Yes, the electrical and magnetic field which is created by elementary particles is really small. The electrical force is proportional to electron charge which is 1.6*10^(-19) Coulon. In addition it is decreased with distance as (1/r^2). Almost the same for magnetic interaction. So if you have a fusion process the main interaction is the force between nuclons inside atomic nuclei (strong interaction). And this strong interaction is more 100 times stronger then electromagnetic interaction.

No, the magnetic fields, i.e. the spins of the nucleons, can not be neglected. If a Triton (spin 1/2) and a Deuteron (spin 1) will fuse they will build an intermediate He5 nucleus, which will decay into He4 and a neutron after a short time. The spin of the He5 nucleus must be 3/2. At low energies like in a Tokamak or in the sun this fusion reaction is s-wave dominated (96 %). Therefore, there are no contributions to the spin of the He5 nucleus. If now the spins of the Triton and the Deuteron are anti-parallel the fusion process is not possible in first order. On the other side, one can use nuclear polarized fuel with aligned spins to increase the total cross section by a factor 1.5 and influence the differential cross section to focus the neutrons on special wall areas. The same effect was even measured for the He3 + d fusion process. The spin-dependence of the d+d fusion reactions is much more complicated, because p- and d- waves must be taken into account. The necessary measurements are under preparation.

Unfortunately, no current project on controlled nuclear fusion envisages utilisation of polarised nuclei. It is difficult to inject them into the plasma, as only neutral particles can enter the latter without being deflected by the plasma magnetic field, and neutralisation of incoming ions usually involves some depolarising collision. It is also difficult to start from scratch with a fuel made of polarised nuclei, as the latter get thermalised during the heating process up to break-even and beyond, no matter what the heating process (Ohmic, radiofrequency, neutral beams) is like. Toghether with the relative smallness of electric currents involved in inter-particle interactions, this is why Coulomb electrostatic collisions only are considered when investigating nuclear fusion cross sections. This is not to say that no self-consistent magnetic field is created by particle motions relative to each other. On the contrary, strongly turbulent magnetic field may spontaneously arise e.g. in the imploding pellet of inertial fusion. Unfortunately, such turbulence raises energy diffusion coefficient further, thus making the achievement of break-even even more difficult. For example, gyro-Bohm, classical diffusion coefficient is replaced by Bohm coefficient.

You can easily get answer that question by comparing the effect of the Coulomb interaction between the two elementary charges and the Lorentz force. acting on one of them in a magnetic field produced by the motion of another. In pure classics. You will receive a ratio of the magnetic force to the electric one as  V/c. 

Fusion power density is propotional to the multiplication of density of Deuterium and that of Tritium in D-T magnetically confined plasma (ex. Tokamak plasma). And the plamsa presure, which is propotional to the density, and magnetic presure (B*B/2mu-o) is balanced. The ratio of plasma pressure to the magnetic pressure is call beta and the beta is limited by stability condition. Increasing the beta is one of the key issues in magnetically confined fusion device.  Combining the two facts, the fusion power is propotional to B**4.

It is also true that the operation limit of plasma current and density is also propotional to the magnetic field. It is a very impotant issue for increasing the fusion power and control of fusion power density.

It is also true that the construction cost of magnetically confined fusion device is propotional to B**2 and the power density is propotional to B**4 and it results that higher the magnetic field, the cost of electicity becomes cheaper too.

For obtaining the temperature and density at tokamaks it is usually used Ohmic heating ( for produce magnetic field ) and there are also auxiliary methods of heating ( NBI, ICRF, ECCD ets.)

It is impossible to heat plasma in such way, because Bremsstrahlung losses:

Article Plasma Thermal Bremsstrahlung and Stefan Boltzmann Radiation...

Stefan-Boltzmann law holds for plasmas at thermodynamic equilibrium with their own photons, a feature of stellar plasmas rather than of lab plasmas - see e.g. the discussion in F. Pegoraro, Fusion Technol. 26, 1243 (1994). Admittedly, weakly collisional, axisymmetric, toroidal, low-β turbulent lab plasmas with negligible radiation losses are locally at thermodynamic equilibrium, in the sense that the typical timescales of thermalization (= ‘relaxation to a Maxwellian of the distribution function’) of ions and electrons in a  region between the magnetic surfaces with poloidal flux ψ and ψ + dψ are much shorter than the typical timescales of particle and energy transport in the direction parallel to ∇ψ - see equations (15) and (16) of A. Rogister and D. Li , Phys. Fluids B 4 804 (1992).

All the same, as far as steady-state, magnetically confined plasmas are concerned Bremsstrahlung prevents achievement of electron temperatures above 4 KeV in Ohmically heated plasmas. This is why auxiliary heating has been actively investigated in the last decades. Unfortunately, the larger the auxiliary heating power, the lower the  energy confinement time - a well-known trend clearly identified still in the Eighties (you may look for "Kaye-Goldston law", "L-mode" amd "ITER89 scaling law" on the web).

Alternatively, unsteady plasmas could achieve break-even with the help of purely Ohmic heating. For the discussion of two different (and possibly unsuccessful) approaches to this goal, see A. Di Vita,  Nucl. Fusion 50 (2010) 115001  and Eur. Phys. J. D  67: 191 (2013).  In both cases, confinement issue seems to be critical. 

Thank you Andrea, that is useful information.

4KeV electrons may be in very low density plasmas.

If you heat ions over electrons, ions will transfer the energy to electrons and they will loss the energy.

Plasma radiates a lot, I measured 200eV plasma which size was less than a millimetre and I had to install the sensors at 6 meters distance to avoid saturate them. The emission coefficient plumbed to 1% thanks to shielding the radiation by compressing the matter 2 folds. That helps a little in inertial fusion.

We reached breakeven conditions in 2009, last year it was verified it in external installations. We did not take in account the compression of the matter because we could not measure it, so the results must be better.

I had a lot of trouble to avoid ohmic heating to save power.

By using our plasma gun we could achieve breakeven with any tokamak over 10 teslas

Ion temperatures larger than electron temperatures are an old dream of nuclear fusion, because it is the ion which is involved in nuclear fusion, not the electron.

Trouble is, the larger the difference between these two temperatures, the larger the power transferred from ions to electrons through collisions and/or turbulence (think of ion-acoustic turbulence, for example) the larger the power lost by electrons through Bremmstrahlung.  

Note that the lower the electron particle density the better, as the rate of collisional energy exchange between ions and electrons is an increasing function of electron density. In contrast, at very large values of electron density plasma becomes optically thick and it starts radiating like a black-body, i.e. much more than usual optically thin plasmas. In a pure hydrogen plasma, for example, radiated power scales as the fourth power of electron temperature in black bodies and as the square root of electron temperature according to Bremsstrahlung. In fact, the solar core looses energy through radiation, not convection; the latter dominates in the less dense photosphere.

Experimental results have been published on peer-reviewed journals about ion temperatures well above electron temperature in Z pinch (M.G. Haines, P.D. LePell, C.A. Coverdale, B. Jones, C. Deeney, J.P. Apruzese, Phys. Rev. Lett. 96, 075003 (2006))  and the Dense Plasma Focus (E.J. Lerner, S.K. Murali, D. Shannon, A.M. Blake, F. Van Roessel, Phys. Plasmas 19, 032704 (2012)). Unfortunately, both reports have received no indipendent confirmation so far. 

Plasma guns have been proposed as an alternative to pellets for plasma fueling in tokamaks for a long time, but as far as I know no planned experiment envisages their utilisation to date.

Could you show peer-reviewed publications confirming your claim of breakeven in 2009?

It is a hard work to have more ion temperature than electron one by using Z-pinches, but it is one of the main specification on next Pulsotron machines, if not we are building expensive bulbs. 

I have not peer reviewed in 2009 (I began to write articles last year), but I have this publication about ignition conditions verification. It is a Technical Report, not a scientific paper.

Data Pulsotron-2 ignition conditions verification

Yes. Fusion power density is proportional to beta^2*B^4. Beta is the plasma confinement efficiency and B is the applied Magnetic field.

They do not neglect.

See Acc. Chem. Res. 47, 417-426 (2014) and references therein

The magnetic field produce loss of energy due Bremsstrahlung.

I have wrote the formula here:

Data Useful formula and excel tables for plasma physics V03

I’m grateful to everyone for the useful and detailed answers.

Javier, it seems very important, that you observed small plasma, less than a millimeter in size. An interesting question is: What is smallest size of the plasma (with emission of neutrons, that is, with the fusion process) which has been experimentally observed anywhere in the world? 

The cross section of nucleus-nucleus collisions is small. To overcome the classical Coulomb repulsion (at 10^{-13} cm) it is required particle velocity greater than the speed of light. At the temperature of T=10^{8}K the velocity of the particle allows to bring distance only to 10^{-6} cm. The quantum tunneling effect to seven orders of magnitude (from 10^{-6} cm to 10 ^ {- 13} cm) did not inspire great optimism. So, all this could be cause to start to look for additional physics which could lead to fusion process. (Am I right?)

In two-dimensional quantum structure, the density of charged particles (each with a charge of e) depends on the transverse external magnetic field B as hc/eB (h is the Plank's constant, c is light velocity).

     The energy of these electrons is quantized (Landau levels). We proposed a simple method for the calculation of the spatial structure of each n-th Landau level. Some of the results are as follows:

    In the conventional two-dimensional metric space, the n-th Landau level splits into n segments. At the end points of each segment the particle velocity is zero, and within the segment particle can oscillate. Thus, the ends of the segments are a quantum turning points [Berkeley Physics Course, Vol.4], where the particle velocity vector changes sign, but there is non-zero probability for the particle to penetrate into the adjacent segment. So, there is possibility for particles to fill gradually all segments. However, two particles can not be in one segment because of infinitely high potential which arises in that case [Berkeley Physics Course, Vol.4].

      Thus, in two-dimensional metric space, each n-th Landau level with energy En can contain from one to n of the charged particles that are isolated one from another. Sum of the energies of the particles is equal to the energy of n-th Landau level. Thus, the more particles in this collective quantum state, the lower the energy of each particle. Furthermore, it is shown that the distance between the particles decreases (an increasing of the density) with an increase in the number of particles and with the increasing of the locally "external" magnetic field.

It is not difficult to understand that the distance between the particles can be reduced to 10^{-13} cm, and for plasma particles it is energetically favorable to form these conditions, if there are two-dimensional (of any form) flow of electrons and two-dimensional ( same form) flow of nuclei, which move toward each other and create each for another the local "external" magnetic field. Such collective states may have dimensions ( is calculated) from less than a micron to largest.

A particle, falling into this collective state, emits a photon. Moreover, if the number of particles of a quantum state is increased from n to n + 1, the energy of each of the n particles decreases, so they emit photons of the same frequency. (This is similar to the laser radiation.) Therefore, the formation of collective quantum states can be the cause of a strong plasma radiation, although fusion process has not yet begun. Only if, together with the emission of soft photons, the peak appears in the emission of hard photons and neutrons, this is a signal of the beginning of fusion process.

So, it's very interesting to know which minimal dimensions of the plasma with the simultaneous emission of hard photons and neutrons are observed in the experiment? What is the spectrum of photons? How to estimate the number of photons from one quantum state (it's theoretical aim)? 

If there is here any truth, we can construct a set of individual micro-size centers of nucleosynthesis with the given collective quantum states. (Perhaps it and there is thing that happening in many plasma experiments now?)

Thus, from the experiment, we need learn more about plasma instabilities with micro(nano)-dimension.

@ Lugovoi. It has been shown that huge magnetic fields have favorable impact on Coulomb logarithm [J.R. McNally, Nucl. Fusion 15, 344 (1975)]. Quantization of angular momentum implies that electrons in a magnetic field can have only discrete energy (“Landau”) levels. Electron Larmor radius cannot, e.g. be smaller than the De Broglie wavelength of the electron. The difference in energy between adjacent Landau levels is h*Ωi/(2*π), where h and Ωi are Planck’s constant divided by 2 and the ion cyclotron pulsation, respectively. Energy transfer from ions (energy Ei, mass mi) to electrons with mass me is reduced (E.J. Lerner, S.K. Murali, A. Haboub, J. Fusion Energy 30, 367 (2011)) if (me/mi)*Ei < h*Ωi/(2*π), where we have neglected the electron motion parallel to the magnetic field line. This seems to be a reasonable assumption, provided that the magnetic field is large enough. This “quantum suppression” effect has been originally investigated in astrophysics (G.S. Miller, E.E. Salpeter, I. Wassermann, Astrophys. J. 314, 215 (1987), U. Das, B. Mukhopadyay, Phys. Rev. Lett. 110, 071102 (2013)). Something similar occurs in degenerate plasmas, where some transitions between electron states are forbidden (S. Son, N.J. Fisch, Phys. Lett. A 329, 76 (2004)). As for boron ions at 300 KeV, the inequality above is satisfied provided that B > 1.3 × 10^5 T.

Andrea, we have the same thing, that is, the charged particles (electrons, ions) has discrete levels in a transverse magnetic field. We have used the wave function of a quantum oscillator (which is also the wave function of a charged particle in a transverse magnetic field), and proposed a method to obtain the trajectory of motion of a quantum ensemble, corresponding to this quantum oscillator.

This trajectory is the line on which there are the quantum turning points, the coordinates of these turning points are calculated (see attached Fig.1). Thus, the trajectory is divided into isolated from one another segments, the ends of which are the quantum turning points. Each isolated segment can be filled (or not be filled) with electrons, the movement of which is limited in the space of one of the segments.

The result looks like the quantization of the metric space, where each isolated cell can be filled by the charged particle. In other words, we propose a method for calculating of the spatial quantum structure of charged particles (electrons, ions). Thus, a wave function of a quantum oscillator with quantum number n corresponds to the energy E_ {n} and simultaneously describes one such quantum structure with the number n of isolated segments, which together can contain from one to n electrons (ions).

If everything is correct, we can conclude that we can quantize the metric space, but still can not obtain the rigorous description of quantization of energy of the electrons entering the quantum structure. (Landau investigated the electron, which is not part of any structure.) Therefore, we simply ASSUME that the energy of each of m electrons (m varies from one to n) is equal to the energy of a quantum oscillator E_ {n} , divided by the number of electrons m (see attached Fig.2). Thus, in the plane transverse to the vector of the external magnetic field B, if we have one electron (m = 1), its energy is equal to the energy of the Landau level. If m> 1, there are strong grounds to seek in this transverse plane an above described quantum structure, where the energy of each electron is E_ {n} / m (see attached Fig.2).

This ASSUMPTION and the quantization of the metric space, it seems, can explain many unusual and conventional mechanical and electronic properties of two-dimensional structures, wherever they arise, including plasma (web of two-dimension tubes-filaments, transverse pinching, huge emission of light, ... ).

In our approach, De Broglie wavelength is always less than the average width of the segment     \Delta х  = sqrt{ hc / (eB2*π)} •\Delta \xi, where  \Delta \xi     is calculated using the proposed method (see attached Fig.1). The dependence of \Delta \xi    on n is given in attached Fig.3. The value of   \Delta \xi   that is, of   \Delta  х   ) decreases rapidly with increasing n ( see attached Fig.3). Thanks to the tunnel effect between one of the pair of adjacent segments, nucleosynthesis can be started before   \Delta х    reaches the value of 10 ^ {- 13} cm. However, in principle, if the configuration of the (local) magnetic field creates conditions for the formation of quantum structures with super huge n, explosion like an explosion of supernova can occur.

It is very interesting to learn that "energy transfer from ions (energy Ei, mass mi) to electrons with mass me is reduced ... in degenerate plasmas ... for boron ions at 300 KeV ..., .. B> 1.3 × 10 ^ 5 T." Within our approach, it is a signal of the beginning of formation of collective quantum structures of ions and of collective quantum structures of electrons (which move in magnetic field in opposite each other directions), which grow (the number of n grows), the particle density grows               ( because of     \Delta xdecreases), and these two types of the collective quantum structures create each other the more and more value of local magnetic field (see   \Delta х   and attached Fig.3). This is the way to the beginning of nucleosynthesis.

Let me emphasize that in our approach, it is important not large external magnetic field (generated by the equipment), but it is important to create local conditions for the emergence of local quantum structures and conditions for their growth. This will lead to a large local magnetic field, to the required greater local density (see above), and for the plasma of millimeter (microns) in size. Seems Javier Luis Lopez describes a similar event. It would be very interesting to learn more about these (and similar) cases because there is all kinds of ideas that we would like to check. Thanks. Vladimir

All good and well, but admittedly I got lost at your words: "it is important not large external magnetic field (generated by the equipment), but it is important to create local conditions for the emergence of local quantum structures and conditions for their growth. This will lead to a large local magnetic field, to the required greater local density (see above), and for the plasma of millimeter (microns) in size." As I mentioned, magnetic-field related quantum effects become measurable at B > 1.3*10^5 T, i.e. at pressure > 10^16 Pa (if beta is not too far from 1, as expected in any successful nuclear fusion experiment). You are therefore on the verge of spontaneous transformation of protons in neutrons (farewell to fusion...) even if the density of your plasma is equal to the density of a solid (10^30 m^-3). At such density, even quantum-depressed Bremsstrahlung would prevent you from achieving break-even - in order to convince oneself, Lawson' s criterion is enough. 

With pulsotron-2 I tried to increase more and more the magnetic field an the pressure, but I realized that I lost the energy by radiation due extreme high density (bremsstrahlung rad=k**Z1*Z2*N1*N2*sqrt(T), where N1,N2 are the particle density) but unfortunately I had no way to introduce small amount of fuel inside to try ignition. The better way to try ignition using 0.2*10^5T would be to build and operate the new  capacitors that I designed using a new version of Pulsotron-2, I would need at least 1 year (and some funding, no much)

Actually we are working in pressure reduction  and increase the temperature to 500keV to go here (see attached diagram). Is a good idea to lower magnetic pressure and work in that coordinates to try non conventional fuels?

Yes, Andrea, you are right for independent each other particles, which are like molecules in almost ideal gas. Plasma is an object which continuously forms different quantum structures, which consists of intensively interacting each another particles. Within our approach, in local transverse magnetic field, the protons continuously form collective quantum structures (see Fig.2 in attached file) where the energy of each of these protons (see right part in Fig.2) GROWS if one of protons leaves this collective quantum structure (for example, it’s transition from Fig.2m to Fig.2k ). Any physical system strives for the minimal energy. Therefore, it is suppressed any physical process which leads to the loss of one of protons from collective quantum structure. From other side, energetically favorable is to get one more proton into that collective quantum structure (for example, it’s transition from Fig.2k to Fig.2m ). Thus, these structures are very stable and have tendency to grow. They continuously are broken , if a local magnetic field changes its direction, but they also continuously are created in the surface transverse to that new direction of the vector of local magnetic field. Probably, this is “esse” of dynamics of infinite diversity of forms of plasma.

         Now the question: “What does create a local magnetic field?”

Protons create one type of collective quantum structure (Fig.2). Electrons create the other type of collective quantum structure (Fig.2). In the magnetic field these quantum structures are moving in opposite each other directions, and so, they create each other these “external” (=local) magnetic fields. Experimenters clearly observe these namely very strongly connected structures, which has very different temperatures (because of the difference in the mass between the protons and electrons). Surfaces of these structures flow around each other (the tubes of two-colored filaments, or two flat surfaces with different temperatures, …).

             Of course, more high temperature can give more high local magnetic field and the described above self-organization of high density plasma can be started. What is concerning the different fuels with different masses. It is difficult to predict.

In our formula \Delta х = sqrt{ hc / (eB2*π)} •\Delta \xi (see above), there is not the mass m of particle. However,  the dependence on the mass m of particle is for cyclical frequency \omega = eB/(mc), which is shown in Fig.2.

          Because of dependence of \Delta x on the charge of particle of e and its independence on the particle mass of m, and dependence of cyclical frequency \omega on the particle mass, our approach for huge mass particles, most likely, is the way to obtain the trap for nano(micro)-particles to do different mechanical manipulations with them. The Fig.2 shows some type of trap. 

           However, self-organization can be started for charged particles with any mass. Therefore, any experimental variations would be interesting.

In order to invoke quantum mechanics the linear size of your system should be not much larger than the De Broglie length. For protons at temperatures of thermonuclear interest (say, 10KeV), this length is about 10^-13 m, i.e. about the linear size of a single nucleus. Unless you find a way to overcome the Coulombian barrier at temepratures

De Broglie’s length of wave was offered for the FREE elementary particle as \lambda = h/sqrt{2mE}. Thus we can estimate the size of FREE elementary particle.

You suppose, the De Broglie’s length of wave could be applied to estimate the size already of any quantum structure. That is, the size of quantum structure “should be not much larger than the De Broglie length”. Well.

A neutron star is a composite multi-particle quantum structure. For example , it can be neutron star, rotating as a part of two-star system. Nevertheless, within formal logic, its (as whole) De Broglie length of wave is on the much more orders less than the size of atom, or on the much more orders less than its linear size. Therefore, within formal logic, the classical approach has to explain the properties of a neutron star, including a vacuum polarization (on the star surface), which is a quantum properties. So, we obtain a logical conflict. Thus, statement that the size of quantum structure "should be not much larger than the De Broglie length " sometimes can lead to contradiction. (Additionally see below)

It is very easy to find the quantum structures in plasma. For example, they are filaments, described above (in frame of our approach). It seems, we should only to find right form of magnetic field, which assists the creation of structures with the maximal possible value of quantum number of n ( Figs.2c, 2g, 2o, …). That is only starting point to self-organization of ANY quantum structure, which can have any size. For samples see deep into Universe, where multiparticle quantum structures could be as a “glue” which holds together the nebula with unusually small mass which is insufficient for the gravitational hold of the nebula as a whole. That is, for an explanation of this phenomenon, the dark matter is not required. Namely, the forms of local fields can create this structure.

Moreover, let us consider a metrical space where there are only two particles in very small external magnetic field. For example, it’s B=10^{-20}T. In this case the averaged width of intervals (see Fig.1) is \Delta x = sqrt{hc/(eB 2)} \Delta \xi = 100 meters ! These two particles can be separated each from another by a few empty intervals (Figs.2k). In sum it can be up to kilometers, but these two particles are described by one wave function, and so, each of them “feels” the second particle without visible interaction each other. It’s like in the quantum non-locality (quantum entanglement), which , it seems, is confirmed in many independent different experiments.

In the multiparticle quantum structure, each particle is coherent (indiscernible) part of this structure. It is so because one wave function describes this multiparticle quantum state. And this is not a common theoretical words. This is because these words reproduce a strict succession of our calculations : we take the wave function of quantum oscillator, put this wave function into well known operator equation p=mv and obtain Figs.1,2 in the metrical space.

The problem of cold synthesis (within of our approach)

is a problem of the Coulomb repulsion of two electrons, each of which is shown in Figure 1a and each of which is in different local transverse magnetic fields. As soon as they appear (by chance) in common one transverse magnetic field, instead of classical approximation the quantum approximation will start working.

That is, each of these two electrons will pass into state of Fig.1b and then they emit the excess energy and inevitably build one energetically favorable state of Fig.1c in the plane transverse to the vector of magnetic field. The quantum state of Fig.1c will exist as long as the change of external transverse magnetic field will not occur.

The obvious proof of this sequence of events is that as soon as the plasma experimenters provide freedom of action for plasma, it begins self-organization (pinch effect, filamentation, and so on). This is nature of plasma.

You need no quantum mechanics to describe spontaneous filamentation in plasmas. See e.g. A. Di Vita Eur. Phys. J. D 54, 451–461 (2009). As for De Broglie, I am afraid I am not able to see any 'logical conflict'. What does matter is the relative length of the (thermal) De Broglie length and interparticle distance (or, as you like, the inverse cubic root of particle density). The larger the mass of the individual particle, the cooler the system and the larger the particle density the easier the feasibility of a quantum-mechanical description. 

I have tried to explain that if two ions are described by one common wave function (see. Figs.2c, 2e, 2h), for each of them separately de Broglie wavelength can not be entered correctly. De Broglie wavelength can be entered correctly only for the states shown in Figs. 2a, 2b, 2d, 2g. Further. What is the de Broglie wavelength of a particle, it is clear. However, I can not correctly determine the de Broglie wavelength of the quantum structure of the two particles which are indistinguishable (described by only one wave function).

In your argument you believe that there is no the quantum structures (there is no Figs. 2c,f,g,k,m,o,...), that is, you consider only Figs.2a,b,d,h,... In frame of this approximation, your reasoning seems as quite well (only in some approximation).

Further. The basic principle of quantization is to limit the particles in space (potential well). In our approach, it is obtained naturally. Therefore, the quantization conditions are fulfilled in 100%, that is, in Fig.2 the quantum systems are shown. Of course, it seems questionably that total linear size of multiparticle quantum structure with fixed quantum number of n grows with quantum number of n. However, this is directly connected with properties of function of quantum oscillator. For example, it is not strange to have non-zero probability to find quantum oscillator in any points....

      Generally speaking, our approaches are different. 

Schematically speaking, in the framework of quantum physics, for individual charged particles , I'm trying to do quantitative calculations, moving in sizes from the (sub) angstroms to millimeters. You work in the framework of hydrodynamics and thermodynamics, which are essentially statistical physics, moreover, this is approximation of a continuous medium. Quantum processes appear as quasi-classical approximation for medium.

You do not care as two, three, five nuclei begin to form filamentation. You start to work with millions of nuclei. Your approach assumes that it is a liquid. In it you learn instability. Our approach shows that the number of such instabilities infinitely many and far not each instability can lead to fusion. Our goal - is to learn how to manage the process of the birth and development of "useful" instabilities starting from several nuclei and electrons in the "correct" magnetic field. To date, we believe that the obtained formula \Delta x = sqrt{hc/(eB 2)} \Delta \xi and Figure 3 will help us.

Perhaps it is good that people are moving from various directions, seeking to achieve the same goal.

Even without additional calculations, already Figure 3 shows that the fusion process can be achieved into the instability, created "correctly" by quantum structures with a very large quantum number of n and with very large conductivity (between the electronic and ionic structures as whole objects). 

It seems, here your approach of continuous medium can begin to work , that is, the plasma will be a liquid with a high density and conductivity. Because of the very high conductivity, the process will begin to develop very quickly in the right direction, if we create the right shape of instability.

I think that if you create instability in a direction you would reduce the confinement time. In the Pulsotron instabilities leads to a dramatic reduction in confinement time so short that the capacitor bank remains totally charged after unsuccessful shots. (In successful shot it discharges about 97% of the energy in the target)

Andrea, in the attached file, it is quantitatively shown that, regardless of the temperature of the plasma, the de Broglie wavelength of particle(proton), belonging to obtained quantum structure, is more than distance between two adjacent particles(protons) of this structure. This proves the validity of the application of quantum mechanics in our approach.

Actually, the quantity En you invoke in your Eq. (3) is the energy of the -nth level of a quantum oscillator involving a single particle. Accordingly, it is the energy of no multiparticle structure (but in the trivial case of n uncorrelated particles, each of them in the fundamental state). Indeed, your words ' this multiparticle quantum structure is described by one wave function of quantum linear harmonic oscillator which corresponds to the energy En' seem to be a circular argument. In fact,  we are allowed to speak of 'multiparticle quantum structure' once we assume that  De Broglie length is larger than the typical interparticle distance in the classical sense, i.e. the reciprocal of the cubic root of the particle density. Smallness of De Broglie length may be an assumption (if any...), not a result, of Eq. 3).

Yes, it’s difficult to understand, if there are jumps in the logic. So let me explain our logical sequence in short form (see an attached “our-logic.pdf” file) and describe an experimental test for the formula E_{e}=E_{n} /m (see an attached “our-calculation.pdf” file).

In fig.2 (see “our-calculation.pdf”) the “circular” variations of the energy E_{e} (with growing “n” and “m”) are connected with the “circular” variations in the density of electrons and with the growth of external magnetic field B.

The absolutely same “circular” variations was observed experimentally for conductive electrons, moving in transverse magnetic field B in two-dimension graphene (see figs.1,4 in attached “experiment.pdf” file or http://lanl.arxiv.org/abs/0705.2180 ).

In attached “our-calculation.pdf” file one can find the method for calculation of values of these density fluctuations and the minimal size and “intrinsic disorder length scale” (in metrical space) where these density fluctuations can occur. In this file , one can find a quantitative agreement between the theoretical calculations and the experimental data.

It seems, a cited (in “our-logic.pdf” file) answer of Philip R. Howie (University of Cambridge) could be explanation that this is not trivial result.

Moreover, our approach predicts that these density fluctuations can occur in different multi-form structures (see Fig.2 in “our-calculation.pdf”), and the observed Electron-Hole Puddles on Graphene surface can be explained by the same way because any real experimental graphene sample has many little local deformations (on the surface of the substrate) which produce different local magnetic fields [Phys. Rev. Lett. 97, 016801 (2006)] and so, the different local structures (Fig.2) can be formed in plane transverse to the vector of every different local magnetic field. Thus, the colored tracery in Fig.3A (“experiment.pdf” , http://lanl.arxiv.org/abs/0705.2180 ) gives information concerning to even little deformations on the graphene surface (when the average carrier density is zero). This can have interesting practical applications..

Here "our-calculation.pdf" file is added

Here "experiment.pdf" file is added

Thank you for your explaination. Indeed, no doubt quantum mechanics only is able to describe electrons in graphene. All the same, I am able to find no support to your step no. 15 in any conceivable thermonuclear plasma. For example, the caption of your Fig. 3 refers to 'best fit to the data using an effective Fermi velocity'. The latter quantity is relevant to a degenerate plasma made of electrons (i.e. fermions); the corresponding kinetic energy is less than 10 eV. In contrast, fusion-relevant plasmas are usually made of nuclei of some isotope of hydrogen, are far from degenerate and are much hotter than 1 KeV.  Then, I am afraid I am not able to grasp your point.

                 You cite experimenters. 

The modern theoretical quantum approach supposes actively to use a Fermi sphere for the momentum (energy) distribution, where the particles are tightly packed and have the minimal allowed energy. We absolutely do not use any idea, connected with the Fermi sphere. However, in our approach [ ISBN 978-3-659-48612-8 ] , in the plane transverse to the external magnetic field, the multiparticle quantum structures, with any fixed quantum number of “n”, can be formed (they were named as United Quantum Oscillator , or UQO, see attached Fig.5), where the particles are tightly packed and the energy per one particle is minimal ( for the given number of particles of system, Fig.5c,g,o ; just for the number of particles, being, namely in the plane transverse to the external magnetic field ). Thus, some properties of Fermi sphere , it seems, look like properties of UQO. The difference between them is that the Fermi sphere approach is working well for not big energies, whereas UQO can be formed at any energy (to obtain UQO, any of theoretical restriction in energy have not been taken into account). Probably, just this was one of a crucial aspects to get quantitative and qualitative explanations of experimental data for conductivity and mechanical properties of two-dimension system like a graphene [ ISBN 978-3-659-48612-8 ], for correlations between the energy per one electron and the density of electrons in the changing external magnetic field, etc.

Thus, UQO can be formed at any energy, and, to find UQO ,we always have to look for the charged particles , being on the locally formed two-dimension surface , which is transverse to (also locally) external (to this surface) magnetic field which can be formed, for example, by other multiparticle quantum two-dimension structure (UQO), which moves in opposite direction. That, possible, could be found just on the surface boundary between the medium consisting only of electrons and the medium consisting of only ions of plasma. Life time of surface boundary of these two mediums has to be enough for formation of UQO.

                         So, two questions arise : 

1) Where can we look at that ?

2) What is a “crucial experiment”, which could unambiguously point on the manifestation of the properties of UQO?

                        An answer on first question is: 

in the patented experiment (see attached EXP.PDF , Fig.1) where the loop of plasma rotates (see Fig.2a) in the transversely intersecting magnetic and electric fields, where the ions and electrons of plasma are moving along well known trajectory, named as cycloid [L.A. Artsimovich, S.Yu. Lukyanov. The moving of charged particles in the electric and magnetic fields. Nauka, Moscow,1972,p.224]. In attached EXP.PDF, in the photo of the plasma loop (see Fig.2a), one can see separately moving (towards each other) two types of flows. One of them (blue) consists of (more hot) electrons. The second one consists of (more "cold") ions. These two type of flows are moving in opposite each other directions (between cathode, in centre, and anode , on the periphery, which is the circle of the coil). A constant accelerating voltage between cathode and anode is 4.5 kV. External (to the plasma loop) constant magnetic field is transverse to surface of this photo in Fig.2a. This loop of plasma is oscillatory circuit, which due to the resonant self-excited oscillations can accumulate the energy. Inductance was done high to get more magnetic energy. The coefficient of pumping of energy equals 390. An approximate temperature of plasma electrons is 80 million degrees.

                           Now an answer on second question is: 

        Some time ago, a very interesting and intriguing “incident” occurred. 

Experimenters tried to reach the maximal values of all parameters of the experimental equipment. During this process, the energy of a loop of plasma was increased. Suddenly, energy of plasma passed a way back, through the electrical circuitry of experimental equipment and further on a high-voltage cable (with a transverse section equal to a thickness of human hand). This high-voltage cable had the length of hundreds meters. Further the transformer substation was spoiled, including all its insulators. The metal wires of cable were speckled with the deep hollows, filled by the metal powder which looked like a dust, but no traces of melting of metal existed. Proportionally big noise from explosion (as it happens in processes with the thin exploding wire) was not too.

It seems, this can not be explained by hydrodynamics or thermodynamics.

Thus, let us remember one important property of UQO.

In attached Figs.5c,g,o , one can see UQO which are totally filled by electrons and correspond to the quantum numbers of n=2,3,4 respectively.

On the right of the Fig.5 it is given the energy per one electron.

In [ ISBN 978-3-659-48612-8 ] it is shown that the electron in Fig,5b can be considered as a valence electron of one atom and this electron in Fig.5c is after formation of molecular bond (by two valence electrons from two atoms). A comparison of the energies per one electron in Figs.5b and 5c shows that it is energetically favorable for the valence electron to form the molecular bond.

Let us imagine, UQO with quantum number of n=3 (and totally filled by electrons, Fig.5g) is moving near a valence electron, which is part of the molecular bond which corresponds to UQO with quantum number n=2 (Fig.5c).

A comparison of the energies per one electron in Figs.5b and 5o shows that it is energetically favorable for each of four of these electrons to form UQO with n=4 (Figs.5o).

As a result, we will have gotten decay of molecular bond. A pure element of periodic table can be obtained from chemical compound.

As a result we have gotten a powder (which looks like a dust) instead of high-voltage cable.

       So, this “incident” can be “crucial experiment”, which could unambiguously point on the manifestation of the properties of UQO.

Here it is necessary to remember that if we create condition to form UQO (see above) , UQO has tendency to grow because it is energetically favorable for the particles to be in UQO (see Fig.5), and it is energetically not favorable for the particles to leave UQO. For the particle, included in UQO, to leave UQO means to interact with any particles which are out of UQO. So, the probability for any transfer of energy between the particles, included into UQO, and the particles (atoms, nuclei, electrons, etc.), which are out of UQO, is strongly suppressed. However, in principle, this suppression is the way to get a HIGH-TEMPERATURE SUPERCONDUCTIVITY and SUPERFLUIDITY.

           (This is my answer also on Javier Luis López's question.) 

please, look at   http://dx.doi.org/10.4236/jmp.2015.67103