Neutron production from the fracture of piezoelectric rocks

Neutron production from the fracture of piezoelectric rocks

A Widom and J Swain, Physics Department, Northeastern University, Boston MA, USA. Y N Srivastava, Department of Physics and INFN, University of Perugia, Perugia, Italy.

Abstract

A theoretical explanation is provided for the experimental evidence that fracturing piezoelectric rocks produces neutrons. The elastic energy micro-crack production ultimately yields the macroscopic fracture. The mechanical energy is converted by the piezoelectric effect into electric field energy. The electric field energy decays via radio frequency (microwave) electric field oscillations. The radio frequency electric fields accelerate the condensed matter electrons which then collide with protons producing neutrons and neutrinos.

I. Introduction

There has been considerable evidence of high energy particle production during the fracture of certain kinds of crystals. In particular, fracture induced nuclear transmutations and the production of neutrons have been clearly observed. The production of neutrons appears greatly enhanced if the solids being fractured are piezoelectric materials. Our purpose is to describe theoretically the manner in which the mechanical pressure in a piezoelectric stressed solid about to fracture can organize the energy so that neutrons can be produced.

The nuclear physics involves a standard weak interaction wherein collective radiation plus an electron can be captured by a proton to produce a neutron plus a neutrino — written in the paper as equation (1), radiation energy plus an electron plus a proton going to a neutron plus an electron neutrino.

The required collective radiation energy may be produced by the mechanical elastic energy storage via the piezoelectric effect. By the definition of a piezoelectric material, the conversions of energy between elastic energy and electric energy are allowed.

In terms of the electric field and the crystal strain tensor, the precise definition of the piezoelectric tensor is discussed in section II. The final result may be expressed as an effective interaction Hamiltonian, equation (3), in which the interaction energy density is minus the piezoelectric tensor contracted with the electric field and the strain tensor, integrated over the volume. The tensor coefficients describe piezoelectricity as shown in figure 1.

Figure 1. Shown is the Feynman diagram exhibiting the change of a phonon described by the tensor strain into a photon described by the vector electric field, and vice versa. The piezoelectric coupling strength tensor is exhibited in the interaction Hamiltonian, equation (3).

Some implications of the conversion from mechanical energy into electromagnetic energy are quite striking. For example, a piezoelectric ignition system can be constructed wherein a sharp mechanical impulse to a piezoelectric material can induce a sharp voltage spike across the sample with the resulting spark igniting a fire in a surrounding gas. More dramatically, the rocks crushed in earthquakes contain piezoelectric quartz. The mechanical impulse causing micro-cracks in the rocks can thereby produce impulse earthquake lightning flashes.

In section III we review the stresses and strains which accompany micro-cracks in rocks that are being fractured. Elasticity theories of such micro-cracks are well known. The central result is as follows. If the bond stress denotes the elastic stress required to break the chemical bonds on an area of a micro-crack and the surface tension denotes the surface tension of the free face of a crack, then the fracture stress required to create a crack of half length a is given by equation (4) — the square root of the product of bond stress and surface tension divided by the half length — so that for brittle fracture the fracture stress is very much smaller than the bond stress.

In section IV, the manner in which the conversion of mechanical to electrical energy takes place is explored. It is shown that copious electromagnetic energy is emitted in the radio frequency microwave regime. The radiation accelerates the electrons allowing for nuclear transmutations in forms following from equation (1). In the concluding section V, the number of neutrons produced by rock fractures will be estimated.

II. Piezoelectric interactions

The energy per unit volume of a piezoelectric material obeys equation (5): the change in energy density equals the temperature times the change in entropy, plus the stress tensor contracted with the change in strain tensor, minus the electric dipole moment per unit volume dotted with the change in electric field.

The adiabatic piezoelectric tensor may be defined, equation (6), as the change of polarisation with electric field at constant entropy and strain, which is equal to minus the change of stress with strain at constant entropy and field. To quadratic order, the mechanical electric field interaction energy follows from equation (6); it is the second cross derivative of the energy density with respect to field and strain, taken in either order, and the interaction energy density is minus the piezoelectric tensor contracted with field and strain — leading to the quantum operator in the effective Hamiltonian and the Feynman diagram of equation (3).

The adiabatic electric susceptibility of the material at constant strain is defined in equation (8) as the change of polarisation with field at constant entropy and strain, while the same susceptibility at constant stress is given by equation (9). The elastic response tensor, equation (10), is the change of strain with stress at constant entropy and field, and it determines the difference between the two susceptibilities: the thermodynamic identity, equation (11), is that the constant-stress susceptibility equals the constant-strain susceptibility plus the piezoelectric tensor contracted twice with the elastic response tensor.

For a complex frequency with non-negative imaginary part, there are dynamical electric susceptibilities at constant stress and at constant strain. The dynamical version of equation (11) is easily obtained. Phonon modes described by the dynamical phonon propagator affect the dynamical susceptibilities via equations (12): the displacement field is the electric field plus four pi times the polarisation, the dielectric tensor is the identity plus four pi times the constant-stress susceptibility, and the constant-stress susceptibility is the constant-strain susceptibility plus the piezoelectric tensor contracted twice with the dynamical phonon propagator. Equation (11) is the zero frequency limit of equation (12).

The dynamical dielectric response tensor appears in the polarization part of the photon propagator. The Feynman diagrams contributing to the polarization part of the photon propagator in a piezoelectric system are shown in figure 2. These are equivalent to equation (12) and explain why mechanical acoustic frequencies appear in the electrical response of piezoelectric materials.

Figure 2. The Feynman diagrams contributing to the polarization part of the photon propagator in a piezoelectric material. The resulting dielectric response has a contribution due to mechanical phonon modes as exhibited in diagrammatic form.

III. Fracture and stress

Shown in figure 3 is a crystal under stress inducing a micro-crack of width 2a and length very much greater than a. The energy required to create a micro-crack of half width b and length L is given by equation (13): the energy per unit length equals four times the surface tension times b, minus pi times one minus the squared Poisson ratio, times the squared stress, times b squared, divided by Young's modulus. Here the surface tension is that of the micro-crack interface, and Young's modulus and the Poisson ratio are those of the material.

Figure 3. A micro-crack is formed in a solid under stress. The width of the micro-crack is 2a and the length (into the paper) is very much greater than a. The half width a is the critical length size for forming the micro-crack as in equation (14).

A. Tensile strength

The maximum of the elastic micro-crack energy per unit length represents the energy barrier to micro-crack creation. In detail, equation (14): the maximum occurs at half width a, where a equals two times the surface tension divided by pi, times Young's modulus divided by one minus the squared Poisson ratio, divided by the squared fracture stress; and the barrier energy equals twice the surface tension times a, which is four times the squared surface tension divided by pi, times the same modulus factor over the squared fracture stress.

The stress level which nucleates a micro-crack is thereby the well known result, equation (15): the fracture stress is the square root of twice the surface tension times Young's modulus, divided by pi times one minus the squared Poisson ratio times the half length a. The tensile strength of the material is then given by equation (4), wherein the broken chemical bond strength, equation (16), is twice Young's modulus divided by pi times one minus the squared Poisson ratio — determined by Young's modulus and the Poisson ratio.

B. Numerical estimates

Employing the values of material constants for fused quartz, we can estimate at least the powers of ten that would apply to piezoelectric rocks such as granite rocks. The values, equation (17), are a surface tension of about 10² erg per square centimetre, a bond stress of about 10¹² erg per cubic centimetre, a fracture stress of about 10⁹ erg per cubic centimetre, and a critical half width of about 10⁻⁴ centimetre — in satisfactory agreement with the elastic theory as reviewed in section III A.

Some comments are in order. (i) For quartz, the value of the critical half width is about 1 micron. (ii) For the brittle fracture of quartz, the macroscopic fracture surface experimentally exhibits micro-cracks with a length of about 20 microns, very much greater than the critical half width. (iii) As is usual in fractures, the fracture stress is very much smaller than the bond stress — here about a thousandth of it. (iv) The velocity of sound compared with the velocity of light obeys a ratio of about 10⁻⁵. The ratio of phonon frequencies to photon frequencies in cavities of similar length scales thereby obeys, equation (18), a ratio of about 10⁻⁵ for similar sized cavities.

The importance of the above equation (18) is that the phonon modes enter into the dynamic dielectric response function in virtue of equation (12).

IV. Neutron production

The neutron production rate at the fracture stress is here considered due to energetic electrons scattering off protons which are naturally present in, say, granite as water or organic molecules. The Feynman diagram in the Fermi theory limit of the standard model is shown in figure 4, described in equation (1).

Figure 4. Neutron production takes place via the standard Fermi weak interaction. The electron energy is renormalized from its rest energy to a higher value by condensed matter microwave radiation present as the stress approaches the fracture value. The coupling strength at the four fermion vertex is the Fermi constant.

A. Electron renormalized energy

To begin to analyze the production of neutrons via the reaction of equation (1), one must calculate the mean energy of electrons in condensed matter when accelerated by an electric field, equation (19): the rate of change of momentum equals the electron charge times the electric field. The electron energy is estimated by the relativistic expression, equation (20), the square root of the squared rest energy plus the squared momentum times the squared speed of light.

If the mean squared electric field strength in a bandwidth is given by a spectral density, then equation (19) implies equations (21): the mean squared field is the integral of the spectral density over frequency, and the mean squared momentum is the squared charge times the integral of the spectral density divided by the squared frequency. If a dominant frequency is present in the spectrum of electric field fluctuations, then equation (21) is more simply written as equation (22): the mean squared momentum is the squared charge times the mean squared field divided by the squared dominant frequency.

So the ratio of the energy to the rest energy of the electron, equation (23), is the square root of one plus the square of the quantity charge times field divided by mass times speed of light times dominant frequency. A value of this ratio greater than one is critical for measuring whether or not there is sufficient radiation energy to allow for the reaction in equation (1).

B. Further numerical estimates at fracture

To estimate the electric field, one notes that the stress at fracture is in large part due to the electric field strength: the stress is about the squared field over four pi, which with the values of equation (17) implies a field of about 10⁵ gauss, equation (24). Since the ratio of electron charge to mass times the speed of light is about 1.75882915 × 10⁷ per gauss per second, equation (25), one finds a charge-times-field-over-mass-times-light-speed value of about 10¹² per second, equation (26).

The frequency of a sound mode localized on a micro-crack of width 2a for a reasonable sound velocity in rock is in the microwave range, about 10⁹ per second, equation (27). One should then observe electromagnetic microwave emission when the sound mode is turned into an electromagnetic mode via the piezoelectric effect.

In virtue of equations (23), (26) and (27) one finds an energy ratio of about 30. The threshold value of that ratio for equation (1) to be possible without radiation is about 2.53, so that the energy renormalized by radiation is above threshold by a wide margin. The electron energies on the surface of a micro-crack in a stressed environment with an external stress at the fracture value obey, equation (28), an energy of about 15 MeV.

The transition rate per unit time for equation (1) by the usual standard has been computed as the square of the Fermi constant times the squared electron mass over the reduced Planck constant times the speed of light, times the electron rest energy over the reduced Planck constant, times the squared energy ratio. Numerically, equation (29), this is about 7 × 10⁻³ times the squared energy ratio in hertz, which is about 0.6 hertz for an energy ratio of about 30.

The transition rate per unit time per unit area of micro-crack surfaces may be found from equation (30), the number of protons per unit micro-crack area in the first few layers of the quartz granite times the transition rate per proton. Typical values, equation (31), are about 2 × 10¹⁴ protons per square centimetre, giving about 10¹⁵ hertz per square centimetre.

If the fracture takes place with hydraulic fracture processes, then the neutron production rate will be about a factor of ten higher due to the higher water concentration on the micro-crack surface areas.

V. Conclusions

It is in the nature of piezoelectric matter that strong mechanical disturbances give rise to strong electromagnetic responses. This is true for piezoelectric rocks such as granite which contain large amounts of quartz. For large scale piezoelectric rock fracturing, as takes place in earthquakes, electromagnetic responses in many frequencies, from radio frequency to gamma ray frequency, are to be expected. Some have attributed earthquake lights and/or lightning to the phenomena discussed in this work.

We have employed the standard model of weak interactions along with the known theory of piezoelectric materials to explain the experimental evidence that fracturing piezoelectric rocks produces neutrons. We have also explained why such fracturing processes produce microwave radiation. The elastic energy micro-crack production ultimately yields the macroscopic fracture whose acoustic vibrations are converted into electromagnetic oscillations. The electromagnetic microwaves accelerate the condensed matter electrons which then scatter from protons to produce neutrons and neutrinos. This work also may have implications for a better understanding of radiative processes associated with earthquakes.