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A simple analytic model valid for all collisionality regimes is developed to describe the power deposition in a cylindrical inductively coupled plasma source with a planar coil. The heating is ohmic at high pressures and remains finite at low pressures. The low‐pressure collisionless heating is due to kinetic nonlocal effects. The model is in good agreement with other calculations of collisionless heating. A diffusion model is then used to determine the plasma density profile and the electron temperature in terms of the gas pressure and the source geometry. The heating and diffusion models are used to determine the scaling of the inductive electric field with applied frequency and input power, and the results are compared with published experimental data to verify the scaling.

An analysis is given of the influence of the electron ponderomotive force on the equilibrium plasma profiles of partially ionized, radio frequency discharge sources, The ponderomotive force can be written as a gradient of a potential varying with the square of the RF field in the plasma and is largest for electrons, The impact of this electron ponderomotive force on density and electrostatic potential profiles is demonstrated using a one-dimensional analytic model with supporting numerical solutions and a two dimensional fluid simulation. For nearly collisionless plasmas the ponderomotive force is valid when /spl omega//sub ce//sup h///spl omega/ 1, the ponderomotive force, is modified by other nonlinear force terms that need to be evaluated. >

The two‐dimensional (R−Z) electromagnetic code fmrz has been used to model inductive sources self‐consistently in time. The code models an argon plasma with momentum transfer, excitation, and ionization as electron–neutral reactions and momentum transfer for the ion–neutral reactions. The electrons and ions are treated as fluid species and a reduced set of Maxwell’s equations is used to advance the electromagnetic fields. The set of equations used in fmrz is not subject to typical numerical constraints present in many time dynamic codes, allowing one to choose appropriate time and space scales to resolve only the frequencies and scale lengths of interest. The model retains nonlinear driving terms which give rise to a ponderomotive force that distorts the density profile. Density and power profiles show the physical effects of various terms in the equations. Trends in average density and temperature compare well with an analytic model and other simulation models.

A 2 MV, 800 mA, K+ injector for heavy-ion fusion studies is under construction. This new injector is a one-beam version of the proposed 4-beam ILSE injector. A new 36-module MARX is being built to achieve a 5 μs flat top. The high-voltage generator is stiff (

The Heavy-Ion Fusion Accelerator Research Program at Lawrence Berkeley Laboratory has proposed building a 10 MeV induction linac systems experiment, ILSE, to investigate accelerator physics and beam manipulations which are needed or desirable for an induction linac driver. This paper describes the experiments proposed for ILSE: transverse beam combining, drift compression, bending of space-charge-dominated beams, final focus, recirculation, and some studies of beam propagation in the environment of the reactor chamber.

Several plasma instabilities driven by a heavy ion beam impinging on a target are considered. Potential instabilities include longitudinal electrostatic (two-stream) instabilities and transverse electromagnetic (filamentation or Weibel) instabilities. Since the ion beam may be accompanied by electrons, both electron beam and ion beam instabilities are considered. The electrostatic beam instabilities saturate by formation of an electron tail. In this paper the filamentation instability driven by this electron energy anisotropy is shown to be benign; this confirms the results of earlier studies in which the slower ion filamentation instability was shown to be unimportant. The beam ion, target electron instability is found to be suppressed by the gradient in the target density. Moreover, even if it were to grow, it would saturate at a benign level. In conclusion, no significant problem due to corona plasma instabilities in heavy ion fusion applications has been found.

Summary form only given. In previous work developing the Complex Particle Kinetics (CPK) method we utilized fragmentation, merging, and self-consistent particle evolution (Hewett, 2003) along with a binary collision algorithm (Larson, 2003) in our pursuit of a method that would simulate the transition from continuum to fully kinetic physics. CPK is built upon simulation particles that carry a Gaussian spatial profile and an internal Maxwellian velocity distribution. We recently incorporated ideas from the Finite Mass Method (Gauger et al., 2000) and have completely changed how the particles interact. The self-consistent particle evolution has also been generalized. The simulation particles move under the influence of internal and external forces and can expand, contract, and rotate as well as overlap and pass through one another. The resulting algorithm captures the continuum limit without prohibitive expense and smoothly transitions to the kinetic limit. Particle fragmentation and merging are still viable options within the algorithm and will be exploited to increase resolution where required and decrease expense where feasible. We will present an overview of the algorithm and the results of some recent simulations.

Summary form only given. A new type of "smart PIC" algorithm, intended to bridge the gap between Eulerian fluid and kinetic regimes, is now being used for a variety of applications in ICF and weapon effects. The CPK method (complex particle kinetic) concept [Hewett, 2003] uses an ensemble of small, fluid-like macro-particles to represent particle distributions in phase space. These macro-particles are Gaussian-shaped in both position and velocity with internal parameters such as width, thermal velocity, total mass and charge in addition to the traditional PIC like Lagrangian location and drift velocity. Time evolution is modeled by a combination of Lagrangian motion and internal evolution within each individual macro-particle. In regions of "high activity", macro-particles are aggressively fragmented in phase space to probe for emerging kinetic features and aggressively merged, for economy, if interesting features fail to materialize. Exploiting the shape of the CPK particle, fragmentation in both position and velocity space can be accomplished without loss of significant phase space information. Fragmentation preserves the kinetic capabilities of PIC; merging dramatically shrinks the number of particles in non-kinetic or collisional regions. In collision-dominated regimes, merging naturally produces a few Lagrangian particles that act much as nodes in free-Lagrangian hydrodynamics. Recent improvements [Larson, 2003], allow inter-particle interactions between those particles within a mean-free-path, thus naturally including partially-collisional effects that go over to collision-dominated physics, as the mean-free-path becomes less that the particle width. Applications relevant to our principal applications will be presented

We have used the 2-D PIC code, GYMNOS, to study the transient behaviors in the Heavy Ion Fusion (HIF) injectors. GYMNOS simulations accurately provide the steady state Child-Langmuir current and the beam transient behavior within a planar diode. The simulations of the LBL HIF ESAC injector experiments agree well with the experimental data and EGUN steady state results. Simulations of the nominal EHF injectors have revealed the need to design the accelerating electrodes carefully to control the ion beam current, particularly the ion loss at the end of the bunch as the extraction voltage is reduced. >

A program is underway to construct a 2 MV, 800 mA, K/sup +/ injector for heavy ion fusion. The electrostatic quadrupole injector configuration consists of a zeolite source, a diode of up to 1 MV, together with several electrostatic quadrupole units to simultaneously focus and accelerate the beam to 2 MV. The key issues of source technology, high voltage breakdown, beam aberrations, and transient effects are discussed. Results from ongoing experiments and simulations are presented. >

Heavy ion fusion experiments require very high current beams with excellent beam quality during a short pulse. Scaled experiments planned at LBL require very short pulses (/spl mu/sec) compared to what one expects in an HIF driver (20-30 /spl mu/s). A 1 MV acceleration column composed of aperture lenses has been constructed at LBL in order to study the propagation effects on such ion pulses. The column is initially empty of space charge but with the full acceleration potential applied. A short current pulse is then injected into the column with a planar diode "current valve". Effects on the pulse propagation due to rise time, pulse duration, and beam size have been studied. Experiments on transported beam current and emittance have been conducted using a carbon arc plasma source (2" and 5" diameter) and a 1" diameter potassium alumino-silicate ion source. Computer simulations using a 2.5D time dependent code are compared with the experimental data. >

Numerical simulation using the r-z code GYMNOS shows the time evolution of the focusing and accelerating potential found in Post's MIRRORTRON. Formed in a strongly anisotropic loss cone plasma, the transient potential is generated by the application of an additional local mirror field produced by a strap coil. The current rise time in the strap coil must be on the order to tens of nanoseconds in order for the plasma ions to remain inertially confined while the hot electrons respond to the local mirror force. The plasma density is low in order to facilitate the penetrations of the magnetic field into the plasma. Typical plasma densities in a MIRRORTRON are 10/sup 10/-10/sup 11/ cm/sup -3/ with electron temperatures in the hundred kiloelectronvolt to several megaelectronvolt range. The magnitude of the potential produced is on the order of the electron temperature. The formation of this potential peak is followed and test ions are projected through the resulting fields in order to evaluate the feasibility of using a MIRRORTRON as a heavy-ion accelerator. >

It is noted that optimization is important to the development of high-current, heavy-ion accelerators for power production based on inertial confinement fusion. Two heavy-ion recirculating linac configurations are examined that eliminate the necessity to provide reset pulses for the cores used in the linac induction accelerator modules. The configurations considered use the reset pulse to accelerate beam bunches traveling in the reverse direction and thus potentially improve system efficiency and/or reduce cost. A three-linac recirculator and a bidirectional recirculator are considered. >

A novel HIF code, FOCI, combines the capabilities of several codes: (1) the x,y envelope equations are integrated in the axial direction, (2) an optimizer varies selected transport system parameters to make the envelope satisfy weighted constraints, (3) transverse particle motion can be followed simultaneously, (4) the E field used for the particle advance is computed from Poisson's equation using the local beam charge density, as input, and (5) longitudinal velocity variations can be included in the particle description. >

An injector is being developed at LBL (Lawrence Berkeley Laboratory) that would serve as the front end of a scaled induction linac accelerator technology experiment in heavy ion fusion. The ion mass being used is in the range 10-18. It is a multi-beam device intended to accelerate up to 2 MeV with 500 mA in each beam. The first half of the accelerating column has been built and experiments with one carbon beam are underway at the 1 MeV level. >

Summary form only given, as follows. Very short duration electromagnetic pulses are important in a wide variety of situations that include exo-atmospheric-generated EMP, ultra short pulse lasers, radar, and cross talk on integrated circuits. In an effort to overcome the grid generation challenge by letting the code build resolution in the most active regions, a new algorithm has emerged that promises adaptive resolution of the pulse and various various geometric elements. These adaptive node techniques, in a model called aETHER, have now been applied to the Lorentz gauge Maxwell's equations and have proven able to identify and concentrate on emerging features. Most recently, these concepts have been applied to a modified set of EM field equations wherein the Poisson equation has been replaced by the zero-electron-mass momentum equation. A discussion of these concepts and the result of test cases are presented.

Heavy ion fusion requires high current density, low‐emittance ion sources that are reliable and long lived. We report experimental and simulation results on the performance of carbon arc ion sources intended for use in a scaled induction linac experiment. These sources use a planar electrostatic plasma switch to prevent plasma from entering the extraction gap before the extraction voltage pulse is applied. This provides good beam optics for short pulse extraction. Measurements of current density and emittance are presented. Both double‐slit and channel plate‐pepper pot techniques are used for emittance measurement. Data presented are from a compact three‐arc source with plasma coupling of the cathodes. Data on lifetime and multiple arc triggering are also presented. The plasma switch performance has been modeled with a 2D explicit electrostatic particle‐in‐cell code. Results showing plasma shutoff phenomena and behavior during extraction are presented. A 2D steady‐state ion flow model is also used to pred...

Responding to a need to include realistic chemistry and ionization in the driven plasmas in plasma-aided manufacture, we became convinced that existing computational tools were not able to fulfill our requirements. In response to this, we invented an entirely new approach to such problems. Our approach is to generalize the capabilities of existing methods: Particle-In-Cell (PIC) contains the required details of the distributions but becomes overwhelming expensive when applied to realistic, multidimensional problems. The common alternative, the hydrodynamic model, has simply lost the ability to model this science because of the simple fluid assumption. Our solution, GAPH (for Grid and Particle Hydrodynamics), retains the kinetic capabilities of PIC while using fluid concepts within each particle to control the incredible expense. The coupling of these two methods has resulted in a very powerful new algorithm that is now being applied to a range of programmatic problems throughout LLNL that now far outstrips the initial applications area.

Low-frequency electromagnetic simulation models have a wide range of industrial applications. We have built several models, differentiated by slightly different physics approximations or computational solution methods, that have proven quite useful in a variety of applications. Our models been used to investigate beam plasma interactions in ICF targets, antenna plasma coupling in plasma processing, and magnetic implosion drive in Z-pinch pulsed power generators. The common feature of these models is that they retain inductive effects but implicitly ignore computationally intensive, fully electromagnetic effects. However, the preponderance of our work has been limited to only two dimensions. We have made significant progress modeling low-frequency electromagnetic physics with a new model in 2-D that is now capable of modeling antenna structures in 3-D. Although LLNL`s interest in plasma processing has diminished, we have certainly added to LLNL`s capabilities. Interestingly, we have already found another application, the magnetic behavior of read/write heads in the magnetic storage industry, that can make use of many of the computational methods described here, rewarding us again for maintaining a strong core competency in low-frequency EM plasmas.

Increased interest in the development of high density plasma sources for ultra large scale integrated circuit manufacturing with varying geometry and power coupling strategies require the development of 2D models for accurate plasma reactor studies. An objective of their research was to develop a computationally efficient numerical code that would be useful as a design tool for inductively coupled plasma reactors. To achieve that goal the authors created a 2D fluid model (INDUCT2D) of an argon discharge. As uniformity is a primary issue in wafer manufacturing they have used INDUCT2D to investigate the effect of pressure and reactor geometry on the spatial uniformity of the etching ion flux. They have found that the optimum pressure for maximum uniformity depends upon the spatial profile of the inductive heating, and decreases with the reactor aspect ratio.

Summary form only given, as follows. We have developed a new algorithm, the Darwin direct implicit particle-in-cell (DADIPIC) method, for particle simulation of low frequency, non-neutral phenomena in plasmas. One of the difficulties in simulating plasmas lies in the enormous disparity between the fundamental scale lengths of a plasma and the scale lengths of the phenomena of interest. We have chosen to combine two methods, Darwin and direct implicit, to eliminate the constraints c/spl Delta/x//spl Delta/t

The inclusion of a velocity distribution within each Finite Size Particle (FSP) has lead to a decrease in the number of particle required to accurately describe Local Thermodynamic Equilibrium (LTE). Although this is appealing in many instances, systems which are dominated by diffusion and/or pressure gradients suffer from the FSP inability to probe the necessary parameter space. Many methods have been introduced to alleviate this shortcoming, such as deformation and fragmentation of a particle volumes or the generation of virtual particles to effect pressure. We have generated a new algorithm that compares well with selected closed form solutions. Grid and Particle Hydrodynamics (GaPH) works by allowing each FSP to internally undergo local hydrodynamics. The resulting distribution contained within each FSP is then cast into a central and expansion particles. The system is now advanced in time and particles that sufficiently overlap in phase space are merged. By allowing the FSP to aggressively probe the available parameter space through fragmentation, new characteristics to the distribution can emerge. By aggressively merging particles, redundant representations of the distribution tend to be eliminated.

In this project, the authors enhanced their ability to numerically simulate bounded plasmas that are dominated by low-frequency electric and magnetic fields. They moved towards this goal in several ways; they are now in a position to play significant roles in the modeling of low-frequency electromagnetic plasmas in several new industrial applications. They have significantly increased their facility with the computational methods invented to solve the low frequency limit of Maxwell`s equations (DiPeso, Hewett, accepted, J. Comp. Phys., 1993). This low frequency model is called the Streamlined Darwin Field model (SDF, Hewett, Larson, and Doss, J. Comp. Phys., 1992) has now been implemented in a fully non-neutral SDF code BEAGLE (Larson, Ph.D. dissertation, 1993) and has further extended to the quasi-neutral limit (DiPeso, Hewett, Comp. Phys. Comm., 1993). In addition, they have resurrected the quasi-neutral, zero-electron-inertia model (ZMR) and began the task of incorporating internal boundary conditions into this model that have the flexibility of those in GYMNOS, a magnetostatic code now used in ion source work (Hewett, Chen, ICF Quarterly Report, July--September, 1993). Finally, near the end of this project, they invented a new type of banded matrix solver that can be implemented on a massively parallel computer -- more » thus opening the door for the use of all their ADI schemes on these new computer architecture`s (Mattor, Williams, Hewett, submitted to Parallel Computing, 1993). « less

The code GYMNOS is a 2.5D magnetostatic PIC code that is distinguished from the multitude of similar codes by its ability to handle boundary conditions and internal structures. This code was started (see next section) during the period starting from December 1987 and the major sections were completed and in routine use by the end of 1990. The code is written in R-Z geometry (axisymmetric m=0 only) but could easily be modified to handle an X-Z configuration. The several competitors of this code, having similar (but we believe less general) capabilities, distinguish themselves in a different way: They have very formal, thick instruction manuals and usually command an army of people to maintain both the code and said manual. We are here attempting to convince the reader that GYMNOS is better in these two regards. Offered here is a description of a code with a great deal of physics capability coupled with no interactive graphics capability and very little system capability. These later two subjects contribute most of the complexity in both the code and manual in the competition! Expressed another way, the user, will almost never be thwarted by unexpected system changes on Friday afternoon. The price extracted ismore » that the user is required to monitor/babysit runs so that output is ``hand-delivered`` to the out I put medium and properly archived to prevent loss. A further spinoff of this simplicity is that ``single pilot`` operation is easy. And, the only manual is this modest document.« less

We report on three experiments performed in connection with the 2MV electrostatic quadrupole (ESQ) injector under construction at Lawrence Berkeley Laboratory. Scaled experiments have been conducted to study possible beam emittance growth due to beam aberrations in an ESQ injector. The experiment uses the SBTE (Single Beam Transport Experiment) accelerator system, quarter-scale ESQ set-up and a potassium ion diode source. Measured emittance growth changes significantly with variations in current and diode energy, in good agreement with theoretical predictions. In addition, beam transport experiments were performed in a 1MV axisymmetric electrostatic aperture column using a zeolite 1″ diameter potassium ion source. Experimental measurements in good agreement with 2-1/2D simulations showed that low-emittance beams can be produced in axisymmetric structures. Finally, ESQ breakdown voltage tests without beam were performed at up to two times the quadrupole working voltage.

Summary form only given. A 1-MV accelerator column composed of aperture lenses that focus and accelerate K/sup +/ ions from a hot-plate is being studied experimentally at Lawrence Berkeley Laboratory. The source must provide adequate current with a sufficiently low temperature so that the beam can be focused to the required small spot at the target. Time-independent codes provide design insight but fail to give any information about beam transients. HIF applications require careful control of the entire beam pulse. Continued refinement of the time-dependent code GYMNOS now allows detailed comparison with experiment and provides guidance for the evolution of the column design.

Summary form only given. The requirements of HIF place severe restrictions on the operating parameters of the ion source. In addition to the usual requirements of high brightness, constant current, and constant energy/nucleon, it is necessary to switch the ion flow on abruptly with enough ion flux so that the full HIF pulse current requirement can be met without the flow becoming source limited. A technique that shows promise is a combination of an arc source with control grids. The grids are biased with a potential to confine electrons and thus switch off any ion flow until the grid voltage is overpowered by the abrupt application of the extraction voltage. The plasma feed from the arc sources must satisfy various criteria so that the plasma electrons neutralize the flow up to this switch grid, but no further, so that space charge limited flow can be maintained between the switch and extraction grid. The plasma-grid interaction has been modeled with the particle-in-cell code GYMNOS in the relevant parameter regime to verify the understanding of the switch-off requirements followed by the switch-on stage. It was found that the arc source must provide plasma with density and flow rates within a small range of parameters; otherwise, the grid cannot properly confine electrons-the essential elements for constant current during the pulse

Summary form only given, as follows. The Mirrortron is studied using the 2-1/2 D cylindrical PIC (particle-in-cell) code GYMNOS. The Mirrortron takes advantage of large transient potentials generated in a mirror cell in order to accelerate and focus an ion beam. Particular attention is given to the behavior of the chamber plasma as an external coil produces a substantial change in the local axial magnetic field. How these changes produce both accelerating and focusing electrostatic fields was investigated. In addition, effects produced by the propagation of a finite-length space-charge dominated ion beam during the pulse were considered. At present GYMNOS is electrostatic and magnetostatic: consideration of magnetoinductive effects will occur with the implementation of a Darwin field model

The results of hybrid simulations of the formation and the evolution to equilibrium of field-reversal experiment (FRX) configurations are presented. The observed rapid reconnection is explained in terms of forced reconnection driven by a Kruskal-Schwarzschild instability and the self-consistent production of toroidal magnetic field.

A multidimensional hybrid simulation model has been developed for use in studying plasma phenomena on extended time and distance scales. The model makes fundamental use of the small Debye length or quasineutrality assumption. The ions are modeled by particle-in-cell techniques, while the electrons are considered a collision-dominated fluid. Some electron inertial effects are retained. The fields are calculated in the nonradiative Darwin limit. The quasineutral counterpart of Poisson's equation is obtained by first summing the electron and ion momentum equations and then taking the quasineutral limit. The resulting elliptic equation correctly includes those electrostatic potentials which occur in sheath or ambipolar phenomena while neglecting the short-range electrostatic fields which give rise to plasma oscillations. This model has been implemented in a two-dimensional code QN2. A lower hybrid drift unstable equilibrium with parameters accessible to both hybrid and full-particle simulation has been selected as a test of the code and a demonstration of the model. Initial results indicate quite good agreement between the two simulation methods in linear growth rate and wave number.

A formulation of the electron momentum equation and Maxwell's field equations suitable for global solution in an r-z hybrid plasma simulation code has been derived. The assumption of zero electron inertia is made in the electron momentum equation and Maxwell's equations are used in the radiation-free or Darwin limit. These techniques make explicit use of the axisymmetric properties of the model to decouple the components of the model equations. Equations to self-consistently advance the electron temperature are not presently included in this scheme. The model equations which result from these considerations are two coupled, nonlinear, second order partial differential equations. These two equations are integrated in time by a noniterative ADI procedure along with the explicit particle-in-cell ion time integration procedure. The resulting nearly implicit electron-field algorithm treats wide variations in the local signal velocity without instability; this consideration is most important since pure vacuum regions are allowed. The global nature of the solution requires boundary conditions only on the boundaries of the simulation region; arbitrary intermixing of plasma vacuum regions requires only the simple detection of low density cells and does not require monitoring of plasma vacuum interfaces.

The formation of an FRC (Field-Reversed Configuration) is considered by using a hybrid simulation model that treats the electrons as a zero-inertia thermal fluid and the ions either kinetically or as a simple fluid. The unexpected spontaneous generation of a substantial toroidal magnetic field Bθ near magnetic reconnection points requires explanation. Although ion kinetic effects are responsible for some of the internal reconnection, simulations with a fluid ion representation also show similar self-generation of the toroidal field. Further investigation reveals that the origin of the toroidal field can be found in some of the terms that arise from retention of the Hall term – an automatic feature of the two-component model. Conceptually, the phenomena can be understood by noting that, if the magnetic field is 'frozen into the electron fluid', a sheared toroidal electron flow will stretch the poloidal field to make the toroidal field.

The stable lifetimes of many field-reversed configuration (FRC) experiments have been observed to be limited by the onset of the m = 2 rotational instability. The origin of the rotation which drives this instability is investigated with the aid of hybrid simulation methods. The loss of particles with guiding centres on open field lines is found to lead to localized rotation near the separatrix and this effect is found to be enhanced by end-shorting of the radial electric field outside the separatrix. In simulations allowing azimuthal variation the velocity shear is eventually relaxed so that the bulk of the plasma begins to rotate. These simulations also show that the m = 2 instability does not appear as long as rotation is localized near the separatrix. It is found that by increasing the separatrix radius it is possible to significantly delay the transfer of rotation from the separatrix, hence delaying the onset of the m = 2 instability.

The numerical time integration of an initial value problem based upon the nonlinear Vlasov equation as applied to an axisymmetric, inhomogeneous plasma column has been developed. All six components of the electric and magnetic fields are advanced self-consistently in time. An assumption has been made which neglects purely electromagnetic modes. Each field contains a contribution from plasma source terms calculated by a Green's function technique as well as a boundary value contribution which may be time dependent. The Vlasov equation has been reduced to a convenient numerical form by an extension of the standard Fourier-Hermite expansion. Radial dependence is discretized. Results are presented on the streaming instability of the electrostatic column.

Particle simulations of strongly inhomogeneous plasma phenomena that model 6-pinch behavior in the implosion and post-implosion phases have been performed using an inhomogeneous, self-consistent Vlasov equilibrium as an initial state. In the hot ion regime a strong flute instability has been observed, whose properties agree well with those predicted by the linear and nonlinear theory of the lower-hybrid-drift instability.

The two‐dimensional Fokker–Planck equation with strong quasilinear diffusion on the tail of the electron distribution function is solved analytically and numerically. The quasilinear diffusion is taken to be parallel to the confining magnetic field of the plasma, as is appropriate for lower‐hybrid current drive. The radio frequency fields that produce this quasilinear diffusion are assumed to be imposed by external sources. It is found that an appreciable broadening of the resonant plateau in the direction perpendicular to the toroidal magnetic field leads to more particles carrying the current. As a result both the current (J) and the power dissipated (Pd) are substantially enhanced, compared with the predictions of one‐dimensional theory. The figure of merit J/Pd is three times larger than the one found from the one‐dimensional theory.