Journal of Modern Physics, 2013, 4, 528550 http://dx.doi.org/10.4236/jmp.2013.44075 Published Online April 2013 (http://www.scirp.org/journal/jmp) KamLANDExperiment and SolitonLike Nuclear Georeactor. Part 1. Comparison of Theory with Experiment V. D. Rusov1*, D. A. Litvinov1, E. P. Linnik1, V. N. Vaschenko2, T. N. Zelentsova1, M. E. Beglaryan1, V. A. Tarasov1, S. A. Chernegenko1, V. P. Smolyar1, P. O. Molchinikolov1, K. K. Merkotan1, P. E. Kavatskyy3 1Department of Theoretical and Experimental Nuclear Physics, Odessa National Polytechnic University, Odessa, Ukraine 2State Ecological Academy for Postgraduate Education and Management, Kyiv, Ukraine 3Research Institute of Sustainable Development Technologies “Live Earth”, Kyiv, Ukraine Email: *siiis@te.net.ua Received December 2, 2012; revised January 5, 2013; accepted February 1, 2013 Copyright © 2013 V. D. Rusov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We give an alternative description of the data produced in the KamLAND experiment, assuming the existence of a natural nuclear reactor on the boundary of the liquid and solid phases of the Earth core. Analyzing the uncertainty of antineutrino spectrum of georeactor origin, we show that the theoretical (which takes into account the solitonlike nu clear georeactor) total reactor antineutrino spectra describe with good accuracy the experimental KamLANDdata over the years of 20022007 and 20022009, respectively. At the same time, the parameters of mixing (25 21 2.5 10m 252 21 7.49 10eVm 2 12 tan 0.436 eV2, ) calculated within the framework of georeactor hypothesis substantially differ from the parameters of mixing (, 2 12 n 0.437 ta ) obtained in KamLANDexperiment for total exposure over the pe riod of 20022009. By triangulation of KamLAND and Borexino data we have constructed the coordinate location of solitonlike nuclear georeactors on the boundary of the liquid and solid phases of the Earth core. Keywords: KamLAND Experiment; Borexino Experiment; Nuclear Georeactor; Neutrino Oscillations; Parameters of Mixing 1. Introduction It is obvious now that the experiments by the Kam LANDcollobaration over the last 8 years [15] are ex tremely important not only for observation of reactor antineutrino oscillations, but because they make it possi ble for the first time to verify one of the most vivid and mysterious ideas in nuclear geophysics—the hypothesis of natural nuclear georeactor existence [620]. In spite of its singularity and long history, this hypothesis becomes especially attractive today because it lets one explain clearly from the physical standpoint different unrelated, at the first glance, geophysical anomalous phenomena the fundamental nature of which is beyond any doubt. First of all it concerns the problem of 3He and 4He iso topes origin in the Earth interior, the concentration ratio of which, as is well known, “mystically” increases towards the center of Earth [21,22]. This is practically impossible to explain by existing models of the origin of the anoma lous 3He concentration and 3He/4He ratio distribution in the Earth interior since they have serious contradictions. For example, Anderson et al. has pointed out [23]: “The model whereby high 3He/4He is attributed to a lower mantle source, and is thus effectively an indicator of plumes, is becoming increasingly untenable as evidence for a shallow origin for many high 3He/4He hot spots ac cumulates. Shallow, low 4He for high 3He/4He are logi cally reasonable, cannot be ruled out, and need to be rig orously tested if we are to be understand the full implica tions of this important geochemical tracer”. Apparently, the most advanced model, which is devoid of the men tioned contradictions, is the socalled GonnermannMuk hopadhyay model, preserving noble gases in convective *Corresponding author. C opyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 529 mantle [24]. However this model ignores the possible high concentrations of 238U and 232Th in the outer core (as it is shown by numerous laboratory experiments [16, 18,20]), and this is the weak point of this model. At the same time, it is shown [17] that, if the nuclear georeactor exists, within the framework of model, which takes into account the georeactor thermal power and distribution of 238U and 232Th in the Earth interior, it is possible also to obtain a good description of the known experimental 3He/4He distributions in the crust and mantle. A potent argument in favour of the nuclear georeactor existence are the results of recent seismotomography re searches of the anomalous high heat flow (13 ± 4 TW) on the coremantle boundary. This heat is much higher than the radiogenic heat in the lower mantle (D''region) [25]. To explain such an anomalous high heat flow the authors of this paper have advanced the hypothesis of young solid core of the Earth with its crystallization energy as a cause of anomalous temperature effect. In full measure it concerns the known problem of na ture of an energy source maintaining the convection in the Earth liquid core or, more precisely, the mechanism of magnetohydrodynamic dynamo generating the Earth magnetic field. It is obvious, that the wellknown 40K mechanism of radiogenic heat production in the solid core of the Earth does not solve the problem on the whole, because it can not explain the heat flows balance on the coremantle boundary (see [26] and refs therein). It is also worth mentioning the socalled mechanism of the Earth magnetic field inversions closely associated with the problem of convection in the Earth liquid core. It seems to be strange, but both these fundamental prob lems have a simple and physically clear solution within the framework of hypothesis of existence of the natural nuclear georeactor on the boundary of the liquid and solid phases of the Earth [17,27]. If the georeactor hypothesis is true, the fluctuations of georeactor thermal power can influence on Earth global climate in the form of anomalous temperature jumps in the following way. Strong fluctuations of georeactor ther mal power can lead to the partial blocking of convection in the liquid core [27] and the change of an angular velo city of liquid geosphere rotation, thereby, by virtue of a conservation law of Earth angular moment to the change of angular velocity of mantle and the Earth surface, respec tively. It means that the heat or, more precisely, dissipa tion energy caused by friction of Earth surface and bot tom layer can make a considerable contribution to total en ergy balance of the atmosphere and thereby to influence significantly on the Earth global climate evolution [27]. However, in spite of the obvious attractiveness of this hypothesis there are some difficulties for its perception predetermined by nontrivial properties which georeactor must possess. At first, natural, i.e. unenriched, uranium or thorium must be used as a nuclear fuel. Secondly, tra ditional control rods are completely absent in the reactiv ity regulation system of reactor. Thirdly, in spite of the absence of control rods a reactor must possess the prop erty of socalled inner safety. It means that the critical state of reactor core must be permanently maintained in any situation, i. e. the normal operation of reactor is auto matically maintained not as a result of operator activity, but by virtue of physical reasonslaws preventing the ex plosive development of chain reaction in a natural way. Figuratively speaking, the reactor with inner safety is the “nuclear installation which never explodes” [28]. It seems to be strange, but reactors satisfying such un usual requirements are possible in reality. For the first time the idea of such a selfregulating fast reactor (so called mode of breedandburn) was expressed in a gen eral form at II Genevan conference in 1958 by Russian physicists Feynberg and Kunegin [29] and relatively re cently “reanimated” as an idea of the selfregulating fast reactor in travelingwave mode of nuclear burning by L. Feoktistov [30] and independently by Teller, Ishikawa and Wood [31]. To interpret the experimental KamLAND antineutrino spectra [35] we consider below the properties of such an unusual reactor. 2. SolitonLike Nuclear Georeactor and the KamLAND Antineutrino Spectra (Experiments over the Period of 20022004) The main idea of reactor with inner safety consists in selection of fuel composition so that, at first, the charac teristic time τ of nuclear burning of fuel active (fissile) component is substantially greater than the characteristic time of delayed neutrons production and, secondly, nec essary selfregulation conditions are fulfilled during the reactor operation (that always take place when the equi librium concentration of fuel active component is greater than its critical concentration [30]). These very important conditions can practically always be attained, if among other reactions in a reactor core the chain of nuclear transformations of the Feoktistov uraniumplutonium cy cle type [30] 238239 239 239 U,U Np Pu ,fission β β n n (1) or the TellerIshikawaWood thoriumuranium cycle type [31] 232233 233 233 Th ,ThPa U,fission β β n n (2) are appreciable enough. In both cases the active components of nuclear fuel are the generated fissile isotopes of 239Pu (1) or 233U (1). The Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. JMP 530 n n characteristic time of such a reaction, i.e. the respective βdecay time, is approximately equal to τ = 2.3/ln2 ≈ 3.3 days and τ ≈ 39.5 days for reactions (1) and (2), respec tively. This is several orders of magnitude greater than the time of delayed neutrons production. uranium cycle (2), but in time τ = 39.5 days. The details of the system of kinetic equations for com ponents of nuclear fuel and neutrons (as a diffusion ap proximation) in such chains are rather simple and were described in detail in our paper [17]. Typical solutions for such a problem have the form of solitonlike concen tration wave of nuclear fuel components and neutrons (Equations (3)(9) in [17]) and are shown in Figure 1. Within the framework of solitonlike fast reactor theory it is easy to show that the phase velocity u of nuclear burning is determined by following approximate equality [32] Selfregulation of nuclear burning process (under indi cated above relation between the equilibrium and critical concentrations of fuel active components [30]) takes place because such a system, being left by itself, cannot pass from a critical state to reactor runaway mode as the critical concentration is bounded above by the finite value of plutonium equilibrium concentration, i.e. Pu > crit . On phenomenological level the selfregulation of nuclear burning is manifested as follows. Increase of a neutron flux due to some reasons will result in rapid burnup, for example, of plutonium, i.e. in decrease of its concentration, and therefore in decrease of neutron flux, while the new nuclei of 239Pu are produced with the same rate during τ = 3.3 days. And vice versa, if the neutron flux is sharply decreased due to external action, the burn up rate decreases too and the plutonium accumulation rate will be increased as well as the number of neutrons produced in a reactor after approximately same time. Analogous situation will be observed for the thorium 6 42 2 2 864 exp , 29π 3π π 4 β crit fis crit uτaa L n ann (3) where is and crit are the equilibrium and critical concentrations of active (fissile) isotope, respectively; L is the average diffusion distance for neutron, τ is the delay time caused by active (fissile) isotope production, which is equal to the effective period of intermediate nuclei decay in the uraniumplutonium cycle (1) or n n Figure 1. Concentration kinetics of neutrons, 238U, 239U, 239Pu in the core of cylindrical reactor with radius of 125 cm and 1000 cm long at the time of 240 days. Here r is the transverse spatial coordinate axis (cylinder radius), z is the longitudinal patial coordinate axis (cylinder length). s Copyright © 2013 SciRes.
V. D. RUSOV ET AL. 531 thoriumuranium cycle (2). Note that Equation (3) automatically contains the self regulation condition for nuclear burning because the ex istence of a wave is predetermined by the inequality iscrit In other words, Equation (3) is a necessary physical requirement for the existence of solitonlike neutron wave of nuclear burning. We indicate for a com parison that, as it follows from Equation (3), the upper bounds of phase velocity of nuclear burning wave are 3.70 cm/day for the uraniumplutonium cycle (1) and 0.31 cm/day for the thoriumuranium cycle (2) at almost equal average diffusion distance (L ~ 5 cm) for fast neu trons (1 MeV) both for uranium and thorium. nn 2 m 2 12 tan 0.56 Finally, we consider some important details and prop erties of such a solitonlike fast reactor, assuming the exi stence of which, we have obtained the theoretical spectra of reactor antineutrino and terrestrial antineutrino which are in good agreement with the experimental KamLAND data [17] corresponding to the first [1] and third [3] ex posures. According to our notions, a solitonlike fast reactor is located on the boundary of the liquid and solid phases of the Earth [17]. The average thickness of such a shell boundary with increased density and mosaic structure is ~2.2 km [33. In our opinion, the most advanced mecha nism for formation of such a shell below the mantle so far follows from the experimental results by Anisichkin et al. [16,18] and XuezhaoSecco [34]. According to these results, the chemically stable highdensity actinide compounds (particularly uranium carbides and uranium dioxides) lose most of their lithophilic properties at high pressure, sink together with melted iron and concentrate in the Earth core as a consequence of the initial gravita tional differentiation of the planet. In the other words, during early stages of the evolution of the Earth and other planets, U and Th oxides and carbides (as the most dense, refractory, and marginally soluble at high pres sures) accumulated from a magma “ocean” on the solid inner core of the planet, thereby activating chain nuclear reactions, and, in particular, a progressing wave of Feok tisov and/or TellerIshikawaWood type. What is the thermal power of such a reactor? As a na tural quantitative criterion of the georeactor thermal pow er we used the wellknown (based on the geochemical measurements) 3He/4He radial distribution in the Earth interior [17]. It turned out that the experimental average values of 3He/4He for crust, the depleted upper mantle, the mantle (minus the depleted upper mantle) and the so called D''region in the lower mantle are in good agree ment with the theoretical data obtained by the model of Feoktistov’s uraniumplutonium georeactor with thermal power of 30 TW [17]. Figure 2 shows the special experi mental investigation of geologically produced antineu trinos with KamLAND [3] and an alternative description of these data by our georeactor model [17]. We need to note that, in spite of the fact that the expe rimental KamLANDdata are well described within the framework of georeactor model [17] (see Figure 2), some geophysicists still have doubts about the existence of georeactor, and the value of georeactor power (30 TW) arouses a special mistrust. In this connection we would like to pay attention to a strange restriction (W ≤ 6.2 TW) on the value of nuclear georeactor thermal power W, which, unfortunately, is frequently met in the scientific literature recently [3,4,35,36], and strongly masks and distorts the clear understanding of problem of georeactor existence, which is intricate enough by itself. Below we ground a complete inconsistency of this restriction. One of the conclusions of the KamLANDcolloboration is the upper bound of nuclear georeactor thermal power (W ≤ 6.2 TW at 90% C.L.), which is a direct consequence of uncertainty of KamLAND experimental data [4]. How ever, it is necessary to keep firmly in mind that the re striction of 6.2 TW on georeactor power is true only for the specific parameters of mixing, i.e. for 21 = 7.58 × 10−5 eV2, , and takes into account the Figure 2. The energy spectra in KamLAND [17]. Main panel, experimental points (solid black dots with error bars) together the total expectation obtained in KamLAND ex periment (dotted black line) [3] and present paper (thick solid blue line). Also shown are expected neutrino spectrum (solid green line) from Japan’s reactors, the expected neu trino spectrum from georeactor 30 TW (red line), the ex pected signals from 238U (dashed red line) and 232Th (dashed green line) geoneutrinos, 13C(α, n)16O reactions (dashed blue line) and accidentals (dashed black line). Inset: ex pected spectra obtained in KamLAND experiment (solid black line) [3] and our paper [17] (solid green line) extended to higher energy. e v Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 532 existence of georeactor within the framework of nonzero hypothesis [4], but absolutely ignores such a nontrivial property of the nuclear georeactor as an uncertainty of georeactor antineutrino spectrum, which in the case of solitonlike nuclear georeactor reaches ~100%. As it will be shown below, the this level of uncertainty within the framework of maximum likelihood function leads (in the minimization of the χ2function) to a considerable expan sion of restriction on the nuclear georeactor heat power (~30 TW) and, accordingly, to the new oscillation para meters (21 , ) for reactor antineutrino. 252 10 eV tan2.5 2 12 0.437 Pu PufnV m Another widespread error is related to determination of the Earth geothermal power WEarth. It is known that there are two estimations of WEarth, i.e., ~33 ± 1 TW [37] and ~44 ± 1 TW [38]. We are not going to participate in the discussion among the authors of these estimations con cerning the necessity of taking into account the hydro thermal circulation. We would like only to emphasize that these estimations are 1.7  2.3 times greater than the radiogenic heat contribution (from the decay 238U, 232Th and 40K in the mantle and crust), which is 19.5 TW [17]. D.L. Anderson [39] refers to this difference as “the miss ing heat source problem” and summarizes the situation in the following words: “Global heat flow estimates range from 30 to 44 TW Estimates of the radiogenic con tribution (from the decay of U, Th and K in the mantle), based on cosmochemical considerations, vary from 19 to 31 TW. Thus, there is either a good balance between current input and output or there is a serious missing heat source problem, up to a deficit of 25 TW”. Be cause of this missing heat, some researchers think that, if a reactor exists, its thermal power must make up the ex istent deficit of geothermal energy. It is correct when it comes to a thermal power of reactor which operated in the remote past, but which does not operate today. The difference between the heat generated now by a reactor in the Earth interior and the experimentally observed geothermal heat [37,38] is very significant due to the high thermal inertia of the Earth. In other words, it is ne cessary to take into account that the heat generated in the Earth interior is not instantly transferred to the surface, but delays (due to a low heat conductivity) in a time of thermal relaxation of the Earth (τE ≈ 109 years) [40,41]. From here it follows that it is impossible to sum up the heat flows which have different spatial and temporal ori gin. 3. The Nonstationary SolitonLike Nuclear Georeactor and KamLAND Antineutrino Spectrum (Experiments over the Period of 20022007) Now we consider the use of idea of solitonlike nuclear georeactor to describe the KamLAND experimental an tineutrino spectra over the period of 20022007 [4]. For this purpose let us estimate an uncertainty of nuclear georeactor thermal power and an uncertainty of georeac tor antineutrino spectrum, respectively. Note that, gener ally speaking, such an uraniumplutonium georeactor can consist of a few tens or hundreds of reactors (with the total thermal power of 30 TW), which represents the in dividual burning “rivers” and “lakes” of an inhomoge neous actinide shell located in the valleys of rough sur face of the Earth’s solid core [17]. In the general case, the fission rate of 239Pu nuclei for the uraniumplutonium cycle (1) in the onegroup approximation can be written down in the form , (4) where φ = νn is the neutronflux density; ν is the neutron velocity; n is the neutron concentration; σf is the fission crosssection for 239Pu; nPu is the 239Pu concentration; V is the volume of burning area. It is easy to see that due to the random character of cri tical and equilibrium concentrations of plutonium in an actinoid shell and also a stochastic geometry of the “riv ers” and “lakes” of actinoid medium, the relative varia tions of neutron flux density φ, the plutonium concen tration n and the volume of burning areas can run up to 50% and more. Then ignoring the possible variations of fission crosssection for plutonium, we can write down the following relation for the relative variation of fission rate: 12 2 22 Pu Pu Pu Pu 0.87, 1 nV nV (5) On the other hand, it is obvious that a kinetics of geo reactor, which operates on the boundary of the liquid and solid phases of the Earth core at the temperature of 5000  6000 K and a pressure of a few hundreds of thousands atmospheres, must necessarily take into account a heat transfer kinetics. This is caused by the fact that under such thermodynamics conditions between these kinetics nontrivial feedbacks can arise, which will significantly change the “traditional” kinetics of neutrons and nuclear reactions. It should be noted that it seems to be the first time that such a problem is being solved within the framework of reactor physics. We have obtained the de pendence of fission crosssection σ239 Pu for the 94 nuclei averaged over the neutron spectrum on the nuclear fuel temperature T by the computational experiment with an allowance for the moderation of neutrons and neutron resonance absorption (Figure 3). Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 533 Figure 3. Dependence of 239Pu fission crosssection averaged over the neutron spectrum on fuel medium temperature for limiting energy (3 kT) of the Fermi and Maxwell spectra. The similar dependence for the 235U fission crosssection is shown for comparison. This dependence has a form of a power function1 in the 4000 to 6000 K range (see Figure 3): Pu ,where 2 fT . (6) From Figure 3 follows that the weak variations of tem perature in the 4000 to 6000 K range can cause the strong variations of fission crosssection σ for 239Pu, which can run up to 100% and more. The variations of fission crosssection σ for 239Pu will, in their turn, cause the variations of neutron flux density φ and neutron concen tration n. It means that an expression for the fission rate of 239Pu in uraniumplutonium cycle (1), which is analo gous to (4), will be more complicated in the multigroup approximation. However, in spite of the difficulties with analytical de termination of the plutonium fission rate variation, it is possible to show (without loss of generality) the lower estimation of relative variation in the case of multigroup approximation Pu Pu Pu Pu 1 f f . (7) Now let us show the uncertainty of georeactor anti neutrino spectrum with oscillations due to the relative error of plutonium fission rate (5). For this purpose we write down the theoretical form of measured total energy spectrum dd ii nEnE ii nE mE in the ith energetic bin , (8) where Pu 2 , , ,, 4π ip iijip ji mt N EEEpEL L (9) 22 21 22 12 π ,1 sin2sin, 2.48 MeV where . eV l pEl L E LE m m (10) Here mλ is the total number of fissions during the expo sure time Δt determined by the fission rate λPu; E i is the average number of detected antineutrino per fission in the ith energetic bin; ε is the detection efficiency for positrons in the inverse βdecay reaction; Np is the num ber of protons in the detector sensitive volume; Δt is the exposure time; ,pEL is the neutrino oscillation prob ability at the appropriate parameters of mixing and ener gy E at a distance of l from the reactor; L is the oscilla tion length; θ21 is the mixing angle; 222 mmm 12 21 is the mass squared difference; (1/4πL2) is the effective solid angle; is the antineutrinoproton interaction crosssection of inverse βdecay reaction with the corre sponding radiation corrections; ii is the nu clear fuel antineutrino energy spectrum in the ith ener getic bin, MeV/fission; E i is the fraction of ith isotope. Here it should be noted that, in general, normalized antineutrino energy spectra corresponding to the different values of reactor heat power may be considered as self similar. This fact simplifies the further analysis consid erably. At the same time, a selfsimilarity takes place only for equilibrium neutrino spectra [42,43], which are typical for stationary processes in reactor core. And vice versa, when processes in the reactor core are nonstation ary, a selfsimilarity of equilibrium neutrino spectra is violated. It means that if, for example, the variations of neutron energy spectrum (and therefore the variations of mass yields induced by the fission of 239Pu) in the reactor core are considerable, the shapes of corresponding neu trino spectra are not selfsimilar. Therefore, the calcu lated (“stationary”, i.e. equilibrium) spectra and corre sponding experimental (nonstationary) neutrino spectra may differ up to 10%  15% and higher [42,43]. The non equilibrium effect of neutrino spectra will be considered in detail in Section 6. Obviously, due to the stochastic change of the nonsta tionary nuclear georeactor power as a consequence of the 239Pu fission crosssection σ and the georeactor neutrino spectrum shape (9) variations, the relative un certainty of georeactor antineutrino spectrum 1It is interesting, that such a behavior of crosssection on the medium grn nE i with oscillations in the ith energy bin (with an allowance for Equations (6)(9)) looks like temperature is appropriate for the fission crosssection and capture crosssection of 239Pu and absolutely is not appropriate for similar crosssections of the 235U nuclei, which have the classical dependence of 1/υ. type. Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 534 12 22 1 ii i Pu Pu grn grn i n n , (11) where 10% ii is the relative uncertainty due to nonstationarity of georeactor neutrino spectrum shape. Therefore the lower estimation of uncertainty of total antineutrino spectrum with oscillations with an allow ance for Equation (11) and the contribution of uncer tainty (4.14%) of antineutrino spectrum Jap i nE from the Japanese reactors [4] takes on the form 12 22 grn i n 22 ,sin 2m 0.0414Jap ii nn . (12) Note that this uncertainty is the one shown in Figure 4 as a violet band. Now we are ready to use our model of uraniumPluto nium georeactor [17] for the alternative description of the data produced in new KamLAND experiment [4]. Obvi ously the standard methods of obtaining consistent esti mates (e.g. the maximumlikelihood method) normally used for the determination of the oscillation parameters 12 12 [15] must take into account one more reactor, or, more specifically, take into account the anti neutrino spectrum of georeactor with the power of 30 TW which is located at a depth of L ~ 5.2 × 106 m. How ever, following [17], we propose here a simple estimat ing approach. The results of its application show that the hypothesis of the georeactor existence on the boundary of liquid and solid phases of the Earth core does not con tradict the experimental data. So, we proceed as in [17]; if CPTinvariance is assum ed, the probabilities of the ee and ee νν 2 2 n 0.437 oscil lations should be equal at the same values L/E. On the other hand, it is known that the variations of Δm2 domi nate over the more stable small variations of angle θ at the spectral distortion (oscillations) of “solar” neutrino spectrum. Therefore we can assume (on the grounds of CPTtheorem) that the angle which is determined by the experimental “solar” equality tan2θ12 = 0.447 [44] may be used as the reference angle of mixing in the Kam LANDexperiment. Finally, following the computational ideology of [17], we give the results of verification of the optimal oscilla tion parameters (21 , 12 ) by comparing the theoretical (which takes into account the georeactor operation) and experimental spectra of re actor antineutrino based on the KamLAND data over the period of 20022007 (Figure 4). We compare also in Figure 5 the χ2profiles for our georeactor hypothesis and KamLAND nonzero hypothesis, which does not take into account an uncertainty of reactor antineutrino spec trum (see Section 6). 25 2.5 10eV ta m In spite of low statistics of neutrino events (≤150 Figure 4. Prompt event energy spectrum of e candidate events (20022007). The shaded background and geoneu trino histograms are cumulative. Statistical uncertainties are shown for the data; the violet band on the blue histo gram indicates the event rate systematic uncertainty. The georeactor power is 19.5 TW. The georeactor is at a dis tance of 5098 km from the KamLANDdetector (see expla nation in the text and Table 1). v Figure 5. Dependences of χ2/NDF on the mass squared dif ference corresponding to KamLANDhypothesis without georeactor (blue line, tan2θ12 = 0.56 [4]) and our georeactor hypothesis (red line, tan2θ12 = 0.437). 2 21 Δm events/bin), the theoretical reactor antineutrino spectrum (which takes into account a solitonlike nuclear georeac tor with the power of 19.5 TW) describes with an accep table accuracy the experimental KamLANDdata (Figure 4) [4]. Below we pay attention to some important mo ments. 3.1. Peculiarities of the Low Antineutrino Statistics Accounting It is obvious, that a low antineutrino statistics is the rea son of high inhomogeneity of filling event density of an antineutrino spectrum (which is continuous by its nature). This inhomogeneity intensifies due to energy discretiza tion of spectrum (0.425 MeV in KamLANDexperiment). As a result, the formal procedure of events integration Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 535 within one energy bin (ΔE = 0.425 MeV) can generate substantial deviations of the number of events (per bin) from its true average value. To observe this effect it would be necessary to decrease the width of energy bin considerably or, that is the same, to decrease the width of detector energy window. However, since it is impossible, we will attempt to show this effect in the following way. As follows from Figure 4, the significant disagree ment between the experimental and theoretical antineu trino spectra is observed for the 5, 7 and 9 bins. There fore, if the apparent condition 57 1, ,PlpElpEl p 9 ,maxEl 830,7410 km , (13) is imposed on oscillations of nuclear georeactor antineu trino spectrum, then by Equation (10) and the average energies of bins Е5 = 2.8 MeV, Е7 = 3.7 MeV and Е9 = 4.5 MeV (see Figure 4) we can obtain a series of the values l for distances from KamLANDdetector to the supposed location of georeactor on the surface of Earth solid core (Figure 6). 5365,5968,6400,6l. (14) Now let us return to the problem of low antineutrino statistics. Fulfillment of condition (13) for given dis tances (14) makes it possible to recalculate a georeactor antineutrino spectrum (Figure 4) for these distances by Equations (8)(10). Proceeding from a low antineutrino statistics (in energy bins Е5 = 2.8 MeV, Е7 = 3.7 MeV and Е9 = 4.5 MeV), the following variants of the location of a georeactor on the Earth solid core surface are most acceptable: 1) a georeactor with the thermal power of 30.7 TW at 6400 km distance from KamLAND detector (Figure 7(a)); 2) a georeactor with the thermal power of 34.7 TW at 6830 km distance from KamLAND detector (Figure 7(b)); georeactors of equal thermal power but Figure 6. The spatial frequency distribution P(l) of oscillat ing georeactor antineutrinos with energies Е5 = 2.8 MeV, Е7 = 3.7 MeV и Е9 = 4.5 MeV. Shaded area corresponds to the continuous series of distances l from the KamLANDde tector to the Earth solid core surface. (a) (b) (c) Figure 7. The theoretical antineutrino spectrum (blue histo gram), which takes into account the nuclear georeactor (a) with thermal power of 30.7 TW (yellow histogram) situated at a distance of 6400 km from the KamLANDdetector, (b) with thermal power of 34.7 TW (yellow histogram) situated at a distance of 6830 km from the KamLANDdetector, (c) with general thermal power 32.6 TW (yellow histogram) situated simultaneously at the distances of 6400 and 6830 km from the KamLANDdetector. In all figures one can see how discrete antineutrino spectra of KamLANDexperi ment (blue histogram) and georeactor (yellow histogram) mask the low statistics effect in corresponding continuous antineutrino spectra (red and yellow oscillations). Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. Co JMP 536 with the total power of 32.6 TW at 6400 and 6830 km distance from KamLAND detector (Figure 7(c)). bases of cones with a vertex in KamLAND and generat ing sides with lengths equal to the corresponding dis tances from KamLAND detector to the Earth solid core surface (Figure 8). These results force us to recalculate the georeactor thermal power obtained for KamLAND data over the period of 20022004 [17]. The parameters of nuclear georeactors obtained by two calculation methods of anti neutrino spectra over the periods of 20022004 and 2002 2007 are collected in Table 1. 3.2. NonStationary Nature of SolitonLike Nuclear Georeactor Analyzing Table 1, we can see that within the frame work of modified method for calculation of antineutrino spectrum (Figure 7), the average thermal power of geo reactor changes from ~50 TW (during the exposure of 749.1 days in 2005 [3], Figure 2) to ~30 TW (at total exposure of 1890.25 days in 2008 [4]). There is no doubt The considered peculiarities of low antineutrino statis tics make it possible not only to determine the possible distances from KamLAND detector to supposed nuclear georeactor on the Earth solid core surface (Figure 6), but to construct the map of located on the Earth surface lines radially conjugate to linescircumferences formed by the Table 1. Nuclear georeactor thermal power depending on a distance to detector and exposure time in the KamLAND and Borexino experiments. Period 20022004 20022007 20052007 20022009 20082009 20082009 Experiment KamLAND Borexino Exposure, days 749.10 1890.25 1141.15 2135 244.75 537.20 Distance, km Nuclear Georeactor Power, TW 5098 30.0 19.5 12.6 17.3 4.7  6400 47.3 30.7 19.8 28.0 7.1  6830 53.4 34.7 22.4 31.6 7.7  6400 + 6830 50.2 32.6 21.1 29.7 7.3  6711      5.0 0 40 5060708090100110150 mW/m2 6400 km 5365 km 5968 km 6830 km 7410 km Figure 8. Distribution of geothermal power density on the Earth [45] superposed with the conjugated “pseudoreactor” cir cumferences, which are generated by “georeactor” circumferences located on the perimeters of the bases of cones with a ver tex in KamLAND (36.43˚N and 137.31˚E) and generating sides with lengths from KamLANDdetector to the Earth solid core urface equal to 7410 (sky blue), 6830 (orange), 6400 (green), 5968 (blue) and 5365 (pink) km. s pyright © 2013 SciRes.
V. D. RUSOV ET AL. 537 that it is the reflection of a nonstationary nature of geo reactor. Taking into account that total exposure (1890.25 days) is a sum of two consecutive exposures (749.1 and 1141.15 days, respectively), in fact the change in the average thermal power of georeactor is even more, i.e. from ~50 TW over the first exposure to ~20 TW over the second consecutive exposure (see Table 1). From the physical standpoint, the decrease of geore actor thermal power (during the two successive expo sures) almost by a factor of two means that the variances of fission crosssection σ for the 94 nuclei during total exposure over the period of 20022007 also changed in two times, or more exactly, went almost 100% down. It means that the variance of fission crosssection 239 Pu σ for the nuclei reaches practically 100% and this is in good agreement with our estimation of variance of fission rate (7). 239 94 Pu At the same time, we asserted before that the main cause of change of fission crosssection σ 239 Pu 252 7.65 10eVm 2 sin 0.304 252 21 2.5 10eVm 2 tan 0.437 2 sin 0.304 Euro i nE for the 94 nuclei under extreme thermodynamics conditions are the temperature variations of fuel medium. Therefore, there is a natural question as to what (except the geore actor) is the reason of the fuel temperature variations, or more exactly, what is the physical nature of independent source of the fuel temperature variations, which in the end plays the role of external modulator of nuclear geo reactor thermal power. The answer to this very important question related to finding out the physical reasons of non stationary nature of solitonlike nuclear georeactor will be given in the second part of the present paper [46]. Briefly summarizing the results of this section, we can say that in spite of the low statistics of neutrino events (≤150 events/bin), the theoretical reactor antineutrino spectrum (which takes into account the solitonlike nu clear georeactor with the power of 30 TW) describes with acceptable accuracy the experimental KamLAND data [4] (see Figures 4 and 7). Here we pay attention to some important moments. First, the average georeactor heat power is changed from ~50 TW at the exposure time of 749.1 days in 2005 [3] (Figure 2) to ~30 TW at total exposure of 1890 days in 2008 [4] (Figure 3). This re flects the nonstationary nature of the georeactor. 4. The Borexino and KamLAND Experiments and Triangulation of SolitonLike Nuclear Georeactors Location As is generally known, the first stage of Borexino experi ment (Laboratory Nationali del Gran Sasso, Italy) [47] was recently completed, ideology of neutrino measure ments in which is practically analogical to that of the neutrino measurements in KamLAND experiment. It means that the joint use of the Borexino and KamLAND data opens up nontrivial possibility for the solving of very important problem of spatial identification of nuclear georeactor location on the Earth solid core surface, or in other words, the triangulation of the solitonlike nuclear georeactor location on the boundary of the liquid and solid phases of the Earth core. 4.1. Borexino Antineutrino Spectrum (Exposure over the Period of 20082009) We give here the alternative analysis of the Borexino data collected between December 2007 and December 2009, corresponding to 537.2 days of live time [46]. The fiducial exposure after all selection cuts is 252.6 ton·yr. The determination of the expected neutrino signal from reactors, which, as usual, was calculated by Equation (8), required the collection of the detailed information on the time profiles of power and nuclear fuel composition for nearby reactors. In Equation (8) the main contribution comes from 194 reactors in Europe, while other 245 re actors around the world contribute only 2.5% of the total reactor signal. Information on the nominal thermal power and monthly load factor for each European reactor origi nate from IAEA and EDF [47]. It is important to note that to describe the antineutrino spectra in the Borexino experiment, the parameters of mixing (21 , 12 [48]) based on the global threeflavour analysis of solar (SNO) and reactor (KamLAND) experimental data were used. At the same time, for the alternative description of the Borexino data (Figure 9) we use the parameters of mix ing , 12, or in other words, 12 obtained by our model, which takes into account the existence of natural nuclear reactor on the boundary of the liquid and solid phases of the Earth core [17]. It is obvious, that in spite of very low statistics of neu trino (reactor) events (1  2 events/bin), the theoretical reactor antineutrino spectrum (which takes into account a solitonlike nuclear georeactor with the power of 5 TW) describes with an acceptable accuracy the experimental Borexino data (Figure 9) [47]. Note that the lower esti mation of uncertainty of total antineutrino spectrum with oscillations (which is headlined in violet colour in Fig ure 9) was calculated by Equation (12) at the uncertainty of antineutrino spectrum from European reac tors equal to 5.38% [47]. 4.2. Peculiarities of the Low Antineutrino Statistics Accounting As follows from Figure 9, the considerable disagreement between the experimental and theoretical antineutrino spectra is observed for 5, 6, 7 and 11 bins. Therefore, if the apparent condition Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 538 Figure 9. The e v energy spectra in Borexino [47]. Experi mental points (solid black dots with error bars) together with the total expectation obtained in Borexino experiment (red solid histogram) [3] and in the present paper (blue solid histogram). Also shown are the expected neutrino spectrum from European reactors calculated using our parameters of mixing (blue dashed histogram) and obtained in Borexinoexperiment (red shaded area). The expected neutrino spectrum from the georeactor of 5 TW located at a distance of 6711 km from the Borexinodetector (green line) is also shown (see explanation in the text). 1,PlpEl pEl p 56 711 1, ,,max, pEl E l 128,7490 km (15) is imposed on the oscillations of nuclear georeactor anti neutrino spectrum, then by Equation (15) for p(E, l) and the bin average energies Е5, Е6, Е7 and Е11 (see Figure 9) we can obtain a series of the values l for possible distances from Borexinodetector to the supposed location of geo reactor on the surface of Earth solid core (Figure 10). 5310,5400,6310,6711,7l. (16) Returning to the problem of low antineutrino statistics, note that the fulfillment of the condition (15) for given distances (16) makes it possible to recalculate by Equa tions (8)(10) a georeactor antineutrino spectrum (Figure 9) for these distances. Proceeding from a low antineu trino statistics (in energy bins Е5, Е6, Е7 and Е11), the most acceptable location of a georeactor on the Earth solid core surface is at 6711 km distance from the Bore xino detector (Figure 10). It is easy to show that the ther mal power of such a nuclear georeactor is ~5 TW. It is necessary to note that the restriction on the nu clear georeactor thermal power obtained within the frame work of the Borexino experiment is ~3 TW [47]. Though this restriction is obtained within the framework of non zero georeactor hypothesis, it does not take into account the high uncertainty of georeactor antineutrino spectrum. The consequences of neglect of this uncertainty come into a question in Section 6. Figure 10. The spatial frequency distribution P(l) of oscil lating georeactor antineutrinos with energies corresponding to 5, 6, 7 and 11 bins in Borexinospectrum in Figure 9. Shaded area corresponds to the continuous series of the distances l from the Borexinodetector to the surface of the Earth solid core. 4.3. Triangulation of the Locations of SolitonLike Nuclear Georeactors By triangulation of the KamLAND and Borexino data we have constructed the ”pseudogeoreactor” coordinate lo cation conjugate to the real location of solitonlike nu clear georeactors on the boundary of the liquid and solid phases of the Earth core (Figure 11). Analyzing Figures 6 and 7 and also Figure 10 (by which it is possible to determine the most probable dis tances between the detector and a nuclear georector), we have divided georeactors into two groupsoperating re actors and probable lowpower reactors (Figure 11). Naturally, a question is the following: What is the cause of sufficiently high degree of correlation between “conjugate pseudoreactors” and the regions of higher geothermal power of the Earth in Figure 11? Below we consider physical reasons causing such a correlation. Here it should be recalled, that according to our as sumption [17], nuclear georeactors are located in the thin uraniumcontaining highdensity layer (about 2.2 km) [33], which is the peculiar physical boundary of the liq uid and solid phases of the Earth core. According to the results of seismic tomography [33], this layer has a mo saic structure with typical size of ~200 km. This means that the spatial history of nuclear burning wave, or in other words “burning spot”, is completely determined by the area of one patch of a mosaic structure. Since the area on the Earth surface conjugate to the single patch of a mosaic structure has the typical size ~1000 km, this value is the size of the uncertainty domain for the “conjugate burning spot” on the Earth surface. On the other hand, we know that the time of heat transmission from the “burning spot” to the “conjugate Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. Copyr JMP 539 burning spot” has the order of 109 years. Taking into ac count the average velocity of nuclear burning wave (~1 m/year), we obtain that for 109 years the “burning spot” will cover the total distance of 106 km on the Earth sur face. Since this distance easily goes into the area of one patch of a mosaic structure in the form of a certain frac talbroken curve, the domain of uncertainty of thermal flow (on the Earth surface) coincides with the domain of uncertainty of the “conjugate burning spot”. This coinci dence is indeed a reason of good correlation between “conjugate burning spots” location and the areas of higher geothermal power in Figure 11. In other words, a map of geothermal power distribution on the Earth is at the same time a rough approximation of the nuclear geo reactors that are acting or had been acting in the past with their locations determined in this case with an accuracy up to 1000 km. It is interesting that if any considerable geothermal heat release is absent in the places, where the nuclear georectors location is supposed (see Figure 11), the large socalled solitary volcanoes (hot spots), e.g. in the Cen tral and SouthEast Africa (Figure 12), or active volca noes, e.g. Erebus in Antarctica (Ross Sea coast), Decep tion (South Scotch Islands) and a recently discovered 0 405060708090100110150 mW/m2 Figure 11. Distribution of geothermal power density on the Earth [45] superposed with the conjugate “pseudogeoreactor” ellipsoidal closed curves, which were built on basis of KamLAND (red lines) and Borexino (blue lines) experimental data. (☆)—operating nuclear georeactors; (О) and (◦)—nuclear georeactors with a power (if they are operating) of an order of magnitude and more less than the thermal power of reactors designated by (☆). Figure 12. The map of spatial location of “pseudureactors”, hot spots and deepfocus earthquakes over the period of 1993 006. The map of “pseudureactors” and earthquakes was built on basis of the plate boundaries map [50]. 2 ight © 2013 SciRes.
V. D. RUSOV ET AL. 540 nameless volcano (Hudson Mountains in the West Ant arctica [49]) are necessarily there. 5. NonStationary SolitonLike Nuclear Georeactor and New KamLAND Antineutrino Spectrum (Exposure over the Period of 20022009) At the time this paper was written, another data of Kam LAND experiment obtained over the years of 20022009 were published [5]. In spite of this, we decided not to change the structure of the paper, but to add the analysis of these data to existing material, because such an analy sis is the natural illustration of inner consistency of the considered above georeactor hypothesis. KamLAND Antineutrino Spectrum We give here the alternative description of KamLAND data [5] collected from March 9, 2002, to November 4, 2009, corresponding to 2135 days of live time. The num ber of target protons within the 6.0mradius spherical fiducial volume is calculated to be (5.98 ± 0.12) × 1031 for the combined data set which corresponds to an expo sure to electron antineutrino e of 3.49 × 1032 proton years. The determination of the expected neutrino signal from reactors, which was traditionally calculated by Equation (8), required the collection of the detailed in formation on the time profiles of power and nuclear fuel composition for nearby reactors. The relative fission yields, averaged over the entire livetime period, for iso topes (235U:238U:239Pu:241Pu) are (0.571:0.078:0.295: 0.065), respectively. In Equation (8) the main contribu tion comes from 56 Japanese nuclear power reactors, while the contributions from Korean reactors and the re mainder of the global nuclear power industry is estimated to be (3.4 ± 0.3)% and (1.0 ± 0.5)% of the total reactor signal, respectively. Information on the nominal thermal power and monthly load factor for each Japanese reactor originate from consortium of Japanese electric power companies [5]. ν It is obvious, that the theoretical reactor antineutrino spectrum (which takes into account a solitonlike nuclear georeactor with the power of 29.7 TW (see Table 1)) describes with an acceptable accuracy the new experi mental KamLANDdata (Figure 13). Let us note that the lower estimation of uncertainty of total antineutrino spec trum with oscillations is calculated by Equation (12) with an allowance for the contribution of the uncertainty 4.5% (which corresponds to the variant DS2 [5]) of total anti neutrino spectrum from the Japanese reactors. It follows from Table 1 that the average thermal pow er Wt of nuclear georeactor sharply decreases in Kam LAND experiments corresponding to the exposures over the periods of 20022004, 20022007 and 20022009 Figure 13. Prompt event energy spectrum of e candidate events (exposure over the years of 20022009). The shaded background and geoneutrino histograms are cumulative. Statistical uncertainties are shown for the data; the violet band on the blue histogram indicates the event rate system atic uncertainty in the framework of georeactor hypothesis. The total georeactor power is 29.7 TW. Georeactors are located at a distance of 6400 and 6830 km from the Kam LANDdetector (see explanation in the text and Table 1). v Such a power jump indicates that the nuclear georeactor is strongly nonstationary. It is a very important fact for the correct calculation of reactor geoneutrinos, which in the end are the integral part of KamLAND antineutrino spectrum (within the framework of georeactor hypothe sis). To illustrate such a strong nonstationarity we give below the change of the georeactor average thermal pow er over the period of 20022009. Using the average values of nuclear georeactor thermal power Wt reconstructed within the framework of geore actor hypothesis (Table 1), which correspond to expo sures over the years of 20022004, 20022007 and 2002 2009, it is possible to determine the values Wt corre sponding to “latent” exposures over the years of 2005 2007 and 20082009 by obvious expression 12 12 12 12 12 , where . tt t tt WWW tt tt tt t (17) The values of nuclear georeactor thermal power Wt extended in that way with consideration of exposure over the period of 20022004 (see Table 1) make it possible to build the time evolution of the georeactor average thermal power W over the years of 20022009 (Figure 14). Thus, the sharp change of the georeactor average thermal power W during the years of 20022009 must be necessarily taken into account in the calculation of the georeactor neutrino spectra, which are the part of the KamLAND antineutrino spectra within the framework of georeactor hypothesis. For that purpose, within the Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 541 Figure 14. Evolution of reconstructed average thermal pow er W of nuclear georeactor over the years of 20022009. framework of traditional calculations of effective neutri no spectra of both individual nuclides and their mixture [42,43], it is necessary to take into account the high de gree of nonequilibrium of neutrino spectra due to strong nonstationarity of nuclear georeactor power [42,43]. It is very important to note, that in the real experiment exactly the high degree of nonequilibrium of neutrino spectra due to the “latent” nonstationarity of radiation source can become the reason of sharp change of the expected “equilibrium” shape of resulting neutrino spec trum. In this sense, ignoring of the high degree of neu trino spectrum nonequilibrium or, in other words, de scription of experimental effective neutrino spectrum by the equilibrium neutrino spectra of individual nuclides or their mixture can result in serious mistakes in fitting the experimental neutrino spectrum. Let us consider this in more detail. 6. On Some Important Features of Alternative Treatment Procedure of KamLAND Experimental Data 6.1. Time Variation of the Reactor Antineutrino Flux and Upper Limit of Georeactor Power In the paper by KamLANDcollaboration [2], where the results of the second exposure (551.1 days) are analyzed, the original and very interesting method for determina tion of antineutrino rate suppression factor, which de scribes the degree of distortion of antineutrino spectrum, is presented. For that purpose the time variations of the reactor antineutrino flux assuming no antineutrino oscil lation were estimated (see Figure 15(a)). Analysis of the linear dependence of the observed e Figure 15. Adapted from [2]. (a) Estimated time variation of the reactor flux assuming no antineutrino oscillation. (b) Observed e event rate versus nooscillation reactor e flux. Data points correspond to intervals of approximately equal e flux. The dotted black line is a fit [2]; the 90% C.L. is shown in gray [2]. The solid black line is a fit constrained to the expected background [2]. The dotted red line is a con tinuation of the dashed black line, the intercept of which in this case is equal to the sum of expected background and e flux from unknown source (for example, a georeactor). The reactor distance distribution for e events in the absence of oscillation is shown in the inset. e v v v v v v main points of this method. The solid black line in the Figure 15(b) is the linear KamLANDfit (90% C.L.) constrained to the expected background [2]. As analysis of this experiment shows, the value of this expected back ground corresponds to the sum of background contribu tions from different background nuclear reactions to the e e signal above 2.6 MeV and is equal to 17.8 ± 7.3 events. Here it is important to note that to obtain such a fit, the authors of Ref. [2] use the strict assumption that “the intercept is consistent with known background, but substantially larger backgrounds cannot be excluded; hence this fit does not usefully constrain speculative sources of antineutrinos such as a nuclear reactor at the Earth core [15] ”. In other words, being applied to the analysis of the observed event rate on the nooscillation reactor e flux and as sumption that the slope can be interpreted as e rate suppression factor and the intercept as the reactorinde pendent constant background rate (Figure 15(b)) are the spectra, this assumption is Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 542 equivalent to acceptance of the socalled zero Kam LANDhypothesis, which a priori rules out the existence of nuclear georeactor or, at least, rejects the significant influence of an additional e flux from the nuclear georeactor (even if it exists) on the observed e e spectra. In the last case, taking into account the parameters of mixing obtained within the framework of zero Kam LANDhypothesis, “the predicted KamLAND rate for typical 3 TW georeactor scenarios is comparable to the expected 17.8 ± 7.3 event background and would have minimal impact on the analysis of the reactor power de pendence signal” [2]. On the other hand, the simple analysis of the dashed black line in Figure 15(b) shows that the alternative, i.e. nonzero, KamLANDhypothesis, which recognizes the existence of nuclear georeactor with considerable heat power, has all the rights to exist. Note that the dotted black line is a fit [2], the 90% C.L. is shown in gray [2] and the dotted red line is our prolongation of the dashed black line, the intercept of which in this case is equal to the sum of expected background and flux from an unknown source. Below we assume that the nuclear georeactor plays the role of unknown e 00.32n 95n 0.6pp source. Let us estimate its heat power W. It will be recalled that for determination of re actor power in neutrino experiments, according to Equa tions (8)(10) and under other conditions being equal, it is necessary to know 1) the location of georeactor, i.e. the distance between the georeactor and detector; 2) fuel composition and the corresponding equilibrium (or non equilibrium2) concentrations of fission products; 3) geo reactor antineutrino spectrum uncertainty; 4) the survival probability p or the neutrino oscillation parameters. Let us assume that for the nuclear georeactors with the heat power W0 ~ 3 TW (the zero KamLANDhypothesis [2] and W (the alternative nonzero KamLANDhypothesis) the conditions 1) and 2) are identical. Then, taking into ac count that the intercepts and slopes of straight lines in Fig ure 15(b) corresponding to the solid black line (the zero KamLANDhypothesis) and the dotted red line (the alter native nonzero KamLANDhypothesis) are , and 0, 0.10.4 , respectively, it is easy to obtain the approximate estimation of the nuclear georeactor heat power W within the framework of alter native nonzero KamLANDhypothesis: 00 22 TW npWpWW . rate shape syst 0 00 pW n 0.4p . (18) It is necessary to notice that the survival probability is not characterized by the neutrino oscillation parameters computed for the zero hypothesis any more. In our opinion, the given example, which shows some details of the reactor antineutrino flux variations applica tion, is extremely obvious and significant since it sub stantiates in a natural way the possibility of existence of the two alternative (but physically equal in rights) hypo theses for the interpretation of experimental KamLAND data. In this sense it is interesting to examine the results by Fogli et al. [51], who analyzed the KamLAND neutrino spectra in energy and time exactly for the second expo sure [2]. They acted on the premise that “while the en ergy spectrum KamLAND events allows the determina tion of the neutrino oscillation parameters, the time spec trum can be used to monitor known and unknown neu trino sources”. By using available monthlybinned data on eventbyevent energies in KamLAND and on reactor powers in Japan, they performed a likelihood analysis of the neutrino event spectra in energy and time, and not only confirmed the results of KamLANDcollaboration, but also set the upper bound on hypothetical georeactor power in the framework of nonzero hypothesis (W ≤ 13 TW at 95% C.L.). Here a natural question arises: “Why do we obtain the different values of the neutrino oscillation parameters and upper bound on hypothetical georeactor power in comparison with Fogli et al. analyzing the same Kam LANDexperiment within the framework of the same nonzero hypothesis?” Obviously it has nothing to do with the likelihood analysis ideology, which is clearly stated in the paper by Folgi [51] in detail, but rather is a cones quence of the completely different understanding of the hypothetical nuclear georeactor physical properties, some of which are discussed above (see derivation of Equation (12)). Let us demonstrate this. In general, the KamLAND unbinned likelihood func tion can be written as [1,2,51,52] LLLL (19) where the three factors embed information on the total event rate, on the spectrum shape and on the systematic uncertainties. According to [51], we remain the second and third likelihood factors in Equation (19) unchanged and con sider only the first likelihood factor, which can be written as (see also [1,2,51,52]): 2 22 12 1 2π ,sin; ,, 1 exp 2 rate rate theor obs rate Nm N L (20) where (δm2, sin2θ12) are the massmixing parameters, is the systematic energy offset, 2For the details of the calculations of nonequilibrium neutrino spectra see below (Section 6). and are free Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 543 (positive) parameters, Nobs is the total number of observ ed events and the total error is the sum of the statistical and systematic uncertainties 2 r theor NsN 2 0, grn sN 2 rate theo , (21) where s is a fraction of systematic uncertainty. Here is the key moment which reveals the physical distinction between our nonzero hypothesis and nonzero hypothesis by Fogli et al. [51]. In general case, when the additional antineutrino source (i.e. a nuclear georactor) is taken into account within the framework of the nonzero hypothesis, it is also necessary to take into account the uncertainty of georeactor antineutrino spectrum. As is shown above (see Equation (12)), this uncertainty ap pears due to the change of fission crosssection of 239Pu (which is the main component of nuclear fuel) with change of nuclear fuel temperature and, in particular, with change of temperature near the Earth solid core sur face (see Figure 3). As a result, in the case of nonzero hypothesis Equation (21) must have, according to Equa tion (12), the following form: 2 2 01, ratetheortheor grn NsNN ss (22) where Ntheor = Njap + Ngrn, NJap is the total number of events from Japanenese nuclear reactors, Ngrn is the total number of events from nuclear georeactor, s0 is a contri bution from systematic uncertainty of the number of georeactor antineutrino. It is obvious that Equation (22) in contrast to Equation (21) allows higher values of the nuclear georeactor heat power. This, in its turn, leads to a change in survival probability and, consequently, to a change in the neutrino oscillation parameters. In this sense, it is clear that even a very accurate account taken of the time variation of the reactor antineutrino flux (for example, monthly or even daily neutrino flux) and of another not less important features of antineutrino spectrum does not lead to a con siderable change of the antineutrino survival probability (see Figure 15(b), solid black line), if the specific un certainty of georeactor antineutrino spectrum is not taken into account as well. And vice versa, taking into account such a feature of georeactor antineutrino spectrum let us obtain the new restrictions on the georeactor heat power and corresponding values of the neutrino oscillation pa rameters (Figure 5) by likelihood analysis of the Kam LAND energy spectrum (Figure 4) and minimization of corresponding χ2function based on Equation (19). Returning to the known KamLAND estimation of geo reactor heat power, we would like to cite Ref. [4]: “The KamLANDdata, together with solar data, set an upper limit of 6.2 TW (90% C.L.) for a e reactor source at the Earth center [15] assuming that the reactor produced a spectrum identical to that of a slow neutron artificial reactor”. Although it does not evidently follow from the paper, we suppose that within the framework of likelihood analysis of the KamLAND neutrino spectra in energy and time the authors used the nonzero georeactor hypothesis by adding a 57th reactor at L = 6400 km to the 56 Japanese nuclear power reactors. At the same time, the nontrivial properties of some components of nuclear fuel (for example, the 239Pu fission crosssection (see Figure 3)) were not taken into account at all in this paper as well as in all other papers by KamLANDcollabora tion. As is shown above, such a neglect of anomalous behavior of the 239Pu fission crosssection at high tem peratures (in the 2500 to 6000 K range (Figure 3)) im plies, according to Equation (22), the ignoring of high uncertainty of georeactor antineutrino spectrum, which within the framework of maximum likelihood analysis will immediately cause the distortion of “true” values of reactor heat power and corresponding values of the neu trino oscillation parameters. Finally it is worth mentioning that a nonstationary re gime of nuclear georeactor operating, caused by the change of the 239Pu fission rate (Equation (5)) mainly due to the strong variation of the 239Pu fission crosssection (Equation (7)), which is a nonlinear function of medium temperature (see Figure 3), is the main reason of high uncertainty of georeactor antineutrino spectrum. Such a nonstationary regime generates yet another, quite deep and nontrivial problem, i.e. the socalled problem of non equilibrium neutrino spectra. Rejection of this problem can lead to serious errors in fitting of the experimental neutrino spectra. Below we consider this in more detail. 6.2. On the Reasons and Degree of the Antineutrino Spectra Nonequilibrium in KamLAND Experiments To describe the nuclear fuel antineutrino radiation, the nuclide equilibrium concentrations of fissionproduct mix ture and, accordingly, equilibrium antineutrino spectra obtained for hypothetical infinite irradiation time, which provides a secular equilibrium of all without exception fission products, are traditionally used as a zeroorder ap proximation. On the other hand, it is obvious that the equilibrium approximation is not true for the nonstationary nuclear georeactor (Figure 14). Therefore, there is a question, how the strategy of calculation of effective neutrino spectra changes in this case and, in particular, how the resulting neutrino spectrum shape changes due to taking into account the nonequilibrium neutrino spectra instead of equilibrium neutrino spectra, which are used for sta tionary nuclear reactors. As is known, a direct summation method of β,  Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 544 spectra of individual nuclides [42,43] composing the fis sionproduct mixture with regard to the specific modes of fuel irradiation in a nuclear reactor, and proper total ef fective β, spectrum of the nuclear system k k j E k jjj NE E , (23) are used as the calculation algorithm when passing from the βspectrum to the antineutrino spectrum. Here λj is the decay probability of jth nuclide; Nj is the number of nuclei of jth nuclide in the system k; j is the total β, spectrum of jth nuclide normalized by the nuclear decay: ,,ijij i EE E ,ij , j j ij i K . (24) where λjNj is the activity of jth nuclide depending on irra diation mode (fuel initial composition, neutron flux den sity, fuel burnup and other parameters influencing on accumulation of each jth nuclide); Kj is the branching factor for βdecay channel, i.e. the number of βparticles per decay; ,ij is the partial βtransition spectrum of jth nuclide; is the βtransition intensity of jth nu clide. A priori knowledge (based on calculation or experi mental estimation) of the initial concentration Nj(t) of jth fissionable actinoid and the onegroup integral neutron flux density makes it possible to determine the ac cumulation of one or another jth fission product in the reactor core by solving the following system of kinetic equations3 1 11 11 d d p j jcjfi ijji i jj mj mcm mj mm NNN N t NN (25) which describes the time change of the jth nuclide con centration in fissionproduct mixture at the initial condition i Nt 0 0 j NN at the time t linked with the activity of the jth nuclide in the following way Qt j ; index “m” relates to precursor nucleus, m < j; · jj Nt ij is the independent yield of the jth nuclide due to the fuel ith component fission averaged over the effective neutron spectrum; λmj is the decay probability of mth nu clide into the jth nuclide due to β, β+decay, electron capture, isomeric transition etc.; fi is the onegroup fission crosssection for the ith fissionable actinoid; ci is the onegroup (n, γ), (n, 2n) reaction crosssec tion for ith nuclide; cmj is the (n, γ), (n, 2n) reaction crosssection for mth nucleus with jth nucleus formation. Finding the time dependence of nuclide concentration of fissionproduct mixture (see (25)) is a sufficiently la borconsuming problem, and its solution depends on the specific conditions of fuel irradiation, i.e. the time depen dence of neutron flux density, neutron flux spectral com position and also the initial composition of the fuel [42, 43]. Therefore the exact solution of the kinetic system of Equations (25) becomes practically unachievable in the study of fissionproduct buildup in nonstationary nu clear reactors with nonconstant or in general unknown parameters. All said above applies in full measure to the nonstationary nuclear georeactor, the thermal power evolution of which is shown in Figure 14. Moreover, in this case a situation is aggravated by the fact that the 239Pu fission crosssection (the main active component of georeactor nuclear fuel) is a strongly nonlinear magni tude, which grows by the power law in the temperature range 3000  5000 K (Figure 3), which is typical for the nearsurface layers of the Earth solid core. How much is better, when the reactor is stationary. In this case, the lefthand sides of the system of the Equa tions (25) can be set equal to zero, and the system itself transforms into the system of linear algebraic equations, the solution of which (the socalled equilibrium nuclide concentrations of fission product mixture) does not de pend on initial conditions and irradiation time. The ob tained equilibrium nuclide concentrations of fission pro duct mixture make it possible (according to Equation (23) to determine equilibrium partial and total neutrino spec tra, which are usually used for description of effective neutrino spectra of stationary neutrino sources and, in particular, stationary nuclear reactors. Finally, returning to the analysis of concrete Kam LAND neutrino spectra, it is necessary to state that with in the framework of georeactor hypothesis an integral fraction of reactor geoantineutrino is sufficiently great and makes up almost the half of integral fraction of anti neutrinos produced by all Japanese reactors in the Kam LANDexperiment (see Figures 4 and 13). It means that the nonequilibrium property inherent to the reactor geo neutrino spectrum is not only delegated to the Kam LAND experimental neutrino spectrum to a considerable extent, but plays a dominant role in natural distortion of this spectrum with respect to the KamLAND theoretical neutrino spectrum, which is based on the ideology of equilibrium neutrino spectra. The question arises, to which degree this nonequilib rium influences on the effective neutrino spectrum shape in the general case and, for example, in KamLANDex periments. As the numerous test experiments related to the nuclear fuel irradiation under unstable conditions 3Note that the index i changes in the range 1 ≤ i ≤ p, and according to the known Russian catalogue of radioactive nuclides total β, spectra [42], the index j changes in the range 1 ≤ j ≤ 1028, i.e. the matter is a necessity to solve the enormous system of enchained differential equa tions. The method of solution of the system (25) based on the deriva tion of recurrence relations for Nj(t) is in detail described in [53] and realized as the AFPA program package (Accumulation of Fission Products and Actinides) in terms of FORTRANIV. Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 545 show [42,53], the nonequilibrium effect manifests itself as the observed distortion of some pieces of the nonequi librium effective spectrum (with respect to an analogical equilibrium neutrino spectrum), the location of which in spectrum energy coordinates is completely predetermin ed by the time dependence of neutron flux density, neu tron flux spectral composition and initial composition of the fuel [42,53]. On the other hand, the analysis of experimental Kam LANDdata obtained over the years 20022004, 2002 2007 and 20022009 shows that in all considered cases the number of recorded events in the fifth (Е5 = 2.8 MeV) and ninth (Е9 = 4.5 MeV) energy bins of experimental neutrino KamLANDspectra differs substantially from the similar data obtained by fitting or, in other words, theoretical equilibrium neutrino KamLANDspectra (see Figure 1 in [4], Figure 1 in [5] and also Figures 4 and 13). At the same time, the number of recorded events in the fifth bin is always substantially less than in the fifth bin of theoretical neutrino KamLANDspectrum, where as an opposite situation is observed for the ninth bin. It is the socalled problem of 5 and 9 energy bins of neutrino KamLANDspectra, which, in our opinion, is caused not only by detection statistics, but is mainly a manifestation of substantial nonequilibrium of neutrino spectra. Ac cording to [42,43], the amount of nonequilibrium effect, i.e. the difference between calculated equilibrium neu trino spectra and corresponding experimental nonequi librium neutrino spectra can reach 10%  15% and more. At last, it is necessary to remind that in this paper all theoretical neutrino spectra (Figures 2, 4, 9 and 13) are built using the ideology of equilibrium spectra. Within the framework of georeactor hypothesis such an appro ach is reasonable, because the possible high degree of nonequilibrium of experimental neutrino spectra, which manifests itself, for example, as the socalled problem of the 5 and 9 energy bins of neutrino KamLANDspectra, is effectively compensated by the high, but reasoned de gree of uncertainty of theoretical neutrino spectra. 6.3. Geological (Magnetic) TimeScale and Time Evolution of the Nuclear Georeactor Power Within the framework of the alternative hypothesis we obtained the time evolution of the average georeactor heat power over the period of 20022009 (Figure 14), which shows that the average georeactor heat power Wt drops sharply from 50 TW to 5  7 TW in KamLAND experiments over the periods of 20022004, 20052007 and 20082009. Here a natural question arises: “What does such a dynamics reflect or what physical mecha nism causes such a dynamics?” In other words, is it a manifestation of some unknown physics or a trivial con sequence of the “luckily guessed calculation rules which do not represent the true nature of things” [54]. Below we try to give a simple and physically clear inter pretation of the possible fundamental mechanism and its impact on the temporal dynamics of the nuclear georeac tor heat power. It is known that in spite of its long history, the nature of the energy source maintaining a convection in the Earth liquid core or, more exactly, the mechanism of the magnetohydrodynamic dynamo (MHD) generating the terrestrial magnetic field has no clear and unambiguous physical interpretation so far [26,55]. The problem is aggravated by fact that none of candidates for an energy source of the geomagnetic field [55] (secular cooling due to the heat transfer from the core to the mantle, internal heating by radiogenic isotopes, e.g. 40K, latent heat due to the inner core solidification, compositional buoyancy due to the ejection of light element at the inner core sur face) can not in principle explain one of the most remark able phenomena in solarterrestrial physics—a strong (inverse) correlation between the temporal variations of magnetic flux in the tachocline zone (the bottom of the Sun convective zone) [56,57] and the Earth magnetic field (Ycomponent)4 [59] (Figure 16). At the same time, supposing that the transversal (radial) surface area of tachocline zone, through which a magne tic flux propagates, is constant in the first approximation, one can assume that magnetic flux variations describe also the temporal variations of magnetic field in the solar tachocline zone. Thus, Figure 16 demonstrates the mir ror correlation between the temporal variations of mag netic field in the solar tachocline zone and the Earth magnetic field (Ycomponent). To obtain such visual correlations we used the socall ed moving interval averaging algorithm. The procedure of averaging the series n in order to derive the av eraged (by means of the moving interval of N points) series N n x was performed using the following ex pression: 12 1 1, where21 3. N N ni nin ni xx x N Nk (26) 4Note that the strong (inverse) correlation between the temporal varia tions of magnetic flux in the tachocline zone and the Earth magnetic field (Ycomponent) are observed (Figure 16) only for experimental data obtained at the observatories where the temporal variations o declination (δD/δt) or the closely associated east component (δY/δt) are directly proportional to the westward drift of magnetic features [58]. This condition is very important for understanding of physical nature o indicated above correlation, since it is known that the motions of the top layers of the Earth core are responsible for most magnetic varia tions and, in particular, for the westward drift of magnetic features seen on the Earth surface on the decade time scale. Europe and Australia are the geographical places, where this condition is fulfilled (see Figure 2 in [58]). Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. JMP 546 Figure 16. Time evolution of the magnetic flux variations at the bottom (tachocline zone) of the solar convective zone ((black dotted line, see Figure 17), variations of the nuclear georeactor thermal power (blue point with bars), geomagnetic field secu lar variations (Ycomponent, nT/year, red dotted line) [59] and prediction (green dotted line) [59]. All curves are smoothed by sliding intervals in 5 and 11 years. The pink area is the prediction region. where k is the positive integer. The smoothed curve of the magnetic flux variations in the solar tachocline zone, which is shown in Figure 17(c) (black dotted line), demonstrates the result of such an averaging of the initial curve (red fill area in Figure 17(c)) by the two sliding intervals in N = 5 and N = 11 years. Physical sense of such a double averaging consists in the “soft” removing of the 11year solar cycle influ ence and obtaining of the socalled amplitudemodulated representation of magnetic flux variations in the solar ta chocline zone. It follows from Figure 16 that the degree of averaging effect on time sample of the variations of solar magnetic field and the Earth magnetic field (Ycomponent) sharply differs in the degree of their smoothing. In our opinion, this is caused by a considerable delay (10  12 years) of the variations of terrestrial magnetic field (Ycomponent) with respect to the solar magnetic field variations, during which the intermediate deep terrestrial processes con trolled by the solar power pacemaker not only activate and maintain the Earth magnetic field, but also smooth out some fine “details” of the solar power pacemaker in fluence (Figure 17). Below we consider the physical me chanism of the one possible intermediate deep terrestrial process by virtue of which a future candidate for an en ergy source of the Earth magnetic field must play not only the role of a natural trigger of solarterrestrial con nection, but also directly generate the solarterrestrial magnetic correlation on its own. Figure 17. Time evolution of (a) geomagnetic secular varia tions (Ycomponent, nT/year) [59], (b) observed sunspot area for cycles 12  23 [57], (c) integral from 0˚  45˚ latitude of simulated toroidal magnetic flux in bottom shear layer (red shadow zone) for cycles 12  23 [57], plus forecast for cycle 24 [57]. Black curves are smoothed by sliding intervals in 5 and 11 years. The green dotted line is the prediction region. We assume that the temporal variation of solitonlike nuclear georeactor heat power can be a global energy cause of such a fundamental geophysical phenomenon as the variations of terrestrial magnetic field (Ycomponent). Copyright © 2013 SciRes.
V. D. RUSOV ET AL. 547 If it is truth, it must be the Solar magnetic field variations that “drive” or cause the temporal variations of the nu clear georeactor heat power, which, according to our hy pothesis, is the energy source of the Earth magnetic field and, consequently, its variations. One of possible mecha nisms generating strong (negative) correlation between the terrestrial magnetic field and solar magnetic field (see Figure 16) can be socalled axion mechanism of solar dynamogeodynamo connection [60]. The essence of this mechanism is that the total energy of 57Feaxions produced mainly in the Sun core is modu lated at first by the magnetic field of the solar convective zone (due to the inverse coherent Primakoff effect [60]) and after that is resonance absorbed by 57Fe in the Earth core. In other words, the higher the solar magnetic field, the greater the number of axoins is converted (by the inverse Primakoff effect) into γquanta, the smaller the number of axoins reach the Earth and are absorbed in the Earth core, and vice versa. It results in the fact that the variations of 57Feaxion intensity play the role not only of heat source, which changes the temperature of the Earth core, but also the modulator of nuclear georeactor thermal power, because the medium temperature in the range 2500  6000 K modulates the value of the 239Pu fission crosssection (Figure 3). In other words, the solar axions mechanism not only explains the nature of heat source in Earth liquid core, which plays the role of the modulator of nuclear georeactor thermal power, but in a natural way explains the cause of experimentally observed strong inverse cor relation (Figure 16) between the magnetic field of the solar convective zone and the Earth magnetic field (Y component). From Figure 16 it follows that the variation of the Earth magnetic field (Ycomponent) lags behind the va riation of the solar magnetic field about 10  12 years. On the other hand, it is known that a magnetic signal prede termined by an extremum of drift velocity of eccentric dipole of the Earth core lags and therefore is detected on the Earth surface 5  7 years late [61,62]. Within the framework of georeator hypothesis this means that the temporal variation of magnetic field energy WSE on the Earth surface also has the delay of 5  7 years with re spect to the temporal variation of magnetic field energy Wcore of the Earth liquid core which, in its turn, is formed with the delay with respect to the temporal variations of the nuclear georeactor thermal power W. At the same time, since Wcore ~ B2, where is the magnetic induc tion vector, we can obtain from the obvious relation Wcore ~ W that B ~BW. (27) On the other hand, we have found the sampling of values for the nuclear georeactor thermal power (Table 1), which is obtained by the experimental KamLAND data handling over the years of 20022009. This sam pling contains three averaged values: 50.2 TW over the years of 20022004, 21.1 TW over the years of 2005 2007 and 7.3 TW over the years of 20082009. Due to delay of temporal variation of the nuclear georeactor thermal power with respect to the variation of the Earth magnetic field (Ycomponent) it is obvious that to smooth out the influence of background processes (the variations of the Earth liquid core temperature, nuclear fuel composition etc.) accompanying geodynamo opera tion, it is necessary to average the sampling of values of the nuclear georeactor thermal power by a sliding time interval with the length of order of a delay time, i.e. N = 5  7 years. It is rather easy to show that, using a maxi mum possible sliding interval with N = 7 transforms this sampling composed of the 7 virtual annual values over the years of 20022009 into the sampling which contains only one term characterizing the average nuclear geore actor thermal power 7 430 TWW. (28) It is obvious that this single term of new sampling corresponds to the year 2006. If, according to the solar axion mechanism, we assume the existence of a strong (inverse) correlation between the variation of the Earth magnetic field and the value W (see Equation (27)) and also take into account the delay time (5  7 years) of the variation of W with respect to the variation of the solar magnetic field, it is easy to find the coordinates of nuclear georeactor in Figure 16. The value of W is at the intersection of vertical line t = 2006 and the slanting grey line passing through the extreme point (t = 2001) on the curve of variation of the solar magnetic field. Let us remind that the slope of the grey straight line in Figure 16 is the effect of delay under the conditions of strong (inverse) correlation of the solar magnetic field and the Earth magnetic field (Ycomponent). Note that all the future measurements of annual varia tions of neutrino flux in the KamLAND and Borexino experiments will generate new theoretical data describing the variations of nuclear georeactor thermal power. If these variations smoothed by movingaverage process will correlate with the variations of the solar magnetic field and variations of the Earth magnetic field (Ycom ponent), the georeactor hypothesis will obtain another strong indirect confirmation. Finally, we would like to remind that within the frame work of georeactor hypothesis, a forecast of the above mentioned fundamental geophysical processes behavior, which have a common cause (the temporal variations of magnetic field in the solar tachocline zone), is possible only up to a corresponding event horizon predetermined by delay time of the nuclear georeactor power variation Copyright © 2013 SciRes. JMP
V. D. RUSOV ET AL. 548 (5  7 years) or magnetic Yfield of the Earth (10  12 years) with respect the magnetic field in the solar tacho cline zone. It is obvious that such a delay effect makes it possible to predict the behavior of the Earth magnetic field (Ycomponent) reliably by experimental observation of georeactor antineutrino, the variations of which char acterize the variations of nuclear georeactor thermal power. 7. Conclusions We should note that although the nuclear georeactor hy pothesis we used for the interpretation of the Kam LANDexperiment seems to be very effective, it can be considered only as a possible alternative variant for de scribing the KamLAND experimental data. Only a direct measurements of the geoantineutrino spectrum in the energy range >3.4 MeV in the future underground or submarine experiments will finally resolve the problem of the natural georeactor existence and will make it pos sible to determine the “true” values of reactor antineu trino oscillation parameters. At the same time, the solu tion of the direct and the inverse problems of the remote neutrinodiagnostics for the intraterrestrial processes which is essential to obtain the pure geoantineutrino spectrum and to determine a correct radial profile of the βsources in the Earth interior [43,63] will undoubtedly help us solve the problem of a natural nuclear reactor existence on the boundary of the liquid and solid phases of the Earth core. 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