Rotational Jumps Research Articles

Diffusion of Ge Donors in β‐Ga2O3

Diffusion of Ge donors in β‐Ga2O3 is studied using a combination of secondary‐ion mass spectrometry, diffusion simulations, and first‐principles calculations, and compared to previous studies on Sn diffusion. Ge is implanted into (01)‐oriented samples and annealed at temperatures from 900 to 1050 °C for a total of 8 h. From previous first‐principles calculations, Sn is predicted to diffuse via the formation of a mobile complex with VGa that migrates through a sequence of exchange and rotation jumps. Herein, it is similarly predicted that Ge diffusion is mediated by VGa. However, the microscopic mechanism differs, as Ge can diffuse more easily through exchange combined with complex dissociation, rather than rotational jumps. This is explained by the difference in Ga‐site preference of Ge compared to Sn, and the three‐split mechanism that enables low migration barriers for VGa. The dissociation mechanism leads to a considerably faster transport for Ge as compared to Sn. The experimentally obtained Ge diffusion profiles are successfully fitted using a reaction–diffusion model based on the predicted diffusion mechanism, yielding a migration barrier of 2.5 ± 0.2 eV for the complex. The 2.72 eV obtained from first‐principles calculations is in good agreement with this value.

Evidence of ferroelectric features in low-density supercooled water from ab initio deep neural-network simulations

Over the last decade, an increasing body of evidence has emerged, supporting the existence of a metastable liquid-liquid critical point in supercooled water whereby two distinct liquid phases of different densities coexist. Analyzing long molecular dynamics simulations performed using deep neural-network force fields trained to accurate quantum mechanical data, we demonstrate that the low-density liquid phase displays a strong propensity toward spontaneous polarization, as witnessed by large and long-lived collective dipole fluctuations. Our findings suggest that the dynamical stability of the low-density phase, and hence the transition from high-density to low-density liquid, is triggered by a collective process involving an accumulation of rotational angular jumps, which could ignite large dipole fluctuations. This dynamical transition involves subtle changes in the electronic polarizability of water molecules which affects their rotational mobility within the two phases. These findings hold the potential for catalyzing activity in the search for dielectric-based probes of the putative second critical point.

Estimation of Ground Reaction Forces during Sports Movements by Sensor Fusion from Inertial Measurement Units with 3D Forward Dynamics Model.

Rotational jumps are crucial techniques in sports competitions. Estimating ground reaction forces (GRFs), a constituting component of jumps, through a biomechanical model-based approach allows for analysis, even in environments where force plates or machine learning training data would be impossible. In this study, rotational jump movements involving twists on land were measured using inertial measurement units (IMUs), and GRFs and body loads were estimated using a 3D forward dynamics model. Our forward dynamics and optimization calculation-based estimation method generated and optimized body movements using cost functions defined by motion measurements and internal body loads. To reduce the influence of dynamic acceleration in the optimization calculation, we estimated the 3D orientation using sensor fusion, comprising acceleration and angular velocity data from IMUs and an extended Kalman filter. As a result, by generating cost function-based movements, we could calculate biomechanically valid GRFs while following the measured movements, even if not all joints were covered by IMUs. The estimation approach we developed in this study allows for measurement condition- or training data-independent 3D motion analysis.

Актуалізація контролю загальної фізичної підготовленості спортсменів віком 12-13 років у таеквон-до ІТФ

The purpose of the study was to define the structure of means of general preparedness control for athletes aged 12-13 in taekwon-do ITF. Materials and methods. The following methods were used: the method of theoretical analysis and academic and methodological references data generalization; analysis of document materials; the method of expert polling; methods of mathematical and statistical data processing. The structure and content of Training programs in taekwon-do ITF, WTF and karate WKF, rules of competitions in taekwon-do ITF were studied. Experts (24 coaches) determined priority variants of shuttle running, tests to control speed qualities, endurance, strength endurance of various muscles groups, explosive power, speed and power endurance, flexibility of athletes aged 12-13 in taekwon-do ITF. Results and discussion. The issue of various aspects of preparedness control in the system of multi-year athletes’ improvement has always been topical. Over 70 separate means suggested by specialists in various versions of taekwon-do to control general physical preparedness have been revealed. The issue of parameter actualization for control of general preparedness of athletes aged 12-13 in taekwon-do ITF has emerged. The battery of tests in general physical preparedness of athletes aged 12-13 in taekwon-do ITF included: shuttle running 3x10 m (agility), 30-meter dash speed, 1000-meter run (endurance), chin-up and body lifting from the sitting position (power endurance), padded ball throw upfront and long jump from the spot (explosive power), skip rope jumping for 60 seconds, squats on one leg (30 sec), multiple jumps on one leg for 20 seconds (speed and power endurance), side splits, forward body lean in standing position, and “bridge” (flexibility).The following tests have been used to control coordination qualities: “8-shapes running with direction change” aimed at determination of general coordination; “rotation jumps (left-right)” – space perception; “the difference between forward and backwards jump” – distance perception; “three rotations forward – three rotations backwards” (time forward, time backwards, overall time) – keeping kinematic and dynamic motion characteristics; “Romberg’s test right/left” – statistical balance; dynamic balance and complex instrumental measurement of complex and specific reactions. Conclusion. Based on the research, parameters for control of general physical preparedness of athletes aged 12-13 years in taekwon-do ITF have been specified

Structural Phase Transitions in closo-Dicarbadodecaboranes C2B10H12.

The crystal structures of three thermal polymorphs (I, II, and III) for each isomer of closo-dicarbadodecaboranes C2B10H12 (ortho, meta, and para) have been determined by combining synchrotron radiation X-ray powder diffraction and density functional theory calculations. The structures are in agreement with previous calorimetric and spectroscopic studies. The difference between rotatory phases (plastic crystals) I and II lies in isotropic rotations in the former and anisotropic rotations of the icosahedral clusters in the latter. Phase I is the cubic close packing (ccp) of rotating closo-molecules C2B10H12 in the space group Fm3̅. Phase II is the ccp of rotating closo-molecules C2B10H12 in the cubic space group Pa3̅. The preferred rotational axis in II varies with the isomer. The ordered phases III are orthorhombic (meta) or monoclinic (ortho and para) deformations of the cubic unit cell of the disordered phases I and II. The ordering in the phase III of the ortho-isomer carrying the biggest electrical dipole moment creates a twofold superstructure w.r.t. the cubic unit cell. The thermal polymorphism for C2B10H12 and related metal salts can be explained by division of the cohesive intercluster interactions into two categories (i) dispersive cohesive interaction with additional Coulombic components in the metal salts and (ii) anisotropic local interaction resulting from nonuniform charge distribution around icosahedral clusters. The local interactions are averaged out by thermally activated cluster dynamics (rotations and rotational jumps) which effectively increase the symmetry of the cluster. The C2B10H12 molecules resist at least as well as the CB11H12– anion to the oxidation, and both clusters form easily a mixed compound. This allows designing solid electrolytes such as Nax(CB11H12)x(C2B10H12)1–x, where the cation content may be varied and the temperature of transition into the disordered conducting phase is decreased.

Utilizing correlation functions to develop a multistate model of P. denitrificans F1-ATPase and enhance single-molecule imaging resolution

F1-ATPase enzyme is a biological motor which hydrolyzes ATP. The enzyme has been observed via single-molecule imaging experiments wherein the enzyme is allowed to rotate freely while being recorded using a gold nanoparticle. During hydrolysis, the enzyme causes Brownian noise when rotating to new chemical states because of a size difference between the 4 nm enzyme and 40 nm probe. The unconvoluted rotary movement has been revealed using techniques including rotational correction, correlation functions, and comparison of average rotational jumps which contribute to developing a multistate model.

A novel upper bound finite-element for the limit analysis of plates and shells

Shear-locking is a classical issue in displacement-based finite-element approaches for thick plates and shells. It is even more important when considering kinematic limit analysis approaches which attempt at producing an upper bound estimate of the structure collapse load. Previous locking-free finite-element discretizations for thick plates either rely on mixed approaches or approximate strain compatibility relations which inevitably loose the upper bound status of the solution or on discontinuous interpolations which are difficult to implement and have a much higher computational cost. In this contribution, we investigate the use of a simple element with a continuous quadratic displacement and a piecewise linear rotation with continuity at the element mid-edges only. We show that this element can either produce strict upper-bound estimates taking into account the contribution of rotation jumps or a pseudo-upper bound when neglecting this contribution. Although the upper bound status is lost a priori in this case, numerical evidence indicate that limit loads usually converge from above and have a very good accuracy. Finally, we also use this element for shell problems and discuss in particular the formulation of strength criteria for thick shells. Illustrative applications show that the proposed element is free from any shear locking and produces very accurate limit load estimates for plate as well as shell problems.

PH-CPF

We present a new approach for computing planar hexagonal meshes that approximate a given surface, represented as a triangle mesh. Our method is based on two novel technical contributions. First, we introduce Coordinate Power Fields , which are a pair of tangent vector fields on the surface that fulfill a certain continuity constraint. We prove that the fulfillment of this constraint guarantees the existence of a seamless parameterization with quantized rotational jumps, which we then use to regularly remesh the surface. We additionally propose an optimization framework for finding Coordinate Power Fields, which also fulfill additional constraints, such as alignment, sizing and bijectivity. Second, we build upon this framework to address a challenging meshing problem: planar hexagonal meshing. To this end, we suggest a combination of conjugacy, scaling and alignment constraints, which together lead to planarizable hexagons. We demonstrate our approach on a variety of surfaces, automatically generating planar hexagonal meshes on complicated meshes, which were not achievable with existing methods.

Cohesiveness and Nondiffusive Rotational Jump Dynamics of Protic Ionic Liquid from Dispersion-Corrected FPMD Simulations.

We have investigated the liquid phase of an ionic liquid (IL), methylammonium formate (MAF), through the first principles molecular dynamics simulations using van der Waals (vdW) corrected exchange and correlation functionals of the density functional theory. The simulations were carried out to obtain a comparative study of various properties of the MAF using two different generalized gradient approximation functionals (Becke-Lee-Yang-Parr (BLYP) and Perdew-Burke-Ernzerhof (PBE)) along with three types of dispersion corrections (D2, D3, and dispersion-corrected atom-centered one-electron potentials), and two values of the plane-wave cutoff (300 and 600 Ry). We have evaluated the effects of various electronic parameters in describing the hydrogen-bonded structure and dynamical properties of MAF by performing 10 sets of molecular dynamics simulations. Thermodynamic properties are found to be sensitive to the details of electronic structure calculations. Our results of PBE functionals with the semiempirical vdW method provide the best agreement with experimental density. The overall density predictions match the cohesive energy trends, and the calculations incorporating dispersion forces exhibit enhanced intermolecular interactions within the hydrogen-bonded IL framework. All of the vdW-corrected BLYP functionals, mainly the dispersion-corrected atom-centered one-electron potential (DCACP) method, illustrate a well-defined structure of liquid MAF. To look into the dynamical perspective of the hydrogen-bond descriptions, we elucidate two possible mechanistic pathways of the hydrogen-bond jump events between the counterions. The hydrogen-bond breaking and forming mechanism along with the collision dynamics can be best described by incorporating dispersion interactions alongside the exchange and correlation functionals within the Kohn-Sham scheme. The rattling dynamics of ions are observed for dispersion-corrected functionals. Hence, an accurate representation of the delicately balanced interactive forces within ionic liquids is a necessary step toward a better description of its thermophysical and structural properties along with the associated ionic dynamics.

Structural Disorder in Li6PS5I Speeds 7Li Nuclear Spin Recovery and Slows Down 31P Relaxation-Implications for Translational and Rotational Jumps as Seen by Nuclear Magnetic Resonance.

Lithium-thiophosphates have attracted great attention as they offer a rich playground to develop tailor-made solid electrolytes for clean energy storage systems. Here, we used poorly conducting Li6PS5I, which can be converted into a fast ion conductor by high-energy ball-milling to understand the fundamental guidelines that enable the Li+ ions to quickly diffuse through a polarizable but distorted matrix. In stark contrast to well-crystalline Li6PS5I (10–6 S cm–1), the ionic conductivity of its defect-rich nanostructured analog touches almost the mS cm–1 regime. Most likely, this immense enhancement originates from site disorder and polyhedral distortions introduced during mechanical treatment. We used the spin probes 7Li and 31P to monitor nuclear spin relaxation that is directly induced by Li+ translational and/or PS43– rotational motions. Compared to the ordered form, 7Li spin–lattice relaxation (SLR) in nano-Li6PS5I reveals an additional ultrafast process that is governed by activation energy as low as 160 meV. Presumably, this new relaxation peak, appearing at Tmax = 281 K, reflects extremely rapid Li hopping processes with a jump rate in the order of 109 s–1 at Tmax. Thus, the thiophosphate transforms from a poor electrolyte with island-like local diffusivity to a fast ion conductor with 3D cross-linked diffusion routes enabling long-range transport. On the other hand, the original 31P nuclear magnetic resonance (NMR) SLR rate peak, pointing to an effective 31P-31P spin relaxation source in ordered Li6PS5I, is either absent for the distorted form or shifts toward much higher temperatures. Assuming the 31P NMR peak as being a result of PS43– rotational jump processes, NMR unveils that disorder significantly slows down anion dynamics. The latter finding might also have broader implications and sheds light on the vital question how rotational dynamics are to be manipulated to effectively enhance Li+ cation transport.

Gaze behavior of trampoline gymnasts during a back tuck somersault.

In trampolining, gymnasts perform a variety of rotational jumping elements and have to demonstrate perfect control of the body during the flying phase. The performance of a somersault should include an opening phase, i.e. the legs are fully extended pointing vertically at 180° called "kick-out". As previous studies have shown, gaze behavior is essential for the controlling during the flight phase and to prepare for a perfect landing. Gymnasts supposedly use the trampoline bed as orientation and differences in gaze behavior can be expected, depending on how a somersault is performed. The present study investigates the gaze behavior of gymnasts during a back tuck somersault on the trampoline. Eleven experienced trampoline gymnasts performed back tuck somersaults with and without a kick-out while wearing a light weight portable eye-tracking device. All subjects fixated their gaze on a specific point at the trampoline bed and thus used visual information to prepare for landing. During the period of fixation, gymnasts' eyes moved continuously downwards to counteract the backwards head movement. The point of fixation differed between each somersault. Apparently, the fixation position depended on the gymnast's landing position in the bed. Performing a somersault with a kick-out allows gymnasts to orient themselves earlier and thus prepare sooner for landing. Unexpectedly, gymnasts of a higher performance class fixated the bed later compared to less experienced athletes. Supposedly, gymnasts of a better class can allow themselves to fixate later in order to optimize the form and execution of a somersault.

Is there any relationship between glitches of Crab pulsar and Einstein-de Haas effect?

ABSTRACT Many pulsars show sudden rotation jumps, glitches, superimposed to the gradual rotation down due to the continued loss of energy by radiation. A lot of models have been proposed to explain glitches. Most of the models are associated with layer structure of neutron stars. We try to investigate possibility of relationship between glitches of Crab pulsar and Einstein-de Haas effect. In the scenarios, spin polarization of neutron matter with Skyrme-type interactions forms a macroscopic spin angular momentum, and the Einstein-de Haas effect – transfer between the macroscopic spin and orbit angular momentum – gives rise to glitches of pulsars. It is shown that the scenario is in agreement with the observations of Crab pulsar on glitches and post-glitches relaxation.

Ab initio study of phosphorus effect on vacancy-mediated process in nickel alloys – An insight into Ni2Cr ordering

The development of long range order in nickel-chromium alloys is of great technological interest but the kinetics and mechanisms of the transformation are poorly understood. The present research utilizes a combined computational and experimental approach to elucidate the mechanism by which phosphorus accelerates the ordering rate of stoichiometric Ni2Cr in Ni-Cr alloys. A series of Ni-33%Cr-x%P samples (in atomic percent) were fabricated with phosphorus concentrations, x = <0.005–0.1 at.% and aged between 373 and 470 °C for times up to 3000 h. The first-principles modeling considers fcc Ni with dilute P as a reasonable approximation for the complex Ni-Cr-P alloy system. Calculation results show a pronounced enhancement of vacancy transport by vacancy-solute pair diffusion via consecutive exchange and rotation jumps of vacancies associated with the phosphorus atom. The energy barriers of these two migration paths are at least 0.35 eV lower than that of vacancy-atom exchange in pure Ni solvent. The analytical diffusion model predicts enhanced solvent diffusion by 2 orders of magnitude for 0.1 at.% P at 400–500 °C. The model prediction is in good agreement with the evolution of micro-hardness. We characterize the micro-hardness result by a kinetic ordering model, showing a significant decrease of the activation energy of ordering transformation. These results help gauge the risk of industrial alloys developing long range order which increases strength but degrades ductility and toughness. Specifically, minor alloying additions that bind with excess vacancies and lower the vacancy migration barrier can greatly accelerate hardening via Ni2Cr precipitation.

State transition identification in multivariate time series (STIMTS) applied to rotational jump trajectories from single molecules.

Time resolved data from single molecule experiments often suffer from contamination with noise due to a low signal level. Identifying a proper model to describe the data thus requires an approach with sufficient model parameters without misinterpreting the noise as relevant data. Here, we report on a generalized data evaluation process to extract states with piecewise constant signal level from simultaneously recorded multivariate data, typical for multichannel single molecule experiments. The method employs the minimum description length principle to avoid overfitting the data by using an objective function, which is based on a tradeoff between fitting accuracy and model complexity. We validate our method with synthetic data from Monte Carlo simulations modeling fluorescence resonance energy transfer and rotational jumps, respectively. The method is applied to quantify rotational jump dynamics of single terrylene diimide (TDI) molecules deposited on a solid substrate. Depending on the substitution pattern of the TDI molecules and the chosen substrate materials, we find significant differences in time scale and geometry of molecular reorientation. From an additional application of our state transition identification in multivariate time series approach, a significant correlation between shifts of emission spectra and the occurrence of rotational jumps was found.

Nonlinear response theory for Markov processes. III. Stochastic models for dipole reorientations

The nonlinear response of molecular systems undergoing Markovian stochastic reorientations is calculated up to fifth order in the amplitude of the external field. Time-dependent perturbation theory is used to compute the relevant response functions as in earlier treatments (G. Diezemann, Phys. Rev. E{\bf 85}, 051502 (2012), Phys. Rev. E{\bf 96}, 022150 (2017)). Here, we consider the reorientational motion of isolated molecules and extend the existing calculations for the model of isotropic rotational diffusion to the model of anisotropic rotational diffusion and to the model of rotational random jumps. Depending on the values of some model parameters, we observe a hump in the modulus of the nonlinear susceptibility for either of these models. Interestingly, for the model of rotational random jumps, the appearance of this hump depends on the way the coupling to the external field is modelled in the master equation approach. If the model of anisotropic rotational diffusion is considered, the orientation of the diffusion tensor relative to the molecular dipole moment and additionally the amount of anisotropy in the rotational diffusion constants determine the detailed shape of the nonlinear response. In this case, the height of the observed hump is found to increase with increasing 'diffusional anisotropy'. We discuss our results in relation to the features observed experimentally in supercooled liquids.

Nondiffusive Rotational Jump Dynamics in Ethyl Ammonium Nitrate.

We examine the hydrogen bond jump mechanism in ionic liquid, ethyl ammonium nitrate (EAN), using classical molecular dynamics simulations. Hydrogen bond jump in EAN can occur through two different nondiffusive rotational jump mechanisms: N-H bond of ethyl ammonium can switch its hydrogen bond between two oxygen atoms of the same nitrate ion or it can break its hydrogen bond with the oxygen of a nitrate ion to form a new hydrogen bond with the oxygen atom of another nitrate ion. We observe the average magnitude of the jump angle of 30° in the first mechanism, whereas the jump angle for the second mechanism is 70°. The in-plane rotation of nitrate ion facilitates the H-bond switch in the first mechanism, whereas the rotation of the ammonium group of cation around the C-N bond facilitates the H-bond switch in the second mechanism. The jump angle observed in the second mechanism qualitatively agrees with experimentally observed large jump angle. We also investigate the effect of temperature on this nondiffusive rotational dynamics of ionic liquid to observe the changes in the jump angle and its distributions.

Nature of Decahydro-closo-decaborate Anion Reorientations in an Ordered Alkali-Metal Salt: Rb2B10H10

The ordered monoclinic phase of the alkali-metal decahydro-closo-decaborate salt Rb2B10H10 was found to be stable from about 250 K all the way up to an order–disorder phase transition temperature of ≈762 K. The broad temperature range for this phase allowed for a detailed quasielastic neutron scattering (QENS) and nuclear magnetic resonance (NMR) study of the protypical B10H102– anion reorientational dynamics. The QENS and NMR combined results are consistent with an anion reorientational mechanism comprised of two types of rotational jumps expected from the anion geometry and lattice structure, namely, more rapid 90° jumps around the anion C4 symmetry axis (e.g., with correlation frequencies of ≈2.6 × 1010 s–1 at 530 K) combined with order of magnitude slower orthogonal 180° reorientational flips (e.g., ≈3.1 × 109 s–1 at 530 K) resulting in an exchange of the apical H (and apical B) positions. Each latter flip requires a concomitant 45° twist around the C4 symmetry axis to preserve the ordered Rb2B10H10 monoclinic structural symmetry. This result is consistent with previous NMR data for ordered monoclinic Na2B10H10, which also pointed to two types of anion reorientational motions. The QENS-derived reorientational activation energies are 197(2) and 288(3) meV for the C4 fourfold jumps and apical exchanges, respectively, between 400 and 680 K. Below this temperature range, NMR (and QENS) both indicate a shift to significantly larger reorientational barriers, for example, 485(8) meV for the apical exchanges. Finally, subambient diffraction measurements identify a subtle change in the Rb2B10H10 structure from monoclinic to triclinic symmetry as the temperature is decreased from around 250 to 210 K.

Interplay between translational diffusion and large-amplitude angular jumps of water molecules.

Understanding the microscopic mechanism of water molecular translational diffusion is a challenging topic in both physics and chemistry. Here, we report an investigation on the interplay between the translational diffusion and the large-amplitude angular jumps of water molecules in bulk water using molecular dynamics simulations. We found that large-amplitude angular jumps are tightly coupled to the translational diffusions. Particularly, we revealed that concurrent rotational jumps of spatially neighboring water molecules induce inter-basin translational jumps, which contributes to the fast component of the water translational diffusion. Consequently, the translational diffusion shows positional heterogeneity; i.e., the neighbors of the water molecules with inter-basin translational jumps have larger probability to diffuse by inter-basin translational jumps. Our control simulations showed that a model water molecule with moderate hydrogen bond strength can diffuse much faster than a simple Lennard-Jones particle in bulk water due to the capability of disturbing the hydrogen bond network of the surrounding water molecules. Our results added to the understanding of the microscopic picture of the water translational diffusion and demonstrated the unique features of water diffusion arising from their hydrogen bond network structure compared with those of the simple liquids.

Kinetics of self-interstitial migration in bcc and fcc transition metals

Radiation damage is a multi-scale phenomenon. A thorough understanding of diffusivities and the migration energies of defects is a pre-requisite to quantify the after-effects of irradiation. We investigate the thermally activated mobility of self-interstitial atom (SIA) in bcc transition metals Fe, Mo, Nb and fcc transition metals Ag, Cu, Ni, Pt using molecular dynamics (MD) simulations. The self-interstitial diffusion involves various mechanisms such as interstitialcy, dumbbell or crowdion mechanisms. Max-Space Clustering (MSC) method has been employed to identify the interstitial and its configuration over a wide range of temperature. The self-interstitial diffusion is Arrhenius like, however, there is a slight deviation at high temperatures. The migration energies, pre-exponential factors of diffusion and jump-correlation factors, obtained from these simulations can be used as inputs to Monte Carlo simulations of defect transport. The jump-correlation factor shows the degree of preference of rectilinear or rotational jumps. We obtain the average jump-correlation factor of 1.4 for bcc metals and 0.44 for fcc metals. It indicates that rectilinear jumps are preferred in bcc metals and rotational jumps are preferred in fcc metals.

A mode-coupling theory analysis of the observed diffusion anomaly in aqueous polyatomic ions

In contrast to simple monatomic alkali and halide ions, complex polyatomic ions such as nitrate, acetate, nitrite, and chlorate have not been studied in any great detail. Experiments have shown that diffusion of polyatomic ions exhibits many remarkable anomalies; notable among them is the fact that polyatomic ions with similar size show large difference in their diffusivity values. This fact has drawn relatively little interest in scientific discussions. We show here that a mode-coupling theory can provide a physically meaningful interpretation of the anomalous diffusivity of polyatomic ions in water, by including the contribution of rotational jumps on translational friction. The two systems discussed here, namely, aqueous nitrate ion and aqueous acetate ion, although have similar ionic radii, exhibit largely different diffusivity values due to the differences in the rate of their rotational jump motions. We have further verified the mode-coupling theory formalism by comparing it with experimental and simulation results that agree well with the theoretical prediction.