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The observed scaling of neutron yield in the Dense Plasma Focus (DPF) as the fourth power of the current in the plasma was the principal driver of the growth of DPF research in its early days. Subsequent discovery of failure of this scaling law was also the principal reason for its abandonment by major laboratories. Attempts to understand this failure of scaling have so far been inconclusive. This letter looks at this failure in the context of the recently introduced Generalized Plasma Focus (GPF) problem and suggests a possible reason that can be experimentally examined using small plasma focus devices. This involves restrictions placed on the drive parameter by conservation laws for mass, momentum and energy. A suggested empirical workaround to the problem of neutron yield scaling failure could also be configured as a method for increasing the pressure range for neutron emission in small DPF devices., Comment: Accepted for publication in Physics of Plasmas on 14th July 2023

Space propulsion is unique among many proposed applications of the Dense Plasma Focus in being critically dependent on the availability of a scaling theory that is well-grounded in physics, in conformity with existing experimental knowledge and applicable to experimentally untested configurations. This paper derives such a first-principles-based scaling theory and illustrates its application to a novel space propulsion concept, where the plasma focus sheath is employed as a power density amplifying mechanism to transport electric energy from a capacitive storage to a current-driven fusion load. For this purpose, a Generalized Plasma Focus problem is introduced and formulated. It concerns a finite, axisymmetric plasma, driven through a neutral gas at supersonic speed over distances much larger than its typical gradient scale length by its azimuthal magnetic field while remaining connected with its pulse power source through suitable boundaries. The Gratton-Vargas equation is rederived from the scaling properties of the equations governing plasma dynamics and solved for algebraically defined initial (insulator) and boundary (anode) surfaces. Scaling relations for a new space propulsion concept are derived. This consists of a modified plasma focus with a tapered anode that transports current from a pulsed power source to a consumable portion of the anode in the form of a hypodermic needle tube continuously extruded along the axis of the device. When the tube is filled with deuterium, the device may serve as a small-scale version of magnetized liner inertial fusion (MAGLIF) that could avoid failure of neutron yield scaling in a conventional plasma focus., Comment: Accepted for publication in the Physics of Plasmas

This paper, Part I of a series, describes a kinematic framework for the theory of a Dense Plasma Focus which is very similar to the GV model in spirit but which differs in its scope in four respects. First, the GV model derives most of its results from the mathematical properties of the solution of a certain partial differential equation derived from assumptions that apparently represent conservation of momentum but are not a rigorous application of the relevant physics. The present model is based on the scaling properties of the standard equations of motion, which lead to mathematical results identical with the GV model. Second, the GV model is purely kinematic in nature. The present model is also kinematic like the GV model but it incorporates additional insights borrowed from other physical theories, models and experiments. Third, the GV model does not take into account the experimentally observed delay between the start of current and start of plasma propagation and the existence of a nonzero thickness of the dense plasma sheath. The present model incorporates both these features in its kinematic structure. Fourth, the unlike the GV model, the present model allows considerations of some modifications of standard Mather type geometry. In addition to the current waveform, the proposed model reproduces the height and radius of the pinch column, the ratio of pinch density to fill density, the general appearance of the umbrella like plasma profile and streak picture and formation of bounded 3-dimensional plasma structures embedded within the pinch plasma without taking into account microscopic details of physical phenomena, Comment: arXiv admin note: text overlap with arXiv:2010.10057

This paper describes in detail the construction of a local right-handed, orthogonal curvilinear coordinate system whose axes are along the local tangent, the local azimuth and the local normal of an analytically defined 3D surface of rotation, whose shape mimics the shape of a plasma focus current sheath. Expressions for various differential operators are derived in a tutorial format for the benefit of young researchers and non-specialists. Physical problems expressed in this coordinate system would benefit from the natural symmetry properties of the plasma focus sheath. For example, the normal component of current density is zero and the velocity has mainly the normal component. This paper is meant to serve as a readily available reference in the hope that it would be found useful.

The existence of axial (poloidal) magnetic field in a plasma focus and its significance in plasma focus phenomenology has been extensively discussed in a recent review paper. The poloidal magnetic field is a part of the transient 3-dimensional magnetic field structures which arise spontaneously, accelerate ions and keep them moving in trajectories that repeatedly cross a dense and warm plasma target. This is the origin of the abnormally high fusion reaction rate of the plasma focus, which has been known since the 1960s but has begun to be understood quite recently. Further progress now depends on explorations of the global aspects of the evolution of poloidal magnetic field. However, well-known experimental difficulties involved in standard techniques of axial magnetic field measurement hamper such research efforts. Taking cognizance of this stalemate, the International Scientific Committee for Dense Magnetized Plasmas launched an initiative to address this state of affairs in an International Video Conference on 24th April 2020. This paper reports on the first experiments that resulted from that initiative, Comment: Presented at the 10th International Workshop and Summer School on Plasma Physics at Kiten, Bulgaria, 4th to 10th July 2022

Direct experimental evidence for the existence of axial magnetic field both in the radial implosion phase and the pinch phase of a plasma focus has raised many questions of fundamental importance. The most fundamental of these is the fact of its existence, which is incomprehensible in terms of a conventional view of plasma physics. The plasma is known to have an axis of symmetry in the radial implosion phase. The axial magnetic field has a defined polarity with respect to the axis. Since the equations of magnetohydrodynamics are unchanged by flipping the sign of the axial coordinate, the polarity of the axial magnetic field must arise from the initial conditions. Exactly how the initial conditions determine the polarity of the axial magnetic field is not clear. More data are needed to guide theoretical developments of this question. There are several technical problems with the conventional techniques of axial magnetic field measurement such as magnetic probes and Faraday rotation measurement. The combined effect of theoretical and experimental difficulties is that researchers avoid research related to the axial magnetic field in axisymmetric plasmas in favour of more result-oriented projects, with the unfortunate consequence that questions related to the presence of axial magnetic field in axially symmetric plasmas remain unresolved. In order to promote research related to the questions related to the axial magnetic field in the plasma focus and other plasmas with an axial symmetry, this paper introduces a new technique that skirts the technical difficulties of measurement by redefining the objectives and scope of measurement. The paper is written in a tutorial format for the benefit of young researchers just beginning their research career and hence will not be published in regular journals. Peer scientists and young researchers are encouraged to contact the author directly.

Recent experiments using 15 frame interferometry on PF-1000 facility in Warsaw confirm the association between neutron emission and spontaneously self-organized, relatively long lasting, finite plasma structures. A crucial aspect of this association is the simultaneous observation of an axial magnetic field, which can allow magnetic flux lines to densely cover closed surfaces creating "magnetic flux surfaces". Evolution of such 3-dimensional (3-D) magnetic field structures is necessarily accompanied by induced electric field that can provide a very long (theoretically infinite) acceleration path length along a trajectory enclosed within the magnetic structure leading to high ion kinetic energy, resulting in a high reaction rate. Associated charge and current densities can be related to electric scalar potential and magnetic vector potential measured outside the plasma. We report our first observations of these fields outside the plasma focus and discuss their general features. The reported technique is capable of unambiguous first-principles interpretation of signals in terms of quantities related to distributions of charge density and rate of change of azimuthal current density ("electromagnetic structure") in the plasma focus. It is non-intrusive and completely insensitive to non-axisymmetric aspects of plasma. Our first results show that axial magnetic field generated by azimuthal current density distribution symmetric about the axis exists before, during and after the pinch phase., Comment: Adapted from a paper presented at the 9th International International Workshop and Summer School on Plasma Physics, held at Kiten, Bulgaria on 7th July 2020

Striking pictures showing filamentary structures in plasma focus have intrigued researchers from the early days of plasma focus research. A definitive understanding of their occurrence, origin, structure and role in plasma focus physics is still not in sight as summarized in a recent comprehensive review. This is because they are often not observed in a "standard mode" of plasma focus operation with pure deuterium, particularly in large installations, but are found in smaller experiments or those with gaseous admixtures. This has led to the suspicion that filaments are not a native feature of the plasma focus phenomenon. Recent success in observation of filaments in PF-1000 in a pure deuterium operation by a novel modification of the interferometer system that allows simultaneous interferometry and schlieren photography changes this situation. This Letter looks at the implications of this development in the larger context of plasma focus physics. Conceptualization of filamentation as a native feature of the traveling current distribution behind an ionizing strong shock wave is shown to be a feasible paradigm that can be formulated as a computable model for filamentation in the plasma focus.

The dense plasma focus is a plasma discharge powered by a capacitor bank. Standard diagnostics include measurement of the time derivative of the current through and the voltage across its connections with the capacitor bank. Interpretation of this diagnostic data often involves some assumptions regarding the representation of the dense plasma focus as a time varying inductance. One of the characteristic features of the current derivative waveform is a relatively sharp dip and an associated sharp voltage spike. This has often been interpreted as a result of a rapid rise in the time varying inductance of the plasma. Sometimes, an anomalous plasma impedance is invoked. This Letter discusses instances where such interpretation creates conceptual difficulties. A first principles approach to the representation of the dense plasma focus as a circuit element reveals some fundamental problems with the traditional representation of plasma focus as a time varying inductance. The anomalous impedance is shown to be necessary to account for the difference in the motional impedance implied by a time-varying inductance in the circuit element representation and a first principles description based on Poynting's Theorem. Dynamo effects that convert post-stagnation local motion of plasma into 3-dimensional magnetic fields are shown to contribute to the effective inductance of the plasma focus and resolve the observed conceptual difficulties

This paper reports on experimental work related to a technology development project that had to be abandoned in the wake of the COVID-19 pandemic. Recognizing that the window of opportunity for timely exploitation of this technology is no longer open and also taking into account the likely future commercial value of the downstream innovations, it was decided to partially release proprietary experimental information which is of current scientific interest to the plasma focus community. This information pertains to the replacement of the traditional plasma initiation method using a sliding discharge on a glass or ceramic insulator with a novel construction. This paper discusses the motivation, theoretical background, practical realization and experimental observations. Some downstream novel plasma focus based technology concepts that are heavily dependent on this innovation are briefly described

This paper demonstrates the application of the Gratton-Vargas model (GV) 2-D snowplow model to arbitrary anode and insulator geometry. This is an abridged version of a larger project of constructing an incremental model of the plasma focus and similar devices that mainly addresses the transfer of energy from the energy storage to the plasma until it reaches the axis since the physics of plasma transformations that follow is still at the experimental discovery stage. The first stage of this incremental model is to construct a kinematic framework that allows calculation of an imaginary 3-D surface of rotation that serves the same purpose as the imaginary center of mass in mechanics. The procedure is illustrated with a simple anode shape.

The potential of the Dense Plasma Focus (DPF) for industrial applications in many fields is well recognized, although yet to be realized in practice. Particularly attractive is the possibility of its use as inexpensive industrial source of nuclear reactions for diverse high value applications such as fast pulsed neutron radiography of hydrogenous materials, non-intrusive neutron interrogation of concealed organic contraband and rapid production of short lived radioisotopes for medical diagnostics and therapy. Recently, it has been suggested that it may be possible to operate the DPF efficiently in a High-Pressure-Optimized (HPO) mode. This paper explores the design parameter space for such HPO-DPF based on the revised Resistive Gratton-Vargas (RGV) model with a view to identify a practicable set of system parameters and their scaling. The current waveform predicted by the revised RGV model for the chosen set of parameters is fitted to the Lee model to estimate the likely neutron yield., Comment: Accepted for publication in the Journal of Fusion Energy on September 19th, 2017

Recent work on the revised Gratton-Vargas model has demonstrated that there are some aspects of Dense Plasma Focus (DPF) which are not sensitive to details of plasma dynamics and are well captured in an oversimplified model assumption which contains very little plasma physics. A hyperbolic conservation law formulation of DPF physics reveals the existence of a velocity threshold related to specific energy of dissociation and ionization, above which, the work done during shock propagation is adequate to ensure dissociation and ionization of the gas being ingested. These developments are utilized to formulate an algorithmic definition of DPF optimization that is valid in a wide range of applications, not limited to neutron emission. A universal scaling theory of DPF design optimization is proposed and illustrated for designing devices working at one or two orders higher pressure of deuterium than the current practice of designs optimized at pressures less than 10 mbar of deuterium. These examples show that the upper limit for operating pressure is of technological (and not physical) origin., Comment: Accepted for publication in Physics of Plasmss

The Gratton-Vargas snowplow model, recently revisited and expanded (S K H Auluck, Physics of Plasmas, 20, 112501 (2013)), has given rise to significant new insights into some aspects of the Dense Plasma Focus (DPF), in spite of being a purely kinematic description having no reference to plasma phenomena. It is able to provide a good fit to the experimental current waveforms in at least 4 large facilities. It has been used for construction of a local curvilinear frame of reference, in which conservation laws for mass, momentum and energy can be reduced to effectively-one-dimensional hyperbolic conservation law equations. Its utility in global parameter optimization of device parameters has been demonstrated. These features suggest that the Gratton-Vargas model deserves a closer look at its supposed limitations near the singular phase of the DPF. This paper presents a discussion of its development near the device axis, based on the original work of Gratton and Vargas, with some differences. It is shown that the Gratton-Vargas partial differential equation has solutions for times after the current singularity, which exhibit an expanding bounded volume, (which can serve as model of an expanding plasma column) and decreasing dynamic inductance of the discharge, in spite of having no built-in hydrodynamics. This enables the model to qualitatively reproduce the characteristic shape of the current derivative in DPF experiments without reference to any plasma phenomena such as instabilities, anomalous resistance or reflection of hydrodynamic shock wave from the axis. The axial propagation of the solution exhibits a power-law dependence on the dimensionless time starting from the time of singularity, which is similar to the power-law relations predicted by theory of point explosions in ideal gases and which has also been observed experimentally.

Direct measurement of axial magnetic field in the PF-1000 dense plasma focus (DPF), and its reported correlation with neutron emission, call for a fresh look at previous reports of existence of axial magnetic field component in the DPF from other laboratories, and associated data suggesting toroidal directionality of fast ions participating in fusion reactions, with a view to understand the underlying physics. In this context, recent work dealing with application of the hyperbolic conservation law formalism to the DPF is extended in this paper to a curvilinear coordinate system, which reflects the shape of the DPF current sheath. Locally-unidirectional shock propagation in this coordinate system enables construction of a system of 7 one-dimensional hyperbolic conservation law equations with geometric source terms, taking into account all the components of magnetic field and flow velocity. Rankine-Hugoniot jump conditions for this system lead to expressions for the axial magnetic field and three components of fluid velocity having high ion kinetic energy., Comment: Accepted for publication in Physics of Plasmas on 14th October 2014

Experimental data compiled over five decades of dense plasma focus research is consistent with the snowplow model of sheath propagation, based on the hypothetical balance between magnetic pressure driving the plasma into neutral gas ahead and wind pressure resisting its motion. The resulting sheath velocity, or the numerically proportional drive parameter, is known to be approximately constant for devices optimized for neutron production over 8 decades of capacitor bank energy. This paper shows that the validity of the snowplow hypothesis, with some correction, as well as the non-dependence of sheath velocity on device parameters, have their roots in local conservation laws for mass, momentum and energy coupled with the ionization stability condition. Both upper and lower bounds on sheath velocity are shown to be related to material constants of the working gas and independent of the device geometry and capacitor bank impedance., Comment: Accepted for publication in Physics of Plasmas on 20th August 2014

Dense Plasma Focus (DPF) is known to produce highly energetic ions, electrons and plasma environment which can be used for breeding of short-lived isotopes, plasma nanotechnology and other material processing applications. Commercial utilization of DPF in such areas would need a design tool which can be deployed in an automatic search for the best possible device configuration for a given application. The recently revisited [S K H Auluck, Physics of Plasmas 20, 112501 (2013)] Gratton-Vargas (GV) two-dimensional analytical snowplow model of plasma focus provides a numerical formula for dynamic inductance of a Mather type plasma focus fitted to thousands of automated computations, which enables construction of such design tool. This inductance formula is utilized in the present work to explore global optimization, based on first-principles optimality criteria, in a 4-dimensional parameter-subspace of the zero-resistance GV model. The optimization process is shown to reproduce the empirically observed constancy of the drive parameter over 8 decades in capacitor bank energy. The optimized geometry of plasma focus normalized to anode radius is shown to be independent of voltage, while the optimized anode radius is shown to be related to capacitor bank inductance., Comment: Accepted for publication in Plasma Sources Science and Technology on 24th July 2014

Recent resurgence of interest in applications of dense plasma focus and doubts about the conventional view of dense plasma focus as a purely irrotational compressive flow have re-opened questions concerning device optimization. In this context, this paper re-appraises and extends the analytical snowplow model of plasma focus sheath evolution developed by F. Gratton and J.M. Vargas (GV) (Energy Storage, Compression and Switching, Ed. V. Nardi, H. Sahlin, and W. H. Bostick, Eds., vol. 2. New York: Plenum, 1983, p. 353) and shows its relevance to contemporary research. The GV model enables construction of a special orthogonal coordinate system in which the plasma flow problem can be simplified and a model of sheath structure can be formulated. The LPP plasma focus facility, which reports neutron yield better than global scaling law, is shown to be operating closer to an optimum operating point of the GV model as compared with PF-1000., Comment: Published in the Physics of Plasmas on November 4, 2013

Some features of neutron emission from dense plasma focus suggest that the participating deuterons have energy in the range of 105 eV and have a directionality of toroidal motion. Theoretical models of these devices assume that the plasma evolves through a purely irrotational flow and thus fail to predict such solenoidal flow on the scale of the plasma dimensions. Predictions of a relaxation theory are consistent with experimental data [S K H Auluck, Physics of Plasmas,18, 032508 (2011)], but the assumptions upon which it is based are not compatible with known features of these devices. There is thus no satisfactory theoretical construct which provides the necessity for solenoidal flow in these devices. This paper proposes such theoretical construct, namely, the principle of constrained dynamics, and describes an experiment which provides support for this idea. The experiment consisted of low inductance, self-breaking spark discharge in helium at a pressure ~100 hPa between two pointed electrodes separated by 30-50 mm distance kept inside a vacuum chamber mounted on a low inductance high voltage capacitor. The current derivative signal showed reproducible sharp dips at all the extrema of the damped sinusoidal discharge. A planar diamagnetic loop centered with and perpendicular to the discharge axis consistently showed a signal representing rate of change of axial magnetic flux. A qualitative explanation of observed phenomena is obtained using a simple model using the proposed principle., Comment: 26 pages, 9 figures. Accepted for publication in IEEE Trans. Plasma Sci. on January 23, 2013

Random electromagnetic fields are ubiquitous in plasmas, the most common example being electromagnetic radiation of thermal origin. They should exert a random force on electrons and ions in a plasma, adding a random component to their motion. Products of randomly fluctuating quantities, such as velocity and magnetic field, which are correlated through the dynamical equations of the two-fluid model of plasma, should then exhibit non-zero average values. Investigation of such effects requires spatial-spectral representation of the non-linear equations of the two-fluid model. Chandrasekhar-Kendall (CK) functions, their generating function and its gradient defined over an infinite domain are shown to simultaneously provide orthogonal basis for solenoidal, scalar and irrotational fields respectively, facilitating transformation from coordinate space to mode number space and back. This paper constructs a theoretical framework for studying coherent effects of random forces due to random electromagnetic fields in a two-fluid plasma and discusses some results which follow from its structure. Azimuthally symmetric modes are shown to be the sole beneficiaries of the cooperation of random modes in generating non-random effects. The formalism also facilitates investigation of the interaction between compressible dynamics, which plays a central role in plasma compression, heating and confinement and incompressible dynamics, which is involved in phenomena like turbulence and self-organization.

Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients of such expansions are integrals over products of Bessel functions in problems involving cylindrical geometry. In this paper, certain identities involving products of two and three general solutions of Bessel's equation have been derived. Some of these identities have been useful in the study of Turner relaxation of annular magnetized plasma [S.K.H. Auluck, Phys. Plasmas, 16, 122504, 2009], where quadratic integral quantities such as helicity and total energy were expressed as algebraic functions of the arbitrary constants of the general solution of Bessel's equation, allowing their determination by a minimization procedure. Identities involving products of three solutions enable expanding a product of two solutions in a Fourier-Bessel series of single Bessel functions facilitating transformation of partial differential equations representing non-linear dynamics problems into time evolution equations by eliminating spatial dependences. These identities are required in an ongoing investigation of fluctuation-driven coherent effects in a nonlinear dynamical system., Comment: 17 pages, 1 table, 1 fig