the volume differential freeze-out scenario. We also checked the correlation among different parameters, which include the correlation of ${T}_{0}$ and ${\ensuremath{\beta}}_{T}$, ${T}_{0}$ and $V$, ${\ensuremath{\beta}}_{T}$ and $V$, $⟨{p}_{T}⟩$ and ${T}_{0}$, $⟨{p}_{T}⟩$ and ${\ensuremath{\beta}}_{T}$, $⟨{p}_{T}⟩$, and $V$, $n$ and ${T}_{0}$, $n$ and ${\ensuremath{\beta}}_{T}$, and $n$ and $V$, and they all are observed to have positive correlations with each other which validates our results.