Posted on 04/26/2013 10:02:25 PM PDT by Kevmo
Central and Tensor Contributions to the Phonon-exchange Matrix Element for the D2/4He Transition
Peter L. Hagelstein
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Irfan U. Chaudhary
Department of Computer Science and Engineering, University of Engineering and Technology. Lahore, Pakistan
Abstract
The biggest theoretical problem associated with excess heat in the Fleischmann–Pons experiment in our view has been the absence
of energetic particles in amounts commensurate with the energy produced. In response we have pursued models in which the large
nuclear energy quantum is fractionated into a great many lower energy quanta. To connect these idealized models to the physical
system we need to evaluate the associated coupling matrix elements. Recently we have found a new coupling mechanism that arises
when a lattice model is derived starting from a Dirac description of individual nucleons; this coupling mechanism can be considered a
generalization of spin-orbit coupling and produces interactions between the center of mass dynamics and internal nuclear degrees of
freedom. In this work we develop a simplified model for 4He and molecular D2 states with which we evaluate the phonon exchange
matrix element for D2/4He transitions based on the new interaction. We restrict our calculation to the central and tensor contributions
of the Hamada–Johnston nucleon-nucleon potential, which are the strongest, and find coupling between ground state 4He and triplet
P and F molecular states. This interaction matrix element can be used in generalized lossy spin–boson models for the calculation of
excess heat production in the Fleischmann–Pons experiment .
Summary and discussion
We have computed central and tensor interaction contributions to the phonon exchange matrix elements for D2/4He transitions based
on the new a • cP coupling between vibrations and internal nuclear degrees of freedom described in [12], using simplified nuclear
models in connection with the Hamada–Johnston nucleon–nucleon potential. We find nonzero coupling to the molecular 3P and 3F
states, with the largest interaction in the case of 3P; the interaction Hamiltonian for both the central and tensor interactions together
in this model for z-directed motion and/or vector potential is
----snip--------------
where “Hc.” indicates the Hermitian conjugate, for z-directed vibrations. We have augmented the center of mass momentum from
the text with the vector potential following the discussion of Section 5. The coupling in the case of the molecular 3F states is more
Table 10. Dimensionless spatial integrals for the tensor potential contribution to the interaction
matrix element for S = 1, MS = 1, l = 3 and m = −1; the matrix elements for the
other case with MS = −1 and m = 1 differ only by a sign .
Integral eS eT oS oT
----snip--------------
than two orders of magnitude smaller, and there is no coupling to the singlet and quintet molecular states due to central and tensor
contributions. The phase factor i that appears is due to the definition of the triplet molecular relative wavefunction as real; other
conventions are possible, and subsequent calculations that might involve this interaction potential are not impacted by this phase
convention .
It is possible to understand this result in connection with a relative volume argument that we have used previously [22]. Not
only do the two deuterons need to tunnel through the Coulomb barrier in order to interact, but they also need to localize from the
molecular scale to the nuclear scale. As a result, we can think of the interaction Hamiltonian for a specific transition as
----snip--------------
Since the ˆa operator of the a • (cP) interaction is a velocity operator normalized to the speed of light [11], the associated transition
matrix element (which has magnitude |(• • • )|) can be no larger than unity. The magnitude of the transition matrix element in the case
of coupling to the deuteron was estimated to be about 0.003(cP). We might have expected a volume corrected interaction matrix
element calculated in this paper to have a similar magnitude (of 0.003); however, the result that we obtained is somewhat larger
(due primarily to the effect of the relative deuteron–deuteron potential). Our basic conclusion at this point is that the magnitude
of the interaction Hamiltonian calculated in this paper seems to us to be reasonable given the previous calculation for the deuteron
transition .
We have focused in this work on the central and tensor contributions to the interaction, since these are the strongest. Our attention
might reasonably have been focused on the spin–orbit contribution, which would produce different selection rules (including singlet
and quintet coupling, with allowed coupling to the l = 0 rotational state). Such a project is of interest, but there are complications .
The a •cP coupling that we are interested in is closely related to spin–orbit coupling, so it will be necessary to examine the derivation
of the new interaction specifically for the spin–orbit interaction (which would involve going to higher order than was done in [11]) .
In the case of the Hamada–Johnston model, the spin–orbit coupling model is not derived as a normal spin–orbit coupling based on
the central and tensor interactions, but is itself independent and empirical. As such, one wonders whether such a model is appropriate
for an a • cP interaction. Nevertheless we have carried out some exploratory computations for the spin–orbit contribution for the
singlet l = 0 case, and the relevant spatial integrals appear to be much smaller than for the central and tensor cases .
The advantage of using simplified wavefunctions and the Hamada–Johnston potential in the calculations presented in this paper
is primarily that we are able to carry out a reasonable first pass at the largest contribution to the interaction matrix element without too
much effort. It is certainly possible to do a better job, and it seems worthwhile to comment on some of the issues that seem important
in the calculation. First and foremost seems to be the deuteron–deuteron interaction potential, since the magnitude of the probability
amplitude at the fermi scale is very sensitive to this potential (for our calculation we have relied on empiricalWoods–Saxon potentials
optimized to match experimental phase shifts). The use of simplified 4He and deuteron wavefunctions with no D-state admixture
is expected to produce errors in the matrix element perhaps at the 50% level, based on our experience with the coupling matrix
element calculation for the deuteron. The simplicity of the assumed product wavefunctions for both the initial and final states will
lead to errors. In addition we expect errors associated with the use of the Hamada–Johnston potential (although these are likely
small compared to those already mentioned) .
It is possible to do a better job in all areas. For example, impressive results have been obtained in recent years with nuclear
calculations based on chiral effective field theory [23]. There are by now many modern calculations of 4He, such as described in
[24–26]. There is a growing literature that make use of modern potentials and methods for four-nucleon scattering and reaction
calculations. Groups that work in this area would have little difficulty in
Sorry I asked . . .
“Central and Tensor Contributions to the Phonon-exchange Matrix Element for the D2/4He Transition”
http://www.moviesoundclips.net/download.php?id=2852&ft=mp3
Well, I’m glad we cleared that up. Now I can sleep at night.
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