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Click the How To Play button to get a simplified slide show presentation. Click on the image to enlarge, and then save it to your computer by right clicking on the image. Systems analyzed using Monte Carlo simulation include financial, physical, and mathematical models. Download the template and follow along with the information in this article. Monte Carlo SS logo vector. Accelerator-driven sub-critical research facility with low-enriched fuel in lead matrix: Neutron flux calculation. The neutron source is generated by an interaction of a proton or deuteron beam with the target placed inside the sub-critical core.

The results of the total neutron flux density escaping the target and calculations of neutron yields for different target materials are also given here. Neutrons escaping the target volume with the group spectra first step are used to specify a neutron source for further numerical simulations of the neutron flux density in the sub-critical core second step. The results of the calculations of the neutron effective multiplication factor keff and neutron generation time L for the ADSRF model have also been presented. Neutron spectra calculations for an ADSRF with an uranium tar get highest values of the neutron yield for the selected sub-critical core cells for both beams have also been presented in this paper.

Accelerator Driven Subcritical systems are being studied worldwide for their potential in burning minor actinides and reducing long term radiotoxicity of spent nuclear fuels. Neutron fluctuations in accelerator driven and power reactors via backward master equations. The transport of neutrons in a reactor is a random process, and thus the number of neutrons in a reactor is a random variable. Fluctuations in the number of neutrons in a reactor can be divided into two categories, namely zero noise and power reactor noise.

As the name indicates, they dominate i. The reasons for their occurrences and utilization are also different. In addition, they are described via different mathematical tools, namely master equations and the Langevin equation, respectively. Zero noise carries information about some nuclear properties such as reactor reactivity. Hence methods such as Feynman- and Rossi-alpha methods have been established to determine the subcritical reactivity of a subcritical system.

Such methods received a renewed interest recently with the advent of the so-called accelerator driven systems ADS. Such systems, intended to be used either for energy production or transuranium transmutation, will use a subcritical core with a strong spallation source. A spallation source has statistical properties that are different from those of the traditionally used radioactive sources which were also assumed in the derivation of the Feynman- and Rossi-alpha formulae. Therefore it is necessary to re-derive the Feynman- and Rossi-alpha formulae. Such formulae for ADS have been derived recently but in simpler neutronic models.

One subject of this thesis is the extension of such formulae to a more general case in which six groups of delayed neutron precursors are taken into account, and the full joint statistics of the prompt and all delayed groups is included. The involved complexity problems are solved with a combination of effective analytical techniques and symbolic algebra codes. Power reactor noise carries information about parametric perturbation of the system.

Langevin technique has been used to extract such information. In such a treatment, zero noise has been neglected. This is a pragmatic. Neutronics design of accelerator-driven system for power flattening and beam current reduction. In the present neutronics design of the Accelerator-Driven System ADS cooled by lead-bismuth eutectic LBE , we investigated several methods to reduce the power peak and beam current, and estimated the temperature reductions of the cladding tube and beam window from the conventional design.

The methods are adjustment of inert matrix ratio in fuel in each burn-up cycle, multiregion design in terms of pin radius or inert matrix content, and modification of the level of the beam window position and the height of the central fuel assemblies. As a result, we optimized the ADS combined with the adjustment of the inert matrix ratio in each burn-up cycle, multiregion design in terms of inert matrix content and deepened window level. The maximum temperatures of the optimized ADS at the surface of the cladding tube and the beam window were reduced by 91 and 38degC, respectively.

The maximum beam current was improved from Monte Carlo studies of accelerator driven systems energy and spatial distribution of neutrons in multiplying and non-multiplying media. The LAHET code system is used to study the behaviour of the spallation neutrons resulting from the interaction of 2. The spatial and energy distribution of the neutrons with presence and absence of a fissile material in Accelerator Driven Systems ADS are investigated. It is shown that the energy spectra of the neutrons in graphite and lead moderators are very different and such difference is expected to result in noticeable differences in the nuclear waste transmutation abilities of the ADSs that use graphite and lead for neutron moderation and storage.

Reactivity determination in accelerator driven nuclear reactors by statistics from neutron detectors Feynman-Alpha Method. The Feynman-alpha method is used in traditional nuclear reactors to determine the subcritical reactivity of a system. The method is based on the measurement of the mean number and the variance of detector counts for different measurement times. The measurement is performed while a steady-state neutron flux is maintained in the reactor by an external neutron source, as a rule a radioactive source. From a plot of the variance-to-mean ratio as a function of measurement time 'gate length' , the reactivity can be determined by fitting the measured curve to the analytical solution.

A new situation arises in the planned accelerator driven systems ADS. An ADS will be run in a subcritical mode, and the steady flux will be maintained by an accelerator based source. Such a source has statistical properties that are different from those of a steady radioactive source. The theory of Feynman-alpha method needs to be extended to such nonstationary sources. There are two ways of performing and evaluating such pulsed source experiments. One is to synchronise the detector time gate start with the beginning of an incoming pulse.

The Feynman-alpha method has been elaborated for such a case recently. The other method can be called stochastic pulsing. It means that there is no synchronisation between the detector time gate start and the source pulsing, i. The analytical solution to the Feynman-alpha formula from this latter method is the subject of this report. We have obtained an analytical Feynman-alpha formula for the case of stochastic pulsing by two different methods. One is completely based on the use of the symbolic algebra code Mathematica, whereas the other is based on complex function techniques.

Closed form solutions could be obtained by both methods. A new target concept for proton accelerator driven boron neutron capture therapy applications. The mechanical designand analysis of the four-module, bolt-together RFQ will be presentedhere. Operating at MHz, the 5. Each of the 1. A specially designed 3-DO-ring will provide vacuum sealing between both the vanes and themodules.

RF connections are made with canted coil spring contacts. Aseries of 60 water-cooled pi-mode rods provides quadrupole modestabilization.

A set of 80 evenly spaced fixed slug tuners is used forfinal frequency adjustment and local field perturbationcorrection. The mechanical design and analysis of the four-module, bolt-together RFQ will be presented here. A specially designed 3-DO-ring will provide vacuum sealing between both the vanes and the modules. A series of 60 water-cooled pi-mode rods provides quadrupole mode stabilization.

A set of 80 evenly spaced fixed slug tuners is used for final frequency adjustment and local field perturbation correction. Experimental study on neutronics in bombardment of thick targets by high energy proton beams for accelerator-driven sub-critical system. The experimental study on neutronics in the target region of accelerator-driven sub-critical system is carried out by using the high energy accelerator in Joint Institute for Nuclear Research, Dubna, Russia. The experiments with targets U Pb , Pb and Hg bombarded by 0.

Hg target is disadvantageous to U Pb and Pb targets to get more neutrons. Neutron yield drops along 20 cm thick targets as the thickness penetrated by protons increases. The lower the energy of protons, the steeper the neutron yield drops.

In order to get more uniform field of neutrons in the targets, the energy of protons from accelerators should not be lower than 1 GeV. The spectra of secondary neutrons produced by different energies of protons are similar, but the proportion of neutrons with higher energy gradually increases as the proton energy increases. A neutron booster for spallation sources--application to accelerator driven systems and isotope production. One can design a critical system with fissile material in the form of a thin layer on the inner surface of a cylindrical neutron moderator such as graphite or beryllium.

Recently, we have investigated the properties of critical and near critical systems based on the use of thin actinide layers of uranium, plutonium and americium. The thickness of the required fissile layer depends on the type of fissile material, its concentration in the layer and on the geometrical arrangement, but is typically in the mu m-mm range.

The resulting total mass of fissile material can be as low as g. Thin fissile layers have a variety of applications in nuclear technology--for example in the design neutron amplifiers for medical applications and 'fast' islands in thermal reactors for waste incineration. In the present paper, we investigate the properties of a neutron booster unit for spallation sources and isotope production.

In those applications a layer of fissile material surrounds the spallation source. Such a module cou Accelerator driven radiation clean nuclear power system conceptual research symposium. It includes the basic principle of ADS, accelerators , sub-critical reactors, neutron physics, nuclear data, partitioning and transmutation. Neutron physics with accelerators. Neutron -induced nuclear reactions are of key importance for a variety of applications in basic and applied science. Apart from nuclear reactors, accelerator -based neutron sources play a major role in experimental studies, especially for the determination of reaction cross sections over a wide energy span from sub-thermal to GeV energies.

After an overview of present and upcoming facilities, this article deals with state-of-the-art detectors and equipment, including the often difficult sample problem. These issues are illustrated at selected examples of measurements for nuclear astrophysics and reactor technology with emphasis on their intertwined relations. Accelerator-driven systems for fissile materials production have been proposed and studied since the early s. Recent advances in beam power levels for small accelerators have raised the possibility that such use could be feasible for a potential proliferator.

The objective of this study is to review the state of technology development for accelerator-driven spallation neutron sources and subcritical reactors. Energy and power requirements were calculated for a proton accelerator-driven neutron spallation source and subcritical reactors to produce a significant amount of fissile material--plutonium.

- The physics of accelerator driven sub-critical reactors.
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Laser- driven accelerators. Several devices for using laser fields have been proposed and they can be classified in three broad categories - 'far-field' accelerators such as the principle of inverse free electron lasers , 'media' accelerators which, for example, use the inverse Cherenkov effect or laser-controlled plasma waves , and 'near-field' accelerators using a loaded guiding structure such as cavities or gratings. These different approaches come from the fact that a particle cannot be accelerated by the absorption of single photons because of momentum conservation and thus some other element has to intervene.

Fission fragment driven neutron source. Fissionable uranium formed into a foil is bombarded with thermal neutrons in the presence of deuterium-tritium gas. The resulting fission fragments impart energy to accelerate deuterium and tritium particles which in turn provide approximately 14 MeV neutrons by the reactions t d,n. Accelerator driven sub-critical core. Systems and methods for operating an accelerator driven sub-critical core.

In one embodiment, a fission power generator includes a sub-critical core and a plurality of proton beam generators. Each of the proton beam generators is configured to concurrently provide a proton beam into a different area of the sub-critical core. Each proton beam scatters neutrons within the sub-critical core. The plurality of proton beam generators provides aggregate power to the sub-critical core, via the proton beams, to scatter neutrons sufficient to initiate fission in the sub-critical core.

Accelerator driven assembly. A point design for the pulsed neutron facility was initiated early in FY94 after hosting a Defense Nuclear Agency DNA panel review and after subsequently visiting several potential clients and users. However, our point designs and parametric studies identify a unique, cost-effective, above-ground capability for neutron nuclear-weapons-effects studies at threat levels. This capability builds on existing capital installations and infrastructure at LANL.

We believe that it is appropriate for us, together with the DNA, to return to the user community and ask for their comments and critiques. We also realize that the requirements of last year have changed significantly. Laser driven particle acceleration. This dissertation summarizes the last ten years of research at the Laboratory of Applied Optics on laser-plasma based electron acceleration. The main result consists of the development and study of a relativistic electron source with unique properties: high energy MeV in short distances few millimeters , mono-energetic, ultra-short few fs , stable and tunable.

The manuscript describes the steps that led to understanding the physics, and then mastering it in order to produce this new electron source. Non linear propagation of the laser pulse in the plasma is first presented, with phenomena such as non linear wakefield excitation, relativistic and ponderomotive self-focusing in the short pulse regime, self-compression. Acceleration and injection of electrons are then reviewed from a theoretical perspective.

Experimental demonstrations of self-injection in the bubble regime and then colliding pulse injection are then presented. These experiments were among the first to produce monoenergetic, high quality, stable and tunable electron beams from a laser-plasma accelerator. The last two chapters are dedicated to the characterization of the electron beam using transition radiation and to its applications to gamma radiography and radiotherapy. Finally, the perspectives of this research are presented in the conclusion.

Scaling laws are used to determine the parameters that the electron beams will reach using peta-watt laser systems currently under construction. Accelerator driven subcritical reactors. ADS concepts have been proposed in the last decade for a variety of applications. However, there is a convergence of interest of several countries and laboratories on the application of ADS to transmutation.

As far as the so-called partitioning and transmutation PIT strategies, it was indicated that they can be clarified according to the option taken with respect to Pu and MA, i. Proposals are being worked out. For example, composite such as ceramic-metallic or ceramic-ceramic fuels are presently under study. The actinide oxide is dispersed in a metallic matrix Zr, or W or Mo or in an oxide matrix e. In these cases, reliable data are required for the matrix materials.

Hard or very hard fast neutron spectrum is required. At present, there is no clear option for their transmutation one needs a high level of thermalized neutrons , support matrixes for target irradiation, isotopic separations, reprocessing techniques, etc. Finally, ADS transmutation will give rise to fuel cycles, where very active materials will be present. Cm and higher mass isotopes up to Cf will be contributors to dose and neutron source strength. This area will deserve attention in future, in order to define the relevant data needs. It is recommended to coordinate work on MA data as a priority.

Experimental verification of neutron phenomenology in lead and transmutation by adiabatic resonance crossing in accelerator driven systems. Energy and space distributions of spallation neutrons from 2. Recent advances in laser- driven neutron sources. Due to the limited number and high cost of large-scale neutron facilities, there has been a growing interest in compact accelerator-driven sources. In this context, several potential schemes of laser- driven neutron sources are being intensively studied employing laser- accelerated electron and ion beams.

In addition to the potential of delivering neutron beams with high brilliance, directionality and ultra-short burst duration, a laser- driven neutron source would offer further advantages in terms of cost-effectiveness, compactness and radiation confinement by closed-coupled experiments. Some of the recent advances in this field are discussed, showing improvements in the directionality and flux of the laser- driven neutron beams. Neutrons from medical electron accelerators. The significant sources of photoneutrons within a linear- accelerator treatment head are identified and absolute estimates of neutron production per treatment dose are given for typical components.

Measured data obtained at a variety of accelerator installations are presented and compared with these calculations. It is found that the high-Z materials within the treatment head do not significantly alter the neutron fluence, but do substantially reduce the average energy of the transmitted spectrum. Reflected neutrons from the concrete treatment room contribute to the neutron fluence, but not substantially to the patient integral dose, because of a further reduction in average energy.

Absolute depth-dose distributions for realistic neutron spectra are calculated, and a rapid falloff with depth is found. Proposal on the accelerator driven molten-salt reactor ATW concept benchmark calculations. STAGE 1 - without an external neutron source.

The first stage of ATW neutronic benchmark without an external source , based on the simple modelling of two component concept is presented. The main purpose of this benchmark is not only to provide the basic characteristics of given ADS but also to test codes in calculations of the rate of transmutation waste and to evaluate basic kinetics parameters and reactivity effects. About using of ion accelerators in accelerator driven systems.

Radiation Physics and Chemistry Inst. The prospects of using deuteron and alpha particle beams in Accelerator Driven Molten Salt Breeder for simultaneous production of uranium and of thermal power are discussed, disregarding the problems of reactor construction and design. It is shown that by replacing the proton beam by beams of deuterons or alpha particles the energy cost of one neutron can be reduced from The average energy of neutrons increases from Thus, the gain in the number of fissile nuclei and in thermal power production of at least 1.

Problems in the neutron dynamics of source- driven systems. The present paper presents some neutronic features of source- driven neutron multiplying systems, with special regards to dynamics, discussing the validity and limitations of classical methods, developed for systems in the vicinity of criticality. Specific characteristics, such as source dominance and the role of delayed neutron emissions are illustrated. Some dynamic peculiarities of innovative concepts proposed for accelerator-driven systems, such as fluid-fuel, are also discussed.

The second portion of the work formulates the quasi-static methods for source- driven systems, evidencing its novel features and presenting some numerical results. Tokai Research Establishment. JAERI is proceeding a design study of the hybrid type minor actinide transmutation system which mainly consist of an intense proton accelerator and a fast subcritical core.

Neutronics and burnup characteristics of the accelerator-driven system is important from a view point of the maintenance of subcriticality and energy balance during the system operation. To determine those characteristics accurately, it is necessary to involve reactions at high-energy region, which are not treated on ordinary reactor analysis codes. The authors developed a code system named ATRAS to analyze the neutronics and burnup characteristics of accelerator-driven subcritical reactor systems.

ATRAS has a function of burnup analysis taking account of the effect of spallation neutron source. ATRAS consists of a spallation analysis code, a neutron transport codes and a burnup analysis code. Utility programs for fuel exchange, pre-processing and post-processing are also incorporated. The paper is concerned in the source-jerk method used to measure the sub-criticality, and the sub-critical experiment facility, which is used for the study on the neutronics of ADS, driven by external neutron source sup 2 sup 5 sup 2 Cf.

The effects of the location of neutron source and material buffer where is at the location of the pipe of proton beam and target of fission-product dispersion on the sub-criticality of reactor are studied by source-jerk method. Linac- driven spallation- neutron source. Strong interest has arisen in accelerator-driven spallation- neutron sources that surpass existing facilities such as ISIS at Rutherford or LANSCE at Los Alamos by more than an order of magnitude in beam power delivered to the spallation target. The approach chosen by Los Alamos as well as the European Spallation Source provides the full beam energy by acceleration in a linac as opposed to primary acceleration in a synchrotron or other circular device.

Two modes of neutron production are visualized for the source. A short-pulse mode produces 1 MW of beam power at 60 pps in pulses, of length less than 1 ms, by compression of the linac macropulse through multi-turn injection in an accumulator ring. A long-pulse mode produces a similar beam power with 1-ms-long pulses directly applied to a target. This latter mode rivals the performance of existing reactor facilities to very low neutron energies.

Combination with the short-pulse mode addresses virtually all applications. Spectrum shaping of accelerator -based neutron beams for BNCT. We describe Monte Carlo simulations of three facilities for the production of epithermal neutrons for Boron Neutron Capture Therapy BNCT and examine general aspects and problems of designing the spectrum-shaping assemblies to be used with these neutron sources.

The first facility is based on an accelerator-driven low-power subcritical reactor, operating as a neutron amplifier. The other two facilities have no amplifier and rely entirely on their primary sources, a D-T fusion reaction device and a conventional 2. Uncertainty assessment for accelerator-driven systems. The concept of a subcritical system driven by an external source of neutrons provided by an accelerator ADS Accelerator Driver System has been recently revived and is becoming more popular in the world technical community with active programs in Europe, Russia, Japan, and the U.

A general consensus has been reached in adopting for the subcritical component a fast spectrum liquid metal cooled configuration. Both a lead-bismuth eutectic, sodium and gas are being considered as a coolant; each has advantages and disadvantages. The major expected advantage is that subcriticality avoids reactivity induced transients.

The potentially large subcriticality margin also should allow for the introduction of very significant quantities of waste products minor Actinides and Fission Products which negatively impact the safety characteristics of standard cores. In the U. Up to now, neutronic calculations have not attached uncertainties on the values of the main nuclear integral parameters that characterize the system. Many of these parameters e.

In this paper we will consider uncertainties related to nuclear data only. The present knowledge of the cross sections of many isotopes that are not usually utilized in existing reactors like Bi, Pb, Pb, and also Minor Actinides and Fission Products suggests that uncertainties in the integral parameters will be significantly larger than for conventional reactor systems, and this raises concerns on the neutronic performance of those systems.

Minor actinide transmutation in accelerator driven systems. Transmutation of radioactive waste, the legacy of nuclear energy use, gains rising interest. This includes the development of facilities able to transmute minor actinides MA into stable or short-lived isotopes before final disposal. The most common proposal is to use a double-strata approach with accelerator-driven -systems ADS for the efficient transmutation of MA and power reactors to dispose plutonium. An ADS consists of a sub-critical core that reaches criticality with neutrons supplied by a spallation target.

Depletion calculations have been performed for both standard MOX fuel and transmutation fuel with an increased content of minor actinides. The resulting transmutation rates for MAs are compared to published values. Special attention is given to selected fission products such as Tc and I, which impact the radiation from the spent fuel significantly. Alternative definitions of kinetic parameters for accelerator driven systems.

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The first part is dedicated to the classic definition of the kinetic parameters and compares different calculation methodologies. The second part considers a new definition of the kinetic parameters for subcritical assemblies, with particular emphasis on the delayed neutron fraction and the prompt neutron lifetime. This new definition takes into account neutrons from the external neutron source and n, xn reactions, which increase the fraction of prompt neutrons.

This facility can be driven by californium, deuterium—deuterium D—D , or deuterium—tritium D—T external neutron sources. For the D—T neutron source, n, xn reactions must be taken into account in order to produce accurate results because the average energy of D—T source neutrons is Transmutation and accelerator driven systems.

Full text: Today, countries who are presently involved in nuclear energy are facing many challenges to maintain this option open for the next few decades. Among them, management of nuclear wastes produced in nuclear reactors and in fuel cycle operations has become a very strong environmental issue among the public. In most countries with sizeable commercial nuclear programs, deep geological disposal of ultimate highly active and long-lived nuclear wastes is considered as the reference long-term management scheme.

But, many questions arise on the possibility to demonstrate that such wastes can be dealt in such a way as to protect the future generations and the environment. The characteristics of nuclear wastes, the various back end policies concerning spent fuels and the nuclear wastes long-term management options will be first described.

Finally, the possibility for the nuclear physics community to play a part in alleviating the nuclear wastes burden will be pointed out. Use of accelerator based neutron sources. With the objective of discussing new requirements related to the use of accelerator based neutron generators an Advisory Group meeting was held in October in Vienna. This meeting was devoted to the specific field of the utilization of accelerator based neutron generators. This TECDOC reports on the technical discussions and presentations that took place at this meeting and reflects the current status of neutron generators.

The 14 MeV neutron generators manufactured originally for neutron activation analysis are utilised also for nuclear structure and reaction studies, nuclear data acquisition, radiation effects and damage studies, fusion related studies, neutron radiography. Efforts in were focussed on the optimisation of a high-performance device with a maximum neutron flux for MA irradiation experiments of 1. Potential applications of an ADS neutron source as well as various engineering aspects are discussed. Laser- driven electron accelerators. The following possibilities are discussed: inverse free electron laser wiggler accelerator ; inverse Cerenkov effect; plasma accelerator ; dielectric tube; and grating linac.

Of these, the grating acceleraton is considered the most attractive alternative. Applications of laser- driven particle acceleration. The first book of its kind to highlight the unique capabilities of laser- driven acceleration and its diverse potential, Applications of Laser- Driven Particle Acceleration presents the basic understanding of acceleration concepts and envisioned prospects for selected applications. As the main focus, this new book explores exciting and diverse application possibilities, with emphasis on those uniquely enabled by the laser driver that can also be meaningful and realistic for potential users.

A key aim of the book is to inform multiple, interdisciplinary research communities of the new possibilities available and to inspire them to engage with laser- driven acceleration , further motivating and advancing this developing field. Material is presented in a thorough yet accessible manner, making it a valuable reference text for general scientific and engineering researchers who are not necessarily subject matter experts.

Bolton, Katia Laser- driven acceleration with Bessel beam. A new approach of laser- driven acceleration with Bessel beam is described. Bessel beam, in contrast to the Gaussian beam, shows diffraction-free'' characteristics in its propagation, which implies potential in laser- driven acceleration.

But a normal laser, even if the Bessel beam, laser can not accelerate charged particle efficiently because the difference of velocity between the particle and photon makes cyclic acceleration and deceleration phase. We proposed a Bessel beam truncated by a set of annular slits those makes several special regions in its travelling path, where the laser field becomes very weak and the accelerated particles are possible to receive no deceleration as they undergo decelerating phase. Thus, multistage acceleration is realizable with high gradient.

In a numerical computation, we have shown the potential of multistage acceleration based on a three-stage model. Progress of Laser- Driven Plasma Accelerators. There is a great interest worldwide in plasma accelerators driven by ultra-intense lasers which make it possible to generate ultra-high gradient acceleration and high quality particle beams in a much more compact size compared with conventional accelerators.

A frontier research on laser and plasma accelerators is focused on high energy electron acceleration and ultra-short X-ray and Tera Hertz radiations as their applications.

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These achievements will provide not only a wide range of sciences with benefits of a table-top accelerator but also a basic science with a tool of ultrahigh energy accelerators probing an unknown extremely microscopic world. Harnessing the recent advance of ultra-intense ultra-short pulse lasers, the worldwide research has made a tremendous breakthrough in demonstrating high-energy high-quality particle beams in a compact scale, so called ''dream beams on a table top'', which represents monoenergetic electron beams from laser wakefield accelerators and GeV acceleration by capillary plasma-channel laser wakefield accelerators.

This lecture reviews recent progress of results on laser- driven plasma based accelerator experiments to quest for particle acceleration physics in intense laser-plasma interactions and to present new outlook for the GeV-range high-energy laser plasma accelerators. Comments to accelerator-driven system.

Accelerator-driven system ADS that was a subcritical nuclear reactor driven by a high power proton accelerator was recently studied by several large organisations over the world. This paper described two comments for ADS: philosophical and technological ones.

The latter was made from a view point of micro ball lightning BL that was newly discovered by the author. Accelerator based neutron source for neutron capture therapy. For a nuclear system in which the entire -eigenvalue spectrum is known, eigenfunction expansion yields the time-dependent flux response to any arbitrary source. Applications in which this response is of interest include pulsed-neutron experiments, accelerator-driven subcritical systems, and fast burst reactors, where a steady-state assumption used in neutron transport is invalid for characterizing the time-dependent flux.

To obtain the -eigenvalue spectrum, the transition rate matrix method TRMM tallies transition rates describing neutron behavior in a discretized position-direction-energy phase space using Monte Carlo. Interpretation of the resulting Markov process transition rate matrix as the operator in the adjoint -eigenvalue problem provides an avenue for determining a large finite set of eigenvalues and eigenfunctions of a nuclear system. Results from the TRMM are verified using analytic solutions, time-dependent Monte Carlo simulations, and modal expansion from diffusion theory.

For simplified infinite-medium and one-dimensional geometries, the TRMM accurately calculates eigenvalues, eigenfunctions, and eigenfunction expansion solutions.