A simple argument that small hydrogen may exist

– We present theoretical argument, based on virial theorem and De Broglie’s idea from 1924, why the small hydrogen may exist. It may have been created during early stages of the Big Bang, near black holes or large explosions in Universe. In principle, it could be formed in high energy experiment. Being neutral and stable, it would play a significant role in a formation of early galaxies. It would not be observable using usual spectroscopic methods. This paper suggests methods to discover the small hydrogen.


Introduction
In 1920, Rutherford suggested that an electron and proton could be bound in a tight state [1].He tasked his team, including Chadwick, with searching for this atom.Following Chadwick's discovery of the neutron in 1932, there was considerable debate about whether it was an elementary particle, or a hydrogen-like atom formed from an electron and a proton [2].For instance, Heisenberg was among those who argued that Chadwick's particle was a small hydrogen atom until 1933.Ultimately, Pauli's argument prevailed: the neutron, with its spin of 1/2, follows Fermi-Dirac statistics, confirming it as an elementary particle.This is a well-established fact and is not the focus of this paper.
It must have been obvious to Schrödinger, Dirac and Heisenberg, that there is a peculiar solution to their equations.This solution, which corresponds to the small hydrogen, was at the end rejected [3] because the wave function is infinite at r = 0. Since nobody has observed it, the idea of the small hydrogen has died.However, its idea was revived again 70-years later, where authors argued that the proton has a finite size, and that the electron experiences a different non-Coulomb potential at very small radius [4], [5].In fact, such non-Coulomb potentials, for example, Smith-Johnson or Nix potentials [6], [7], are used in relativistic Hartree-Fock calculations for very heavy atoms where inner shell electrons are close to nucleus.Using this method, authors retained solutions for the small hydrogen which were previously rejected.However, in a follow up paper [8], it was recognized that considering these two potentials does not satisfy virial theorem, and that one needs to add much stronger potential to hold the relativistic electron stable.
Brodsky pointed out that one should not use the "1930 quantum mechanics" to solve the problem of the small hydrogen; instead, one should use the Salpeter-Bethe QED theory [9].Spence and Vary attempted to find such electron-proton bound state using QED theory [10], which includes spin-spin, field retardation term and Coulomb potential, assuming the point-like proton.They suggested a possible existence of a bound state.
Bethe and Salpeter presented a theory of the normal hydrogen atom using the Dirac equation [14].However, they did not consider the small hydrogen.
There are two reasons why the small hydrogen idea was not investigated theoretically further: (a) nobody has found it experimentally, and (b) the correct relativistic QED theory is too complicated 1 at small distances.
One hundred years ago, Louis De Broglie published his famous paper [11], which sparked the quantum mechanics revolution; this was before Schrödinger or Dirac equation existed, only the Bohr model was known to him.De Broglie model of normal hydrogen atom, shown in Fig. 1, demonstrates that stable states occur only when the number of standing electron waves is an integer.
We argue, similarly as De Broglie, that a stable "electron standing wave" in a small hydrogen atom can exist only for certain frequencies of the electron wave and that its radius is determined by virial theorem and potential.Basic parameters of this model are shown in Table 1 for both normal and small hydrogen.Although De Broglie's model is based on old quantum mechanics, it provides a reasonable 2 approximation for normal hydrogen.
Our approach is a potential-based calculation.Virial theorem is important for judging whether a bound system is stable.It can draw conclusions about the dynamics of bound states without solving differential equations.We propose to solve the problem using a simple model based on two basic physics principles: (a) Virial theorem, 1 Private communication with Prof. James Vary. 2 For exact values one needs to use Dirac equation.(b) De Broglie's classical quantum mechanics principle stating that the only allowed atomic states are those with integral number of electron wavelengths on a given atomic orbit.These two assumptions are sufficient to judge stability of the normal hydrogen.We will assume De Broglie model is also a good approximation for small hydrogen.The fact that it does not entirely explain the complexity of normal hydrogen is not crucial for this paper; the key argument is that it explains its stability.De Broglie, exactly 100 years ago, wrote a paper arguing that electron motion in an atom can only be stable if the phase wave is tuned with the length of the path, according to the line integral: where λ is the electron wavelength, n is the number of periods, and ds denotes an element of the path of a wave moving from one crest to the next.In its simplest form, the De Broglie wavelength is constrained by the radius r through the equation where n is an integer defining an integral number of wavelengths in the circumference.This provides a reasonable description of normal hydrogen.We will use the same arguments for small hydrogen.Previously, we have suggested a simple numerical iterative method, where we step through radius in small steps until virial theorem is satisfied [12].The procedure to find a solution is as follows: 1.The electron's De Broglie wavelength is constrained by radius r through the equation λ = n2πr (where n is an integer defining the integral number of wavelengths in the circumference).
2. The electron momentum is determined from the De Broglie equation p = h/λ (from which we get the relativistic kinetic energy T kinetic , β, and γ).
3. The stable electron radius is determined by numerically stepping through values of r until virial theorem is satisfied, balancing electron relativistic kinetic energy and potential energy.
Reference [12] demonstrated that three different methods can satisfy virial theorem for small hydrogen: • The electron kinetic energy must balance with the expected virial kinetic energy derived from the potential: • Lucha showed that this equation must be satisfied [13]: where p is the electron relativistic momentum, r is the electron radius, and U (r) is the total potential the electron feels.Since we are dealing with periodic motion, we can drop averaging over time, rewriting equation (3a) as follows: • One can also search for a minimum in the total electron energy E = T kinetic + U : In this paper, we will mainly use the method according to equation ( 2), and use equations ( 3) only as a cross-check.
The electron kinetic energy is calculated as follows: where λ = 2πr/n is the De Broglie wavelength for electron radius r, and n is the number of wavelength periods.
Virial theorem states that for a general potential energy V (r) = αr k , the expected electron kinetic energy T virial is related to potential energy as: For example, for Coulomb potential (U1 = −k1/r), k = −1, and the kinetic virial energy behaves as T virial → −(1/2)U1 as γ → 1, and as T virial → U1 as γ → ∞.

Virial Theorem with Coulomb Potential
Applying equation ( 2) to small hydrogen, one finds that the Coulomb potential V Coulomb = −KZe 2 /r alone cannot hold the electron in a stable deep orbit in small hydrogen, as illustrated in Figure 2, although normal hydrogen can have a stable solution.This is also the case if we add the Smith-Johnson or Nix potentials, used in high-Z atom calculations.These two potentials were also used in references [4], [5], providing the first hint of small hydrogen existence.We argue that a stronger potential acting at a small radius is necessary.
Figure 2: There is only one region of stability for Coulomb potential in the e-p system, corresponding to the normal hydrogen.The small hydrogen is not stable using this potential.

Virial theorem with V (Spin.B) potential
We assume that the following relativistic spin-orbit potential energy at small radius (consider it as ansatz): where µ0 = 5.788 × 10 −9 eV/Gauss is the Bohr magneton, g = 2.0023, and B is the self-induced magnetic field.To understand the origin of this magnetic field, we assume a simple equivalent model where the electron is at rest and the proton is moving around it at this radius.We estimate the magnetic field value as follows: where I is the circular loop current due to the proton's velocity v, and Z is the atomic number.The magnetic field is B ∼ 5.95 × 10 15 Gauss at a radius of ∼ 2.84 Fermi, making the spin term in equation ( 7) dominant and equal to V (Spin.B) ∼ 69.04 MeV, while the Coulomb energy contribution to the balance is only ∼ 0.5075 MeV at the same radius.
Although this paper uses the electron radius r in the following formulas, it should be looked at from a quantum mechanical point of view, i.e., the electron has a distribution of radii with some mean value of ⟨r⟩, determined by its wave function.
Figure 3 demonstrates that adding a potential V (Spin.B) to the Coulomb potential helps satisfy virial theorem at r ∼ 2.84 Fermi.Table 3 shows that the mass of small hydrogen is M (p,e − ) = mproton + γ m electron − |U | = 938.272MeV/c 2 , with a binding energy EBE = T kinetic − |U | ∼ −507.5 keV for n=1. 3n normal hydrogen, where the electron is far from the proton, the spin-orbit interaction V (Spin.B) is a small perturbation.However, at small distances it becomes a significant force, providing stability of the bound system.

Virial theorem with V eff potential
Adamenko and Vysotskii [15] proved, starting from Dirac equation, that the effective potential energy of a relativistic electron in Coulomb field can be expressed as: Paillet and Meulenberg [16] used this potential and concluded that the small hydrogen may exist.Using the iterative method described in chapter 2, I confirm their results, as demonstrated on Figure 5. Quantitative results are shown in Table 3.One can see that for large values of n, the binding energy approaches 511 keV.
Figure 6 shows the V Coulomb , V (Spin.B) and V eff potential shapes as a function of radius close to the proton.One can see that the V (Spin.B) and V eff potentials are almost identical and much stronger than the Coulomb potential at a distance if a few Fermi from the proton.Figure 6 shows the V Coulomb , V (Spin.B) and V eff potential shapes as a function of radius close to the proton.One can see that the V (Spin.B) and V eff potentials are almost identical, which is not obvious from their definitions, and much stronger than the Coulomb potential in the vicinity of the proton.
Figure 7 shows the same conclusion using Lucha's virial stability condition of equations (3), i.e., the stability of small hydrogen occurs at r ∼ 2.84 Fermi for n = 1.Tables 2 and 3 show that both potential choices yield almost the same result for the small hydrogen.They also show that small hydrogen is stable, based on the argument that M (pe − ) mass is smaller than sum of masses proton and electron.Notice also that binding energy EBE values are close to EDDL values presented in Refs.[8], [9], obtained using the relativistic Schrödinger and Dirac equations, i.e., using completely different calculations.Another interesting conclusion is that the mass of the small hydrogen M (pe − ) is slightly smaller than the mass of a neutron.
The small hydrogen cannot be formed spontaneously since the electron can obtain only ∼ 507.5 keV at a radius of 2.84 Fermi from the available static Coulomb potential energy.This means that energy must be supplied to the electron externally to form the small hydrogen (in this respect, this is similar to the electron capture on a proton p + e − → n + νe, which requires external energy of at least 782.33 keV).
The small hydrogen will remain in the n = 1 state, as any excitation to higher n requires too much energy, which is not available in typical collisions in Universe.The small hydrogen will appear optically "dark" to an observer.
Figure 8 shows strengths of various potentials considered in this paper.

Interactions of small hydrogen
In gas medium its dE/dx deposit will be ∼ 10 7 -times smaller than that of a typical charged particle because the small hydrogen will interacts via a slight electric dipole [17].It will not be ionized by collisions with light nuclei or with collisions with another small hydrogen at velocities typical in the Universe, such as, for example, the Bullet Cluster galaxy collision assuming velocity of 4500 km/sec, which corresponds to a kinetic energy of small hydrogen of only ∼ 105 keV.One will be able to recognize an existence of small hydrogen only through gravitational effect.
However, at thermal velocities, small hydrogen can be captured by positively charged nuclei since the Coulomb barrier in this case is significantly smaller than when two positively charged nuclei collide.At energies slightly higher than ∼ 0.511 MeV it could be ionized, and at very high energies it can initiate hadronic shower just like a neutron.
3 Accelerator test to find small hydrogen A free thermal electron, when approaching a thermal proton, is captured on the highest level of the normal hydrogen first and subsequently gains a total energy of ∼ 13.6 eV from the available electrostatic potential energy.It then latches onto the ground level with the correct De Broglie wavelength, where the electron has a radius r ∼ 0.529 Å and a De Broglie wavelength λ ∼ 3.222 Å, which corresponds to an electron kinetic energy of E kinetic ∼ 13.6 eV.If there is a large mismatch in energies, the electron and proton will not form normal hydrogen.
I will use the same argument for the small hydrogen.Table 3 tells us that the electron radius is approximately 2.828 Fermi, the De Broglie wavelength is approximately 17.762 Fermi, electron kinetic energy E kinetic is approximately 69.302 MeV, β = v/c ≈ 0.999973212 and γ ≈ 136.620.For this condition, the electron will latch to the small hydrogen orbit if the proton has the same velocity as the electron; this corresponds to proton kinetic energy of 128.189GeV.
I suggest to send electrons and protons in the same direction, as shown on Figure 9.The required beam kinetic energies are shown in Table 4. Figure 9: Schematic concept to prove that the small hydrogen exists.Proton beam is brought tangentially to electron beam so that both beams travel parallel to each other for some distance.If the small hydrogen is formed, it will emit a 508.6 keV gamma in the two-particle rest frame, while electrons are deflected by a magnet.
Figure 10: Gamma energy is boosted due to the Doppler effect to high values at very forward direction from a source traveling at velocities close to velocity of light -see equation (10).
If the small hydrogen atom is formed, a ∼ 508.6 keV gamma is created in the two-particle rest frame.In the lab frame, the photon energy is very sensitive to the choice of ϕ angle due to the Doppler effect.The ϕ angle is the angle between the direction of motion and the gamma detector position: Figure 10 shows this dependency on the angle ϕ, based on equation (10).The gamma detector should be positioned at ϕ ∼ 6.9 • to measure E Observed /ESource ∼ 1.Energy of gammas produced at ϕ = 0 • will be boosted from 0.5086 MeV to 138.9 MeV.
Therefore detector, shown on Figure 9, would detect gamma in the electromagnetic calorimeter and the small hydrogen will create a large collinear hadronic shower in the hadronic calorimeter, both well separated in space.The idea is to tune beam energies and find a peak at expected electron and proton energies.This result would be a direct proof of small hydrogen existence.This measurement is a high-energy physics equivalent to what were the 1920's bench-top experiments.
At present, the only places where such an experiment is feasible are Brookhaven National Lab (BNL), Fermilab, or CERN (protons of ∼ 128.2 GeV already exist at the CERN SPS, and a ∼ 69.3 MeV electron accelerator may not be that difficult to construct).

The 511keV signal from the Galaxy center
The first gamma-ray line originating from outside the solar system that was ever detected is the 511 keV emission from the center of our Galaxy.The accepted explanation of this signal is the annihilation of electrons and positrons.However, despite 30 years of intense theoretical and observational investigation, the main sources of positrons have not been identified.Ref. [17] has proposed an alternative explanation: the observed signal is due to atomic transitions to "small hydrogen atom".

Collapse of very large stars
De Broglie's hydrogen model says that electron wave is undergoing periodic orbital motion and it does not radiate provided the shell radius is an integral multiple of waves; the orbit need not be circular nor even planar, it can be a vibration in 3D.For example, for a choice of potential according to Tables 3, r stable ∼ 2.828 Fermi, and the proton is surrounded by a "standing electron wave" with a De Broglie wavelength of 17.762 Fermi, oscillating with a very high frequency of ∼ 1.688 × 10 22 Hz.That is a very high number, but applying the same idea to the normal hydrogen, the frequency of the electron wave is ∼ 6.6 × 10 15 Hz, still a very large number.In this picture, the small hydrogen is just a different hydrogen atom with the electron oscillating at a higher frequency.
One could ask if the small hydrogen could be formed in plasma oscillation, which is a coherent oscillation of electrons relative to relatively stable nucleons.We make an ansatz that if the electron plasma frequency reaches values required by the De Broglie model, the small hydrogen could be formed.To reach a plasma frequency of fe ∼ 10 22 Hz, the required electron plasma density is ne ∼ 10 35 /cm 3 .We calculate the oscillation frequency as: where e is the electric charge, ne is the electron density, ϵ0 is the permittivity of vacuum, and me is the electron mass.Table 4 shows examples of plasma parameters for various plasma densities.It seems presently impossible to reach high enough density in typical lab conditions on the Earth to create the small hydrogen, the laser fusion being the highest, but still not enough.It is, however, possible in a collapse of very large stars capable of producing neutron stars.When the pressure in the core of a star becomes high enough after the collapse, it is energetically favorable for electrons to fuse together with protons to form neutrons p + e − → n + νe, and a neutron star is born.To make it energetically possible, one must supply an external energy of at least 782.33 keV to electron in a form of gravitational pressure.The pressure is lower at larger radius, electrons cannot fuse with protons to form neutrons, but the small hydrogen can be formed via the oscillation mechanism.
It would be a discovery with far reaching consequences if such high frequency can be achieved within some materials under an ordinary condition on Earth, or any other simple way to create small hydrogen.

Neutron capture signal in Integral satellite
Figure 11 shows the analysis of low energy spectra, including the nuclear capture signals, by the Integral satellite, which cannot detect thermal neutrons coming from the Sun in its location.The only possible explanation is that neutron capture peaks are caused by cosmic ray proton interactions with the satellites structure, producing neutrons, which then capture and produce multi-MeV Gammas.Quoting Ref. [21], the only puzzling conclusion is this: "Thermal neutron capture is responsible for numerous and strong lines at several MeV; their unexpected presence poses a difficult challenge for our physical understanding of instrumental backgrounds and for Monte Carlo codes."The presence of the thermal small hydrogen in outer space, and its capture on nuclei, could explain these so far unexplained capture signals.
We suggest searching for the thermal small hydrogen in the outer space far away from the Sun and Earth.The satellite should have small mass in supporting structure to minimize neutron production by cosmic protons.The detector could be like the one the Integral satellite used.

Conclusion
We have used a simple iterative virial theorem to conclude that the small hydrogen may exist.To form the small hydrogen atom, electron's energy must be supplied externally, which is a process like the electron capture on proton p + e − → n + νe, which also requires an external energy, i.e., small hydrogen cannot be formed spontaneously.This explains the stability of our world.It can be formed only in high energy physics experiments, in vicinity of large black holes or collapse of large stars, which are also producing neutron stars, and perhaps, during the Big Bang.If it was produced during the Big Bang, it would decouple from hot plasma sooner than CMB time, i.e., sooner than the normal hydrogen was liberated; small hydrogen may have provided a seed to early galaxies.

Figure 1 :
Figure 1: Schematic picture of normal hydrogen.Only frequencies giving integral number of electron wavelengths are allowed in both cases.

Figure 3 :
Figure 3: Two regions of hydrogen atom stability where T kinetic = T virial , one for normal hydrogen and one for small hydrogen, for a choice of potential of V Coulomb + V (Spin.B) for n = 1.

Figure 4
Figure4shows the same conclusion using Lucha's virial stability condition of equations (3), i.e., the stability of small hydrogen occurs at r ∼ 2.84 Fermi for n = 1.

Figure 5 :
Figure 5: Two regions of hydrogen atom stability where T kinetic = T virial , one for normal hydrogen and one for small hydrogen, calculated for potential energy V eff for n=1.

Figure 6 :
Figure 6: V Coulomb , V eff and V (Spin.B) potential shapes as a function of radius close to proton.

Figure 8 :
Figure 8: Relative strengths of various potential energies considered in this

Table 4 :
Electron and proton kinetic energies needed to form small hydrogen in flight (both particles have the same β = v/c = 0.9999732): Potential Electron kinetic energy [MeV] Proton kinetic energy [GeV] Veff 69.301 128.189

Figure 11 :
Figure 11: The evidence for the thermal neutron capture signals detected by the Integral satellite [21].

Table 1 :
Basic parameters for ground level (n = 1) of normal and small hydrogen:

Table 2 :
Small Hydrogen Parameters for U = V Coulomb + V (Spin.B) potential:

Table 5 :
Densities, temperature, and plasma frequencies for different types of plasma examples: