Pontecorvo (1957) , Wolfenstein (1978) , Mikheyev and Smirnov (1989) investigated the effects of neutrino oscillations on matter and derived a formula for the conversion probability of enhanced neutrino flavors in matter. In the case of a two-flavor neutrino (e.g., ${\nu }_e\leftrightarrow {\nu }_{\mu }$), the conversion probability of the neutrino ${\nu }_e\to {\nu }_{\mu }$ as it propagates through matter with a constant density is

$P_{{\upsilon }_e\to {\upsilon }_{\mu }}={\sin }^{2}2{\theta }_m{\sin }^2\left(\frac{\Delta m_m^{2}L}{4E}\right).$(1)

The survival rate of the electron neutrino is

$P_{{\upsilon }_e\to {\upsilon }_e}=1-{\sin }^{2}2{\theta }_m{\sin }^2\left(\frac{\Delta m_m^{2}L}{4E}\right).$(2)

In Equations (1) and (2), $E$ is the energy of the neutrino, $L$ is the length of the oscillation baseline, ${\theta }_m$ is the mixing angle of the effect on matter, and $\Delta m_m^2$ is the squared effective mass difference. The correlation between these values is given by the following equations:

$\tan 2{\theta }_m=\frac{\Delta m^2\sin 2\theta }{\Delta m^2\cos 2\theta -A_{CC}},$(3)

$\Delta m_m^2=\sqrt{{\left(\Delta m^2\cos 2\theta -A_{CC}\right)}^2+{\left(\Delta m^2\sin 2\theta \right)}^2},$(4)

where $\theta$ is the mixing angle in vacuum, $\Delta m^2=\left|m_2^2-m_1^2\right|$ is the squared mass difference between the two mass eigenstates, $A_{CC}\equiv 2E{V}_{CC}=2\sqrt{2}E{G}_F{N}_e$ is the material potential of the charged matter, $G_F$ is the Fermi constant, and $N_e$ is the number density of the electrons in the matter.

When the following conditions are met, neutrinos will resonate with the atoms in the matter.

$V\left(N_e\right)=V_{CC}=\sqrt{2}{G}_F{N}_e=\frac{\Delta m^2\cos 2\theta }{2E}.$(5)

According to the conditions of MSW resonance, in nature, only atmospheric neutrinos (whose energies are 0.1 - 10⁴ GeV ( Honda et al., 1995 ; Gaisser and Honda, 2002 ; Winter, 2016 ) are able to form resonance with matter in the Earth when they propagate inside the Earth. The energy of an atmospheric neutrino under this resonance is as follows ( Winter, 2016 ):

$E_{res}=\frac{\Delta m^2\cos 2\theta }{2\sqrt{2}{G}_F{N}_e}\approx \frac{\Delta m_{31}^2\left[{\text{eV}}^2\right]\cos 2{\theta }_{13}}{7.6\times {10}^{-14}\rho \left[\frac{\text{g}}{{\text{cm}}^{\text{3}}}\right]}\text{eV}.$ (6)

The relationship between $N_e$ and the density of the matter $\rho$ is $N_e=N_A{Y}_e\rho$, in which $N_A$ is Avogadro’s constant, and $Y_e$ is the electron fraction. MSW resonance can be formed when the atmospheric neutrino energy $E_{res}$ and the density of matter in $\rho$ satisfy Equation (6).

When neutrinos oscillate, the weak interaction field formed by the neutrinos reacts against the atoms in the matter. Based on the effective potential of matter acting on neutrinos (i.e., the matter potential of neutrinos), the effective potential of electron neutrinos acting on matter, i.e., the (charge-flow) neutrino potential felt by the matter, can be obtained:

$V\left(N_{{\upsilon }_e}\right)=\sqrt{2}{G}_F{N}_{{\upsilon }_e},$(7)

where $N_{{\upsilon }_e}$ is the number density of the neutrinos. Assuming that the electron neutrino flux is constant and has magnitude ${\varnothing }_0$, the electron neutrino flux after propagating a distance of $L$ is

${\varnothing }_L={\varnothing }_0{P}_{{\upsilon }_e\to {\upsilon }_e}={\varnothing }_0\left[1-{\sin }^{2}2{\theta }_m{\sin }^2\left(\frac{\Delta m_m^{2}L}{4E}\right)\right].$(8)

Since the speed of the neutrino is close to the speed of light $c$, the distance traveled by the electron neutrino per unit time is $c$, and the number density of the electron neutrinos is

$N_{{\upsilon }_e}=\frac{{\varnothing }_L}{c}=\frac{{\varnothing }_0{P}_{{\upsilon }_e\to {\upsilon }_e}}{c}=\frac{{\varnothing }_0}{c}\left[1-{\sin }^{2}2{\theta }_m{\sin }^2\left(\frac{\Delta m_m^{2}L}{4E}\right)\right].$(9)

Substituting Equation (9) into Equation (7) yields

$V\left(N_{{\upsilon }_e}\right)=\frac{\sqrt{2}{G}_F{\varnothing }_0}{c}\left[1-{\sin }^{2}2{\theta }_m{\sin }^2\left(\frac{\Delta m_m^{2}L}{4E}\right)\right]$.(10)

At resonance, ${\theta }_m\to \pi /4$, and Equation (10) can be simplified to

$V\left(N_{{\upsilon }_e}\right)=\frac{\sqrt{2}{G}_F{\varnothing }_0}{c}\left[1-{\sin }^2\left(\frac{\Delta m_m^{2}L}{4E}\right)\right]=\frac{\sqrt{2}{G}_F{\varnothing }_0}{c}{\cos }^2\left(\frac{\Delta m_m^{2}L}{4E}\right).$(11)

Equation (11) is the neutrino potential of the electron neutrino action on the atoms of the matter in the resonance region. This effective potential is weak, but since the resonance is capable of accumulating energy, the necessary excitation energy can eventually be obtained.

Fermi (1934) derived an expression for the decay constant for the beta decay of atomic nuclei:

$\lambda ={\displaystyle {\int }_0^{P_m}\frac{g^2{\left|M_{if}\right|}^2}{2{\pi }^3{c}^3{\hslash }^7}F\left(Z,R,P\right)P^2{\left(E_m-E\right)}^2\text{d}P},$(12)

where $g$ is the weak interaction coupling constant, $P$ and $E$ are the momentum and energy of the radiating electrons, respectively, and $P_m$ and $E_m$ are the maximum electron momentum and maximum electron energy of the radiating electrons, respectively. $F\left(Z,R,P\right)$ is the Coulomb correction factor, which describes the effect of the Coulomb field of the nucleus on the emitted β-particles, and is a function of the subnucleus charge $Z$, radius $R$, and energy of the β-particle $E$. $\left|M_{if}\right|$ is the leapfrog matrix element associated with the subnucleus and the parent nucleus. The magnitude of the lepton probability is mainly determined by the magnitude of the lepton matrix elements $\left|M_{if}\right|$.

When $E_m\gg m_e{c}^2$ and $F\left(Z,E\right)\approx 1$, $F\left(Z,E_m\right)=A{E}_m^5$ (where $A$ is a constant),

$\lambda \propto E_m^5$.(13)

Geiger and Nutall (1911) derived the following empirical formula for the alpha decay constant:

$\lambda \propto E_{\alpha }^{86.25}$,(14)

where $E_{\alpha }$ is the kinetic energy of the $\alpha$ particle.

Studies have shown that melts in the Earth’s interior are mainly distributed in the upper mantle ( Debayle et al., 2020 ). The most dominant radioisotopes in the upper mantle are ²³⁸U, ²³²Th, and ⁴⁰K, which have contents of about 0.13 × 10⁻⁶, 0.75 × 10⁻⁶, and 0.23% × 0.0117%, respectively ( Li, 1976 ). The decay reaction equations of which are as follows ( Fiorentini et al., 2007 ):

${}^{238}\text{U}\to {}^{206}\text{P}\text{b}+8\alpha +6{\text{e}}^{-}+6{\bar{\nu }}_e+51.7\text{MeV}$,(15)

${}^{232}\text{T}\text{h}\to {}^{208}\text{P}\text{b}+6\alpha +4{\text{e}}^{-}+4{\bar{\nu }}_e+42.7\text{MeV}$,(16)

${}^{40}\text{K}\to {}^{40}\text{C}\text{a}+{\text{e}}^{-}+{\bar{\nu }}_e+1.31\text{MeV}$.(17)

The energy released by the decay of radioactive elements can be calculated as follows:

$E=N_A\frac{M}{g}Q,$(18)

where $M$ is the mass of the radioactive element, $g$ is its molar molecular weight, $N_A=6.022\times {10}^{23}{\text{mol}}^{-1}$ is Avergadro’s constant, and $Q$ is the energy released by the decay of a radioactive atom.

Assuming that in the upper mantle, there is a layer of material with a thickness of dr, that all of the energy $\text{d}\Omega$ released by the radioactive material is used for heating and melting the material in this layer, that the heat consumed by the mantle for heating during this process is $\text{d}{\Omega }^{\prime }$, that the heat consumed for melting, i.e., the latent heat of fusion, is $\text{d}\Lambda$,

$\text{d}\Omega =\text{d}{\Omega }^{\prime }+\text{d}\Lambda .$(19)

In this small region, the pressure and temperature are almost constant, so Equation (19) can be approximated as

$\Omega ={\Omega }^{\prime }+\Lambda .$(20)

For example, if this small region of upper mantle material has a mass of 1000 kg, Equation (20) can be used to calculate the total heat required to melt the material: $\Omega ={\Omega }^{\prime }+\Lambda =5.1\times {10}^8\text{J}$. For 1000 kg of the upper mantle material in the main radioactive substances, the total decay-produced heat of up to 1.69 × 10¹⁰ J is much higher than the amount required to completely melt the upper mantle material.

In the above calculations, the constant-pressure specific heat capacity of the rocks is taken to be 1000 J/(kg∙K) ( Shui and Watanabe, 1982 ; Zhang, 2001 ; Chen, 2013 ). The average temperature of the upper mantle is taken to be 1000˚C (or 1273 K), the melting temperature of the material at the same depth is taken to be 1300˚C (or 1573 K), and the latent heat of fusion of the upper mantle is taken to be 210 kJ/kg ( Shui and Watanabe, 1982 ).

The origin of the effect of neutrino oscillations on matter is caused by forward coherent scattering between the neutrinos and matter. Although this coherent scattering is capable of delivering a weak momentum (or energy) to matter ( Freedman, 1974 ; Akimov et al., 2017 ), we were unable to obtain the exact energy delivered to the radioactive nuclei by the neutrino oscillations (MSW resonances) through theoretical calculations. Thus, the estimation of the radioactive decay rate is very rough. Therefore, it is necessary to obtain the real value of the change in the decay rate caused by neutrino oscillation-induced radioactive decay, and experimental probes are needed to determine it. In this study, we established the simplest experimental method. Considering that the density of the water body is the most homogeneous and it is easy for MSW resonance to occur, several samples of a certain radioisotope can be placed in the range of 2.0 - 7.0 km in the ocean depth, and the change in the decay rate with respect to that at the surface can be obtained to determine the required value. This can also be used to test the theory developed in this study.

Zhang and Zhang (2024a) have shown that although the MSW effect of neutrino oscillations is a weak interaction resonance, it can also have an effect on alpha decay. The reason for this is that in neutrino oscillations, there are some couplings and connections between the weak interactions and the strong interactions ( Higgs, 1964 , 2014 ; Weinberg, 1967 ), which are able to induce oscillations in the Higgs field and thus excite α decay. Indeed, as a result of this weak interaction, the weak effective potential is converted into energy when the MSW resonance is generated. This energy is transferred to the atoms in the entire medium involved in the resonance in the form of resonance, which transitions the atoms in an excited state ( Figure 1 ), and since α-decay is very sensitive to energy, MSW resonance is able to increase the probability of α-decay.

A melt pocket is a micro-melting zone originating from the mantle. Its size is in the nanometer to micrometer range, and it is considered the beginning or basic unit of magma. A melt pocket usually consists of neonate minerals + molten glass + residual minerals after melting ( Figure 2 ). It has various shapes, including

droplet, spherical, fan, cowl, vein, and indeterminate shapes ( Liu, 2020 ). The development of melt pocket is found all over the globe and has been studied by many scholars ( Du, 1998 ; Su et al., 2010 ; Liu, 2020 ). It has been found that in the same rock section of the same orthorhombic pyroxene (Opx), some particles are completely melted, while others are not melted, and the degree of melting varies greatly within a thin layer, indicating that they are not in the same thermal field ( Du, 1998 ). In addition, the prevalence of glassiness in the micromelting zone ( Liu, 2020 ) indicates that the melt pocket underwent rapid cooling after melting and that there is a large temperature difference between the melt pocket and the surrounding environment. Obviously, this melt pocket could not have been produced via plate subduction or mantle upwelling, and it should have a different origin. According to Du (1998) , the uneven infiltration of mantle fluids leading to the melting of material is an important factor in the formation of magma (a melt pocket). However, the melt pocket is not connected to the surrounding material by fissures or magma veins, and the flow of mantle material (including gases) cannot penetrate into the melt pocket.

Figure 2. Jiaohe melt pocket. (a) Melt pocket 1 occurs between olivine and orthopyroxene crystals. Its diameter is about 2.5 mm. (b) Melt pocket 2 occurs in the gaps between orthopyroxene and spinel crystals. It has an ox-horn shape and the particle size is 1.5 - 2.0 mm. The images are from Liu (2020) .

We believe that the only factor that can cause the melt to penetrate deep into the minerals without destroying them and cause melting in the microzones in the middle of the minerals is atmospheric neutrinos. Since neutrino oscillations can only induce radioactive decay, melting can only occur in areas enriched in radioactive elements, and the first material to melt is the area enriched in radioactive elements. A melt pocket consists of typical radioactive element-enriched microregions. Du and Wang (2005) showed that in the mantle, radioactive U is the least abundant in mineral crystals, in which it does not exceeding 10 × 10⁻⁹, and it is more abundant in crystal boundaries and cleavage planes (generally tens to hundreds of 10⁻⁹). The melt pocket is the most uranium-enriched material. For example, the uranium mass fraction of the melt pocket of sample JH4 is (323.85 - 1125.1) 10⁻⁹. In addition, the melt pocket is unusually enriched in K ( Du, 1998 ; Liu, 2020 ), and ⁴⁰K is also the major radioisotope in the Earth’s interior. This is a strong indication that melting is closely related to the enrichment of radioactive elements. When atmospheric neutrinos propagate through the mantle and generate MSW resonance, the regions enriched in radioactive elements are strongly heated, causing the material to melt. At this time, the temperature of the molten microregion is significantly higher than the ambient temperature. When atmospheric neutrinos stop passing through the melting microregion due to certain effects (such as interference from geomagnetic storms), the decay of the radioactive elements returns to its natural state, the heat generation is suddenly reduced, and the temperature of the melting region drops rapidly. As a result, some of the molten material will crystallize into minerals due to the drop in temperature, and some of the molten material will solidify into glassy material before it can be crystallized. If atmospheric neutrinos propagate through the solidified micro-area (melt pocket) again, the micro-area will melt again, and new minerals will crystalize. This is the fundamental reason for the diversity and complexity of the new minerals and their combinations in molten microzones ( Du, 1998 ; Liu, 2020 ).

The distribution of the radioisotopes in the mantle is heterogeneous ( Dai et al., 2005 ; Wu et al., 2021 ). The regions where the radioactivity is perturbed by neutrino oscillations, which generate heat and cause melting, are consequently inhomogeneous, with melting occurring only in the small regions that are relatively enriched in radioactive elements, forming a series of melt pockets ( Liu, 2020 ). As the heat generated by the radioactive elements via perturbation caused by neutrino oscillations increases and the temperature increases, the melt pocket grows gradually, or the melt pockets grow and then converge together to form a larger melt. When the proportion of melt reaches 0.02% - 0.20%, the melt is interconnected between mineral particles, and due to the high temperature and low density of the melt, a large buoyancy force is generated and the melt can be easily extracted from the minerals and transported upward due to this buoyancy force ( Zhu et al., 2011 ; Sawyer et al., 2011 ).

Due to the good peristaltic properties of the mantle, melts migrate and converge along the edges of particles in a permeable manner in the mantle. When melt penetrates along the edge channels of particles, there is a power law relationship between the melt ratio and the permeability ( Wark and Watson, 1998 ; Bons et al., 2004 ). Therefore, as the amount of melt increases slightly, the permeability significantly increases. When the local mantle melt ascends, migrates, and converges with rock layers with poor peristalsis and rigid crust, it is usually difficult for the melt to ascend and migrate in a permeable manner, and it can only migrate along cracks or faults, can form magma chambers or mush in suitable areas, or can reactivate previously formed cold mush ( Jackson et al., 2018 ; Ma et al., 2020 ).

The process of the ascent of the magma is also the process of continuous evolution of the magma. In this process, the magma continuously exchanges material and energy with the surrounding environment through merging, fusion, accumulation, alteration, crystallization, and dissolution, which ultimately leads to the continuous melting of fusible materials along the magma’s path (clearing of the path), while refractory materials continue to crystallize and accumulate, and some of the structures in the deep part of the Earth also constantly change ( Figure 3 ).

In addition, from the formation and evolution of the magma, it can also be concluded that the process of magma formation is actually the extraction of fusible materials and elements such as radioisotopes from the upper mantle and asthenosphere, as well as the continuous sorting out of materials in the lower crust (possibly including the middle crust) ( Takazawa et al., 1992 ). After a long geologic time period, the result of these processes is the formation of refractory ultramafic rocks with losses of corresponding elements in the upper mantle, basaltic rocks (coexisting with residues and precipitates) in the lower crust, and evolutionarily mature silicic rocks enriched in the relevant elements in the upper crust.

Calculations have shown that atmospheric neutrinos are able to form MSW resonance with matter in all of the layers of the Earth, and accordingly melts should be widely distributed in all of the layers of the Earth. However, geophysical surveys indicate that melts are mainly concentrated in the upper mantle and the asthenosphere ( Figure 4 ) ( Debayle et al., 2020 ). This may be due to a number of reasons. 1) Since there is a resonance initiation process, it is difficult to produce MSW effects in the shallow crust, and therefore the radioactive materials in the crust are not perturbed by MSW resonance to generate heat. Thus, there is usually no molten material in the crust. 2) Since the upper mantle has a greater concentration of radioactive elements, the radioactive decay due to MSW resonance is able to generate enough heat to cause some of the material to melt. By contrast, the lower mantle has a lower concentration of radioactive elements, and coupled with the increased pressure, the radioactive heat generation promoted by the MSW effect is not sufficient to cause the material to melt. The aforementioned calculations reveal that in the upper mantle, the situation in which 3.02% of the radioactive elements are induced to decay by the MSW effect is sufficient to lead to melting of the material. In the lower mantle, the contents of radioactive elements ²³⁸U, ²³²Th, and ⁴⁰K are about 0.003 × 10⁻⁶, 0.005 × 10⁻⁶, 0.03% × 0.0117%, respectively ( Li, 1976 ). Using these data, it can be calculated that the heat of decay of all of the radioactive material contained in 1000 kg of lower mantle material is only 2.62 × 10⁸ J. This heat is less than the heat required to melt 1000 kg of upper mantle material (i.e., 5.1 × 10⁸ J). Because the pressure in the lower mantle is high relative to that in the upper mantle, melting the same amount of material requires more heat, so it is difficult for the heat produced via decay of the radioactive elements in the lower mantle to completely melt the lower mantle material. 3) Atmospheric neutrino oscillations and the radioactive decay they promote are both probabilistic events, and such probabilistic events are generally normally distributed. That is, from the spatial perspective, a more concentrated region of heat generation and melting exists, and the amounts of heat generation and melting gradually decrease upward and downward with this region as a base point. This region of melt concentration is the asthenosphere. 4) The uneven distribution of atmospheric neutrinos is generated via cosmic ray perturbations by the geomagnetic field, and therefore the radioactive heat generation promoted by their oscillations exhibits an uneven distribution. This can result in an uneven distribution of melt in the lateral direction. 5) Due to the arch tectonic effect ( Zhang and Zhang, 2024b ), melt beneath the basins (flats) on either side of the range converges toward the range, resulting in more melt beneath the range. 6) There may also be other unknown influencing factors.

Once the melt is produced in the upper mantle and the asthenosphere, it ascends and converges under the effect of buoyancy. As the height of the ascent increases, the temperature of the melt decreases, the viscosity increases, and the creep of the mantle decreases. Thus, when the melt arrives in the upper mantle lithosphere, due to the permeability barrier at the lower boundary of the lithosphere, it is difficult for the magma to ascend by means of permeability, and it can only be transported upward along fissures and fractures. As a result, a large amount of melt is trapped at the LAB, leading to a large increase in the melt content in this region and thus forming a sharp seismic wave discontinuity interface ( Schmerr, 2012 ; Zhang et al., 2024 ). Since the melt in the asthenosphere is the product of in situ melting, the melt is more silicon-poor and iron-rich, and the melt that converges in the LAB is usually considered asthenospheric melt that has ascended to the base of the lithosphere along grain boundaries ( Schmerr, 2012 ). This melt crystallizes and precipitates some of the ultramafic minerals due to the occurrence of heat dissipation during the migration process, resulting in the occurrence of a relatively silicon-rich and iron-poor melt in this area ( Zhang et al., 2024 ).

As the convergence of high-temperature melts at the LAB increases, the lithosphere gradually thins via melt erosion ( Hyndman and Canil, 2021 ; Sato & Ozawa, 2023 ). The lithosphere is also subjected to increasing thermal pressure, which ultimately leads to uplift and tearing of the thinner and reduction of the compressive strength of the oceanic lithosphere. As a result, the melt that was trapped at the LAB ascends along the fractures and eventually erupts onto the ocean floor, forming a new oceanic crust. When the thermal pressure is released and the melt (magma) is not replenished, the magma stops erupting. The damaged oceanic crust is then filled with the magma, and when the magma is completely solidified, the lithospheric and oceanic crust are repaired, and the gravitational force exerted by the overlying rocks and the thermal pressure are balanced again. However, the radioactive decay induced by atmospheric neutrino oscillations does not stop, and magma is still gradually generating and ascending, converging toward the LAB. When the thermal pressure exerted by the converging magma is greater than the gravitational force exerted by the overlying rocks, once again the lithosphere and oceanic crust are damaged and fractured, the magma once again erupts along the fractures, the thermal pressure is released, new oceanic crust is formed, and the lithosphere and the oceanic crust are repaired. As a result, the height of the mid-ocean ridge increases, and movement of the oceanic lithosphere and oceanic crust is promoted. Of course, if there are weak structures such as cracks and fractures within the oceanic plate, the melt present under the lithosphere can also ascend and erupt along the cracks and fractures, forming a number of seamounts of varying sizes on the seafloor ( Machida et al., 2015 ; Pan et al., 2021 ).