Resonant formation

Observations were made in Dubna in the early 1960s for the temperature dependence of fusion yields in muon catalyzed dd fusion [52]; this was a strange phenomenon at the time since nuclear reactions of the MeV scale appeared to be affected by the target temperature which is of the meV scale. An Estonian graduate student, Vesman, proposed a resonant process for muonic molecular formation to explain the observations [53]. According to the Vesman mechanism,

$$\mu x + YZ_{\nu _{i}, K_{i}} \rightarrow [(x\mu y)^{S}_{Jv}zee]_{\nu _{f},K_{f}},$$ (9)

where the kinetic energy of $\mu x$ and the energy released upon formation of a loosely bound $(x\mu y)_{Jv}$, with (J,v) the ro-vibrational quantum numbers, is absorbed in the rotational (K) and vibrational ($\nu$) excitation of the molecular complex $[(x\mu y)zee]^{*}$, a hydrogen like molecule with $x\mu y$ playing a role of one of the nuclei. Since the final states have discrete spectra corresponding to ro-vibrational levels ( $\nu _{f},K_{f})$, the process takes place only when the collision energy matches the resonant condition. The temperature dependence is derived from the varying degree of overlap between the Maxwellian distribution (assuming $\mu x$ is thermalized) of $\mu x$ atoms with the resonance energy profile at different temperatures. For the Vesman mechanism to occur, the muonic molecule has to have a level with the binding energy smaller than the dissociation energy ($\sim 4.5$ eV) of the host molecule. This is extremely loosely bound in comparison with the typical muonic energy scale of a few keV. Indeed such states have been shown to exist theoretically by Ponomarev and his co-workers for $d\mu d$ and $d\mu t$ molecules, with quantum numbers (J,v)=(1,1), where J and v denote rotational and vibrational states, respectively.

Muonic molecular ions are three-body systems interacting with both electromagnetic and strong forces, but to a good approximation ( ${\mbox{\lower 1mm \hbox{$\stackrel{\textstyle <}{\sim}$ }}} 10^{-4}$ eV), direct effects of the strong force can be neglected in calculating energy levels of loosely bound states, although other nuclear properties such as the charge form factors and polarizabilities play non-negligible roles. Now modern calculations have achieved amazing accuracies, as discussed in Section 2.1. Table 1.3 shows the nonrelativistic Coulomb bound state energy levels of muonic molecules.

  

Table 1.3: Non-relativistic Coulomb molecular binding energies - $\epsilon _{Jv}^{nr}$ (eV).

J,v	$p\mu p$	$p\mu d$	$p\mu t$	$d\mu d$	$d\mu t$	$t\mu t$
0,0	253.15	221.55	213.84	325.07	319.14	362.91
0,1				35.84	34.83	83.77
1,0	107.27	97.50	99.13	226.68	232.47	289.14
1,1				1.97	0.66	45.21
2,0				86.45	102.65	172.65
3,0						48.70

The molecular complex $[(x\mu y)dee]^{*}$ is metastable, and can decay into several channels. An Auger transition of the $x\mu y$ molecule is an important step, which stabilizes the molecule, leading to fusion. This competes with the back decay process, which returns the excited molecular complex to $\mu x$ and YZ. The details of the formation and back decay processes are discussed further in Section 2.2 and in Appendix B.