^{1}

^{*}

^{2}

In a previous JMP article published May 2013, a comprehensive calculation was presented for all properties of a number of long-life s-state Gailitis resonances lying just above the PS( n = 2) formation threshold in a positron-Hydrogen scattering system. The six open-channel calculation was carried out by solving a set of four hundred thousand coupled linear equations. The modified Faddeev equation was used to obtain the wave-amplitude for each of the six open channels. Details can be found in reference [1] . This note presents some qualitative properties of Gailitis resonances in the scattering systems of d + tu just above the thresholds of the opening of a new channel of the muonic atoms tu(n) or du(n), n > 2 is the principal quantum number. u is a negatively charged muon, d and t are the nuclei of the two isotopes of the Hydrogen atom with one and two neutrons in the nucleus respectively. We study the possible decay channels of some of the long-life Gailitis resonances. Of particular interest is a transition directly from a Gailitis (3-body) resonance to the bound states dtu molecular ions via a radiative emission of a photon or an external auger ejection of a nearby electron. Possible experimental evidence will be presented.

Muon catalyzed fusion has had a long and exciting history since the discovery of the Vesman resonance formation mechanism [

The Vesman resonance is a six-body system. A typical process is

D_{2} is the molecule of heavy Hydrogen (deuteron atom). _{2},

This process is accomplished after the

Subsequent radiative decays lead to the ^{8}/s.

W. Beunlich et al. [^{8}/s.

This large transient fusion rate is attributed to the epithermal energy collision in Equation (1) [

The first indication of the existence of the Gailitis resonance was provided by Gailitis and Damburg in their calculation of the electron-Hydrogen scattering system [

A complete set of the properties of the resonances are calculated directly including their wave functions. As a result, it becomes possible to identify the simple physical mechanism responsible for these resonances. Namely, it is a “dynamic” Stark effect [

or

wave packet in Jacobian coordinates. The atom is initially in an excited state with principal quantum number

Initially, the wave packet has a width equal to the DeBroglie wavelength of the incoming charged particle,

Thus

The life-time of the resonances (or the wave packet) can be determined using the uncertainty principle [

The small energy width

Some useful information of the Gailitis resonance can be seen from

In principle, the calculation of ref. [

time, especially when the current interest for these systems involves

The resonant series in Equation (3) must be truncated when

For the purpose of estimating the lower bounds,

In the following, a special case of

Equation (4), gives

The wave length is in muonic Bohr radius au.

A similar table for t + du should have a difference of only ~1% from the numbers in

n | ||||
---|---|---|---|---|

2 | 5.408 (−6) | 2.702 (3) | 1.370 (−7) | 0.8861 (−12) |

10 | 5.125 (−8) | 2.776 (4) | 1.298 (−9) | 0.9351 (−10) |

20 | 6.531 (−9) | 7.779 (4) | 1.654 (−10) | 0.7345 (−9) |

30 | 1.9475 (−9) | 1.4238 (5) | 4.933 (−11) | 0.2461 (−8) |

40 | 0.8242 (−9) | 2.1890 (5) | 2.0869 (−11) | 0.5816 (−8) |

50 | 0.42278 (−9) | 3.0565 (5) | 1.0709 (−11) | 1.1334 (−8) |

Can a photon or an auger electron carry away the excess energy between a Gailitis resonance and its bound state 3-body molecular ions, such as dtu in a d + tu scattering system?

The only satisfactory answer must be the precise solution of a time dependent multichannel 3-body quantum equation that is not possible at this time. Nevertheless, it is possible to find experimental evidence using the vast volume of experimental data on muon catalyzed fusion research [

We use the perturbative formula of photon emission by a muonic atom (or one electron atom) [_{2}, is

According to reference [

We have:

For the purpose of providing a lower bound, we propose a very simple model for the dipole matrix element in Equation (7).

This is less than that of Equation (8), but is of the same order of magnitude.

This simple model can be used to provide a lower bound for the three body radiative decay.

Equation (7) must be modified for three-body bound state transitions:

Consider the case when the initial state is a time dependent Gailitis resonance. As qualitatively described in Section 2 from the behavior of the phase shift, it is clear the perturbative formula (11) cannot be applied near the singular point which can be excluded from the following average over the life time of the resonance

From the qualitative behavior of the Gailitis resonance described in Section 2 the major contribution to the integral must come near the end of life of the resonance. It is possible to provide a lower bound to this integral using the model provided by the two-body radiative decay, Equation (9). The

We are interested in the principle quantum number

thus the “size” of the initial state is of the order of the resonant wavelength, the final state is one of the bound states of the dtu molecular ions. Their “size” ranges from 3 au to 10 au (from unpublished calculations).

Consider the “size” of the final state

Equation (12) becomes

This value is comparable to that of Equation (8), thus supporting the possibility of dtu molecular formation via the Gailitis resonances.

Muon catalyzed fusion utilizing the six-body resonance Equation (1) is carried out in various mixtures of molecules of heavy Hydrogen and their nuclei, such as D_{2}, DT and t. According to cascade models [

From ^{−9} sec. It is stable against all other pro- cesses [

Sakamoto et al. [

Clearly the contribution is not significant in these experiments.

A deuteron beam of appropriate range of energies that covers the energy levels of

The advantages of this approach are:

1) The experiment can be carried out in room temperature.

2) Such low energy deuteron beam can be produced easily.

3) If it becomes necessary, precision quantum three-body.

Scattering calculations for this system is possible.

In view of the long-term needs for dependable clean nuclear energy, this simple mechanism for muon catalyzed fusion must be given a chance to be tested!