Journal of Applied Mathematics and Physics, 2014, 2, 1-9
Published Online April 2014 in SciRes.
How to cite this paper : Fisenko, S.I., et al. (2014) Thermonuclear Plasma Steady States Generation. Journal of Applied
Mathematics and Physics, 2, 1-9.
Thermonuclear Plasma Steady States
S. I. Fisenko, I. S. Fisenko, R. T. Rymkulov
Rusthermosynthesis JSC, Moscow, Russia
Received N ovemb er 2013
This report is a systematic and complemented summary of the earlier published works by the au-
thors [1-4]. The concept of gravitational radiation as a radiation of one level with the elec trom ag-
netic radiation is based on theoretically proved and experimentally confirmed fact of existence of
electrons stationary states in own gravitational field, characterized by gravitational constant K =
1042 G (GNewtonian gravitational constant) and by irremovable space-time curvature. The re-
ceived results strictly correspond to principles of the relativistic theory of gravitation and the
quantum mechanics. The given work contributes into further elaboration of the findings consi-
dering their application to dense high-temperature plasma of multiple-charge ions. This is due to
quantitative character of electron gravitational radiation spectrum such that amplification of gra-
vitational radiation may take place only in multiple-charge ion high-tem perature plasma.
Electro n, Stationary States, Stable States, Pulse High-Current Discharge, Thermonuclear Fusion
1. Introduction
Last yearsastronomical observations have brought to general relativity based astrophysics and cosmology such
notions as “inflation”, “dark matter”, and “dark energy” thus urging to elaboration of the major number of recent
alternatives to GR. New theories offer interpretation of these experimental data not invoking those notions for
they seem to be wrong or artificial to the authors of these theories. The basic concept implies that gravity must
agree with GR at least within Solar System at present epoch but may be essentially different on galaxy scale or
higher as well as in early Universe. However all experimental attempts to detect gravitational radiation (based
both on GR views and on alternative theories) yield no results. In elaboration of the relativistic theory of
gravitation (namely the relativistic theory of gravitation rather than its particular case such as GR) the authors
have obtained a model of gravitational interaction at quantum level having no equivalents and making
gravitational radiation spectrum computing possible.
2. Gravitational Radiation as a Radiation of the Same Level as Electromagnetic
For a mathematical model of interest, which describes a banded spectrum of stationary states of electrons in the
S. I. Fisenko et al.
proper gravitational field, two aspects are of importance. First, in Einsteins field equations,
is a constant
which relates the space-time geometrical properties with the distribution of physical matter, so that the origin of
the equations is not connected with the numerical limitation of the κ value. Only the requirement of conformity
with the Newtonian Classical Theory of Gravity leads to the small value
, where G, c are, respec-
tively, the Newtonian gravitational constant and the velocity of light. Such requirement follows from the primary
concept of the Einstein General Theory of Relativity (GR) as a relativistic generalization of the Newtonian The-
ory of Gravity. Second. The most general form of relativistic gravitation equations are equations with the Λ term.
The limiting transition to weak fields leads to the equation
∆Φ=−+ Λ
is the field scalar potential,
is the source density. This circumstance, eventually, is crucial for neg-
lecting the Λ term, because only in this case the GR is a generalization of the Classical Theory of Gravity.
Therefore, the numerical values of
in the GR equations are not associated with the
origin of the equations, but follow only from the conformity of the GR with the classical theory.
From the 70s onwards, it became obvious [5] that in the quantum region the numerical value of G is not
compatible with the principles of quantum mechanics. The essence of the problem of the generalization of rela-
tivistic equations on the quantum level was thus outlined: such generalization must match the numerical values
of the gravity constants in the quantum and classical regions.
In the development of these results, as a micro-level approximation of Einsteins field equations, a model is
proposed, based on the following assumption [1,2]:
“The gravitational field within the region of localization of an elementary particle having a mass m0 is cha-
racterized by the values of the gravity constant K and of the constant Λ that lead to the stationary states of the
particle in its proper gravitational field, and the particle stationary states as such are the sources of the gravita-
tional field with the Newtonian gravity constant G”.
The most general approach in the Gravity Theory is the one which takes twisting into account and treats the
gravitational field as a gauge field, acting on equal terms with other fundamental fields ([6]).
Complexity of solving this problem compels us to employ a simpler approximation, namely: energy spectrum
calculations in a relativistic fine-structure approximation. In this approximation the problem of the stationary
states of an elementary source in the proper gravitational field will be reduced to solving the following equa-
( )
ffe KeKf
′′ ′
++ +−−=
( )( )
2 222
22 2
11 21 n
elf eKKfe
rr r
λ λλ
− −−
 
22 22
22 2
11 21
elf KKeef
rr r
λ νλ
−++ +Λ=+−+−
 
( )
fr =
( )( )
0 00
= =
frdr =
Equations (1)-(3) follow from Equations (9)-(10)
S. I. Fisenko et al.
0g gK
xx x
µνµν α
µν α
∂∂ ∂
∂∂ ∂
−+ Γ−Ψ=
( )
RgRT g
µνµνµν µν
− =−−
, (10)
after the substitution of
in the form of
( )()
, exp
El lm
f rY
Ψ= 
into them and specific c omputa-
tions in the central-symmetry field metric with the interval defined by the expression [7]
( )
2 2222222
sindSc e dtrddedr
θ θϕ
=−+ −
The following notation is used above:
is the radial wave function that describes the states with a definite
energy E and the orbital moment l (hereafter the subscripts El are omitted),
are spherical functions,
K cm=
( )
( )
β κπ
Condition (4) defines rn,, whereas Equations (10) through (7) are the boundary conditions and the normaliza-
tion condition for the function
, respectively. Condition (4) in the general case has the form
() ( )
RKr RGr=
. Neglecting the proper gravitational field with the constant G, we shall write down this
condition as
( )
RKr =
, to which Equality (4) actually corresponds.
The right-hand sides of Equations (2)-(3) are calculated basing on the general expression for the energy-mo-
mentum tensor of the complex scalar field:
( )
,, ,,,0
µνµ ννµµ
++ ++
The appropriate components Tµν are obtained by summation over the index т with application of characteristic
identities for spherical functions [8] after the substitution of
( )()
, exp
f rY
Ψ= 
into (12).
In its simplest approximation (from the point of view of the original mathematical estimates) the problem on
steady states in proper gravitational field (with constants K and
) is solved by [1]. The solution of this prob-
lem provides the following conclusions.
1) With the numeric values K 5.1 × 1031 N·m2·kg2 and Λ = 4.4 × 1029 m2 there is a spectrum of steady
states of the electron in proper gravitational field (0.511 MeV0.681 MeV). The basic state is the observed
electron rest energy 0.511 MeV.
2) These steady states are the sources of the gravitational field with the G constant.
3) The transitions to stationary states of the electron in proper gravitational field cause gravitational emission,
which is characterized by constant K, i.e. gravitational emission is an emission of the same level as electro-
magnetic (electric charge e, gravitational charge
). In this respect there is no point in saying that gravita-
tional effects in the quantum area are characterized by the G constant, as this constant belongs only to the ma-
croscopic area and cannot be transferred to the quantum level (which is also evident from the negative results of
registration of gravitational waves with the G constant, they do not exist).
Existence of such numerical value Λ denotes a phenomenon having a deep physical sense: introduction into
density of the Lagrange function of a constant member independent on a state of the field. This means that the
time-space has an inherent curving which is connected with neither the matter nor the gravitational waves. The
distance at which the gravitational field with the constant К is localized is less than the Compton wavelength,
and for the electron, for example, this value is of the order of its classical radius. At distances larger than this
one, the gravitational field is characterized by the constant G, i.e., correct transition to Classical GR holds.
There is certain analytic interest in
-decay processes with asymmetry of emitted electrons [9], due to (as it is
supposed to be) parity violation in weak interactions.
-asymmetry in angular distribution of electrons was re-
gistered for the first time during experiments with polarized nucleuses 27Со60, β-spectrum of which is character-
ized by energies of MeV. If in the process of
-decay exited electrons are born, then along with decay scheme
n pe
→+ +
there will be also decay scheme
S. I. Fisenko et al.
( )
n pee
ν γν
→ ++→++
 
is a graviton.
Decay (14) is energetically limited by energy values of 1 MeV order (in rough approximation), taking into
consideration that the difference between lower excitation level of electron’s energy (in own gravitational field)
and general <100 keV and the very character
-spectrum. Consequently, 27Со60 nucleuses decay can proceed
with equal probability as it is described in scheme (13) or in scheme (14). For the light nucleuses, such as 1Н3
-decay can only proceed as it is described in scheme (13). At the same time, emission of graviton by electron in
magnetic field can be exactly the reason for
-asymmetry in angular distribution of electrons. If so, then the
phenomenon of
-asymmetry will not be observed in light
-radioactive nucleuses. This would mean that
-asymmetry in angular distrib ution of electrons, which is interpreted as parity violation, is the result of elec-
tron’s gravitational emission, which should be manifested in existence of lower border
-decay, as that’s where
-asymmetry appears to be.
Using Kerr-Newman metric for estimation of the numerical value of K one can obtain the formula [7]
222 24
(//)( //)
KmcrLLmcmrcer c
whe r e r, m, e, L are classical electron radius, mass, charge, orbital momentum respectively, and c is the speed of
Despite the fact that we used external metric and orbital momentum in deriving the formula (15), its use is le-
gitimate for the orbital momentum of a particle in internal metric equal to the electron spin by an order of mag-
nitude. The estimation of K from the formula (15) using the numerical values of the abovementioned arguments
agrees with the estimation that stands in correspondence with numerical values of electron energy spectrum in
proper gravitational field. This may suggest that the physical nature of spin is possibly such that these are just
values of the orbital momentum of a particle in proper gravitational field.
3. Spectral Lines Widening of the Radiation of Multiple-Charge Ions
Figures 1 and 2 show characteristic parts of micropinch soft X-ray radiation spectrum. Micropinch spectrum
line widening does not correspond to existing electromagnetic conceptions but corresponds to such plasma
thermo-dynamic states which can only be obtained with the help of compression by gravitational field [2,3], ra-
diation flashes of which takes place during plasma thermalization in a discharge local space. Such statement is
based on the comparison of experimental and expected parts of the spectrum shown in Figure 2(a) and (b). Ad-
justment of the expected spectrum portion to the experimental one (see [1]) was made by selecting average val-
ues of density ρ, electron temperature Te and velocity gradient U of the substance hydrodynamic motion.
Figure 1. Measured Fe He-δ line at 8.488 keV (broken curve)
compared to calculation (smooth curve) [10] .
S. I. Fisenko et al.
Figure 2. Experimental (a) and calculated (b) parts of a micropinch
spectrum normalized for line Lyβ intensity in the area of the basic
state ionization threshold of He-like ions. The firm line in variant (b)
corresponds to density of 0.1 g/cm3 , the dotted lineto 0.01 g/cm3;
it was assumed that Te = 0.35 keV, (see [1]).
As a mechanism of spectrum lines widening, a Doppler, radiation and impact widening were considered. Such
adjustment according to said widening mechanisms does not lead to complete reproduction of the registered part
of the micropinch radiation spectrum. This is the evidence (under the condition of independent conformation of
the macroscopic parameters adjustment) of additional widening mechanism existence due to electron excited
states and corresponding gravitational radiation spectrum part already not having clearly expressed lines because
of energy transfer in the spectrum to the long-wave area.
That is to say that the additional mechanism of spectral lines widening of the characteristic electromagnetic
radiation of multiple-charge ions (in the conditions of plasma compression by radiated gravitational field) is the
only and unequivocal way of quenching electrons excited states at the radiating energy levels of ions and excit-
ing these levels by gravitational radiation at resonance frequencies. Such increase in probability of ion transi-
tions in other states results in additional spectral lines widening of the characteristic radiation. The reason for
quick degradation of micropinches in various pulse high-currency discharges with multiple-charge ions is also
clear. There is only partial thermolization of accelerated plasma with the power of gravitational radiation not
sufficient for maintaining steady states [4].
4. Direct Experimental Measurements
4.1. Emission Spectra of Electron Beams Interacting with Cathode
Technically, the most accessible is the use of electron beams, taking into account the above made reservation
(narrow width of the energy levels of stationary states of the beams electrons in its own gravitational field). In
rough approximation, to estimate the numerical values of the stationary states energies in its own gravitational
field, the following values were obtained:
0.511 MeV,0.638 MeV,.....0.681 MeV.Е ЕЕ
= ==
Certainly, numerical values of the main stationary state
10.11 MeVE=
are exact, as well as the energy
range of stationary states itself, equal to 171 keV, that is, as we approach the
, numerical values of spectrum
energies are evaluated more accurately. The largest error occurs in the evaluation of the numerical value of the
first stationary state (and, surely, of the next few states); after that, an error decreases as we approach the
The idea of the experiment is that during deceleration of the electron beam on target, along with the well known
S. I. Fisenko et al.
and well studied continuous spectrum of the deceleration electromagnetic emission and linear spectrum of cha-
racteristic radiation, a gravitational emission of a linear spectrum will take place. Because of the impact of emis-
sion, the linear spectrum of gravitational emission energies will be offset relative to transition energies
21 32
( ),()EEEE−−
, etc. The magnitude of this offset
for electrons with good accuracy is given by the
0.98 [MeV]EE∆≅
It follows from (16), that the energies of the gravitational emission spectral lines are offset relative to the
transition energies for about 6 to 30 keV (again, with good accuracy for top energy levels of an electron in its
own gravitational field). Therefore, the gravitational emission linear spectrum is guaranteed to stay in an energy
range of 70 keV to 140 keV (perhaps, even shifted down by energy value). Unlike the well known methods of
recording electromagnetic emission (throughout all its range), no such methods of recording exist for gravita-
tional emission. For linear spectrum of gravitational emission the above-mentioned feature (lineation of spec-
trum) can be used, compared to continuous spectrum of the deceleration electromagnetic emission (in matching
by energies ranges). While doing so, of course, imposition of lines of characteristic electromagnetic emission on
continuous spectrum of deceleration emission should be taken into consideration (taking into account the anode
specific material).
The action of the gravitational quantum on the detector, in terms of energy, in principle, is similar to action of
the photon. The main problem will be the resolution of the detector, taking into account narrow width of the
gravitational emission lines. Thats the fact with which will be connected the main difficulty of spectrum lines
recording in the wavelength range of (0.01 - 0.04) Å (corresponding to electron gravitational emission spectrum).
Gravitational emission spectrum lines will be recorded either in form of narrow maximums (similar to characte-
ristic emission lines), or in form of minimums, depending on resolution of the detector. It is also clear that the
measurements should be conducted at various values of the accelerating potential of electrons, in any case pro-
viding for emission range of (0.01 - 0.04) Å. Again, known character of the characteristic emission spectra of
the used anode metals will allow avoiding possible errors in identification of the gravitational emission spectrum
4.2. Emission Spectra of Electron Beams on Foil for Various Materials and Their Energy
Spectra after Passing the Foil
This data, of course, should be completed both by electron gravitational emission lines and by stationary states
energy spectrum of an electron in its own gravitational field. Previously, the following measurements were car-
ried out. Figure 3 shows the energy spectra of electron beams in a pulsed accelerator, measured with a semicir-
cular magnetic spectrometer. Presence of two peaks on energy spectra is associated with an operating mode fea-
ture of the pulsed electron accelerator, i.e., the presence of the secondary peak of lower power. This leads to the
second (low-energy) peak in energy spectral distribution. Measurement error in the middle and soft part of the
spectrum does not exceed ±2%. The energy spectrum of electrons passing through the anode mesh, and spectra
of electrons passing through the foil attached above the mesh anode of the accelerator were measured with a
magnetic spectrometer. This data, as well as a calculated range, are shown in Figure 3. Similar measurements
were taken for Ti foil (foil thickness was 50 microns) and Ta foil (of 10 microns thickness). For Ti and Ta upper
limits of measurements were 0.148 MeV and 0.168 MeV correspondingly; above these limits the measurement
errors substantially increase (for the accelerator used). Figur 4 shows the difference of normalized spectral den-
sities of the theoretical and experimental spectra of electrons after passing through the Ti, Ta and Al foils. These
data indicate the presence spectrum of energetic states of electrons in their own gravitational field, which are
excited while passing through foil.
Emission spectra in this series of experiments were not measured, and the measurements should be performed
with detectors of higher sensitivity with simultaneous measuring of emission spectra. Provided data is not suffi-
S. I. Fisenko et al.
Figure 3. Electron energy spectra: 1after passing the grid, 2after passing the Al foil 13 μm thick; 3
spectrum calculation according to ELIZA program based on the database [11] for each spectrum 1. The
spectrum is normalized by the standard.
Figure 4. Difference of spectral density for theoretical and experimental spectra of electrons passed
through Ti, Ta and Al foils.
cient for numerical identification of the energy state spectrum of the electrons in their own gravitational field,
but the very existence of the spectrum, according to this data, is unquestionable.
5. Thermonuclear Plasma Steady States Generation
The problem of controlled fusion realization is directly connected to obtaining steady state of dense high-tem-
perature plasma.
0,04 0,05 0,06 0,07 0,08 0,090,10,11 0,12 0,130,14 0,15 0,16 0,17 0,18 0,190,20,21 0,22 0,23 0,24
E, MeV
0,04 0,06 0,080,10,12 0,14 0,16 0,180,20,22
E, MeV
S. I. Fisenko et al.
It can also be unambiguously stated that the present state of the art (retaining plasma by magnetic fields of
various configurations, squeezing by laser radiation) does not solve the problem of dense high-temperature
plasma retention for a time required for the reaction of nuclear fusion but only solving the problem of heating
plasma to the state when these reactions can exist. In the offered method of forming dense-high temperature
plasma steady states for nuclear fusion a new fundamental concept is used, namely retaining plasma by radiated
gravitational field as radiation of the same kind as electromagnetic:
Forming and accelerating binary plasma with multivalent ions by accelerating magnetic field in a pulse high-
current discharge.
Injection of binary plasma from the space of the accelerating magnetic field:
exciting stationary states of an electron in its own gravitational field in the range of energy up to 171 keV
with following radiation under the condition of quenching lower excited energy levels of ion electron shell of a
heavy component (including quenching excited state of electrons directly in nuclei of small sequential number
as carbon) when retarding plasma bunch ejected from the space of the accelerating magnetic field. Cascade tran-
sitions from the upper levels are realized in the process of gravitational radiation energy transit to long-wave
range .
The sequence of the operations is carried out in a two-sectional chamber of MAGO installation (Fig ur e 5,
developed in Experimental Physics Research Institute, Sarov, [12]; the structure of the installation is most suita-
ble for the claimed method of forming steady states of the dense high-temperature plasma) with magnetody-
namic outflow of plasma and further conversion of the plasma bunch energy (in the process of quenching) in
the plasma heat energy for securing both further plasma heating and exciting gravitational radiation and its tran-
sit into a long-wave part of the spectrum with consequent plasma compression in the condition of radiation
blocking and increasing [4].
Of interest, there are two modes of the installation operations depending on the work gas composition [13].
1) A composition with hydrogen and (for example) xenon providing for achieving steady states of plasma
with consequent realization of thermonuclear reactions for compositions of (d+t) + multi-charge atoms type.
Fusion reaction creating helium generates neutrons
234 1
112 0
17.6 MeVHHHen+ →++
Figure 5. Outline drawing of the discharge chamber (MAGO chamber) and X-ray diagnostic system.
S. I. Fisenko et al.
and was embodied in the well-known Teller-Ulam design with radiation implosion.
An application of the compression-by-the-rad iate d -gravitati onal-field design, unlike the Teller-Ulam design,
is not limited by the minimum discharge power attendant to the usage of a plutonium nucleus. This means both a
feasibility of this fast fusion reaction a steady mode for low-power discharges, and (under certain conditions) a
feasibility of explosive energy release without the use of fissile elements (like plutonium and uranium) for high-
power discharges.
2) A composition with hydrogen and carbon providing thermonuclear reactions of carbon cycle in plasma
steady state mode, including energy pick-up in the form of electromagnetic radiation energy
CNO cycle is a set of fusion reactions resulting in conversion of hydrogen into helium using carbon as a cata-
l yst.
In a compact notation, this cycle would be written as
( )
( )
()( )
( )
12 1313 14151512
C p,N eC p,Np,O eN p,C
γνγ γνα
( )
15 12
N p,C
reaction rounds the cycle out. The net result is that four protons turn into an α-parti cle
4He nucleus with no neutrons among end products of the cycle. Producing one helium nucleus releases 25 MeV,
the produced neutrinos carry away about 5% more of that energy. A peculiarity of such fusion reaction cycle is
that it occurs in natural conditions of astrophysical objects. At the same time, e.g. the design features of MAGO
facility allow achieving such a volt-ampere characteristics (VAC) mode that boosts the speed of carbon cycle
reactions. An implementation of carbon cycle in the gravitational compression design may possibly become a
basic design to form steady states of the thermonuclear plasma, bearing in mind low abundance of hydrogen and
lithium isotopes that react with no neutrons being produced either.
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