Nuclear fusion

The rate of nuclear fusion in the $d\mu t$ molecule is of the order of 10¹² s^-1, hence it is too fast to measure experimentally. The reaction is strongly dominated by the resonance ⁵He $^{J,\pi = \frac {3}{2} +}$, which lies about 55 keV above $\mu t+ d$ threshold. Because of the centrifugal barrier of the J=1 states of the molecule, fusion is expected to take place mainly from J=0 states, i.e., (J,v)=(0,1) and (0,0), into which the initial state (1,1) is quickly converted via fast E1Auger transitions.

For $d\mu d$, on the other hand, the transition from $J=1\rightarrow 0$ is strongly suppressed (since it requires spin flip, analogous to the ortho-para transitions in the homonuclear hydrogen molecule), and fusion takes place mainly from the J=1 states, if the molecule is resonantly formed in the (1,1) state. This in fact offers a unique opportunity to study the p wave fusion reaction at very low energy, which is difficult to do in a beam experiment. An interesting feature of this reaction is that the fusion width ratio of the (n+ ³He) channel to the (p + t) channel is about 1.4, apparently violating charge symmetry.

In a simple approach, the fusion rate is calculated by treating three-body Coulomb physics (molecular wave functions) and nuclear physics separately [54], a method known as factorization. For $d\mu t$, we have

$$\lambda _{d\mu t}^{f} = A \lim _{R_{dt}\rightarrow 0} \int S\vert\Psi _{d\mu t}(r,R_{dt})\vert^{2} d^{3} r,$$ (10)

where A is related to the zero energy limit of the astrophysical S factor,

$$A=\frac {\hbar}{\pi e^{2} \overline{M}_{dt}} \lim _{E\rightarrow 0} S(E),$$ (11)

with $\overline{M}_{dt}$ being the reduced mass of d and t. S is obtained from the fusion cross section with Coulomb repulsion factored out, hence it contains all the nuclear physics. Thus in this approach, the overlap of the d-t wave functions in the united-atom limit is multiplied by a single number describing the strong interaction.

More accurate approaches treat the nuclear force dynamically by directly incorporating it as a complex potential in the three-body Hamiltonian (optical potential method) [55,56], or as complex boundary conditions at the nuclear surface (R-matrix method) [57,58,59]. These elaborate approaches agree rather well with each other, as well as with the simple factorization approach.

In $\mu x +y$ collisions, fusion ``in flight'', i.e., without forming a bound $x\mu y$ molecule, is also possible, and its rates have been calculated by various methods including, most recently, Faddeev equations (see [26] and references therein). The reactions are enhanced, compared to bare nuclear collisions, especially if virtual states exist, due to the increased overlap of the nuclear wave functions.