# Nature’s Pick-Up Tool, the Stark Effect Induced Gailitis Resonances and Applications

^{1}

^{2}

^{*}

## Abstract

**:**

^{7}Li low energy nuclear fusion, d-d fusion on a Pd lattice and the initial transient fusion peak in muon catalyzed fusion. Hopefully, these examples will help to identify Gailitis resonances in other systems.

## 1. Uncovering the Truth from the Nature Takes Time

_{0}

^{2}was found, where a

_{0}is the Bohr radius. The S-partial wave portion of reaction (1) is plotted in Figure 1 in the energy range between the Ps(n = 2) threshold to the $\overline{H}$(n = 3) threshold.

_{ii}), i = 5,6, are presented in Figure 2 below.

_{max}≈ 1000a

_{0}used in our calculations is too short, the third resonance get cut in half. In spite of such defects, these graphs provide enough information to reveal real physics.

_{ii}), i = 5,6, as a function of the energy E

_{1}, the collision energy with respect to channel 1. Other channel energies are determined in term of E

_{1}. For example the channel energy for channel ε

_{5}and ε

_{6}are measured from Ps(n = 2), while E

_{1}measured from H(n = 1).

## 2. Physics Revealed from Figure 2a

^{−}to −π/2. That means that the attractive electric dipole field from Ps(n = 2) suddenly turned strongly repulsive when the energy of the proton matched the electric dipole flipping energies, thus forcing the proton to give up all its energy, and it then turns into an expanding wave packet centered on y

_{m}, where m is a quantum number. From Figure 2b–d, y

_{1}, y

_{2}can be measured directly, but not y

_{3}. The proton stripped off its energy and turns onto an expanding wave packet. That represents the first of two stages of the lifetime of the Gailitis resonance. The second stage begins when the phase shift suddenly drops from π/2 to 0

^{+}, indicating a strong attractive force from the target. What will happen during the second stage, which depends only on the host system, is here the Ps. It produces the resonant reaction represented by p + Ps(n = 2) → e

^{+}+ H(n ≤ 2). Thus, Figure 2a reveals the pick-up action of the Gailitis resonances.

## 3. Other Common Property of the Gailitis Resonances

^{2}= (1/λ)

^{2}. Then, the uncertainty principle is given by Δy Δp = 1. Applied to the Gailitis resonance when Δp→p, we find Δy ≈ 1/p = λ and Δε = (Δp)

^{2}= p

^{2}.

_{m}/Δε

_{m}= 1 for all m. Due to the incoherent use of units in some of our previous calculations [5], this formula was listed incorrectly as ε

_{m}/Δε

_{m}= 4π

^{2}.

_{m}= (1/Δε

_{m}) × 2.42 × 10

^{−17}sec

_{m}is given in Rydberg.

## 4. Resonant Condition Read from Faddeev Amplitudes

_{1}| of the target Ps(n = 2) in the mass normalized Jacobi coordinate system in Figure 2b–d to the resonance energies. The subscript 1 represents the electric dipole moment only, where y

^{2}

_{m}= <y

^{2}

_{m}> are directly measured at the peaks of the wave packets in Figure 2c,d. However, due to the energy we used to calculate, 2d is too far from the resonant energy. There are two peaks in the wave packet. It is easy to identify which one produced the m = 1 resonance (see [3,4]). Numerically, the resonant equation below comes directly from the wave functions 2c and 2d

_{m}= m|μ

_{1}|/y

^{2}

_{m}, where m = 1, 2.

_{3}cannot be measured from Figure 2b with our limited computer resources. Using a numerical extension of Figure 2b beyond 1000a

_{0}, along with help from Equation (4), y

_{m}, for m = 3 was determined. The properties of the three resonances are listed in Table 1. All quantities are in mass normalized Jacobi coordinates and |μ

_{1}| = 47.94778.

_{m}resonant energy measured just above the Ps(n = 2) threshold is very small. They are within the range of the fine structure energy width, where the Coulomb degeneracy is removed by the small relativistic perturbation in the pure Coulomb force Hamiltonian that split the Coulomb energy level into a number of independent energy levels depends on the angular momentum quantum number ℓ. The energy width of this fine structure energy levels is very small. All ε

_{m}must lie within this width, thus three-body scattering calculation involves 6-open channels represented in Equation (2).

^{+}will induce a constant electric dipole moment in the target atom. Consequently, an attractive inverse square force exists that support Gailitis resonance listed in Table 1.

_{m}= m|μ

_{1}|/y

^{2}

_{m}, m = 1, 2, 3.→

_{m}/Δε

_{m}= 1 indicates that, in the traditional measurements of energy and width of resonances, in the complex energy plane, this ratio must be close to one and all Gailitis resonances lying on this straight line [5]. The present method provides a complete set of properties for all the Gailitis resonances outlined above. These properties are very useful in the search for Gailitis resonances for more complex systems. Such searches have already solved a number of decades old outstanding problems. A few of them are outlined below.

## 5. Lifetime Rate of Gailitis Resonance in Muon-Catalyzed Fusion

^{4}He + n + μ + 17.6 MeV.

## 6. Long-Lived Gailitis Resonance Composed of the Electron-Rydberg Hydrogen Atom

_{m}and λ

_{m}even for the n = 2 system found in Table 1. For Rydberg Gailitis resonances, the lifetime can be expected to exceed its radiation decay lifetime. That seemed to be the case for the earlier stage of research in antihydrogen production when the antihydrogen was created in Rydberg states. The expectation for such states is to radiatively cascade down to low excitation states failed.

## 7. Nuclear Fusion in (p+^{7}Li) -> ^{8}Be^{*} -> 2α

^{7}Li reaction overlaps with that of the compound nuclear energy level centered at 19.9 MeV, with a decay width that extends even below the proton separation energy of

^{8}Be (see the arrow in Figure 4).

^{+}, the

^{7}Li has I

_{1}= 3/2

^{−}and the proton has I

_{2}= ½, with l = 1

^{−}contribution.

^{+}, that makes a perfect match between the resonance and this compound nuclear energy level Be* (shown by the arrow).

## 8. S-State Gailitis Resonance (d, Dp)—The “Quasi-Particle”

^{+}+ H calculation, it is expected that the activity of the resonance is not strong enough to disturb the lattice. In this section, we assume that the deuteron atom is bound on Pd lattice.

_{p}), Figure 5 shows that the wave packet of d is a spherical layer of probability density with maximum located on a spherical surface with resonant radius r

_{0}. A deuterium atom on the lattice is represented by D

_{p}, and it has an electric dipole moment induced by the lattice vibrating phonon, indicating that the electric dipole can be tuned to locate the resonances. Tuning, for example, can be performed using new laser technology to induce a phonon state.

_{0}remains unchanged during the first stage of the life of the resonance, but the probability density expands both inward and outward from this surface.

_{p}, the spherical surface with maximum probability density begins to shrink.

_{p}remains on the lattice, unperturbed by the activities of the resonance. As usual, the resonant energies of Gailitis are very small for the long-lived resonances. When the probability cloud begins advancing over the Coulomb barrier, no matter how little, the quasi-particle (d, D

_{p}) is already in a shallow negative attractive tail of the central nuclear potential, far away from the complication of the core nuclear forces.

_{p}) fusion.

_{p}) have in common with

^{4}He

^{+}? They both have one electron and two deuterons d. The

^{4}He

^{+}is the lowest possible energy system involving these three particles. The most important difference between these two systems concerning this study is the energy difference. Neglecting all small energies involving the lattice and the Gailitis resonance the (d, D

_{p}) is 23.85 MeV above

^{4}He

^{+}. This is the energy needed to separate the two d from the α-particle, the nucleus of

^{4}He

^{+}. Unfortunately, there is no compound nuclear energy level of α to match the nuclear properties of an S-state (d, D

_{p}). Instead of becoming a compound nuclear energy level of α at excitation energy 23.85 MeV (see Figure 6), during the second stage of the life of the (d, D

_{p}), resonant action leads the quasi-particle (d, D

_{p}) into a shallow negative tail of the nuclear potential of D

_{p}, which, in an attempt to expel the intruder, injects it with energy equal to the potential energy drop, as the cloud keeps shrinking.

_{p}) reaches the energy of a lattice normal mode phonon, the nuclear energy begins to flow into the lattice, one phonon at a time, until the size of the “quasi-particle” shrinks close to the region where the core nuclear force dominate. Then, the nuclear force takes over the dynamics. The (d, D

_{p}) either reaches the ground state of

^{4}He

^{+}and still remains on the lattice, or gets kicked out of the lattice as a quasi-particle where background electrons can slow it down, until reducing it to a free

^{4}He

^{+}. This leakage of one phonon at a time is a slow process and the amount leaked each time, one phonon, is negligible for nuclear energy. Namely, only a classical conservation of energy need apply.

## 9. Discussion

_{0}and y = 1306 a

_{0}. Any computer would be hard pressed to accommodate such a large calculation correctly as seen even in some most recent calculations. In other words, the long range Coulomb force is producing unexpectedly long range physics. Present computational methods needs substantial improvements.

^{+}+ H(n ≤ 2), presented earlier in this report. Table 1 shows that the first Gailitis resonance shows up at y

_{1}= 296.8a

_{0}. Had we used an effective cut-off, y

_{max}= 300a

_{0}, all the physics of the Gailitis resonances would have remained hidden.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References and Notes

- Gailitis, M.; Damburg, R. The Influence of Close Coupling on the Threshold Behavior of Cross Sections of Electron-Hydrogen Scattering. Proc. Phys. Soc.
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**2014**, 5, 18. [Google Scholar] [CrossRef] - Figure 4 was taken from the Evans: The Atomic Nucleus; McGraw-Hill Inc.: New York NY, USA, 1955; p. 203.

**Figure 1.**Total S-state antihydrogen formation cross section. Taken from reference [2]. The relatively large cutoff radius of 450 a

_{0}used enabled two Gailitis resonances appear near the threshold of $Ps\left(n=2\right)$.

**Figure 4.**Some of the known energy levels of Be

^{8}and reactions involved in their formation and dissociation, see reference [8].

**Figure 5.**Cross section view of the wave-packet: an expanding spherical layer of probability density of the approaching charged particle.

m | ε_{m} (Ryd) | ε_{m}/Δε_{m} | λ_{m}(a_{0}) | y_{m}(a_{0}) |
---|---|---|---|---|

1 | 5.4436(−4) | 1 | 380.85 | 296.8 |

2 | 0.19436(−3) | 1 | 637.35 | 702.4 |

3 | 0.84344(−4) | 1 | 967.54 | 1306.0 |

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**MDPI and ACS Style**

Hu, C.-Y.; Caballero, D. Nature’s Pick-Up Tool, the Stark Effect Induced Gailitis Resonances and Applications. *Atoms* **2020**, *8*, 32.
https://doi.org/10.3390/atoms8030032

**AMA Style**

Hu C-Y, Caballero D. Nature’s Pick-Up Tool, the Stark Effect Induced Gailitis Resonances and Applications. *Atoms*. 2020; 8(3):32.
https://doi.org/10.3390/atoms8030032

**Chicago/Turabian Style**

Hu, Chi-Yu, and David Caballero. 2020. "Nature’s Pick-Up Tool, the Stark Effect Induced Gailitis Resonances and Applications" *Atoms* 8, no. 3: 32.
https://doi.org/10.3390/atoms8030032