Coupling between the lattice and internal nuclear degrees of freedom Peter Hagelstein 1 and Irfan Chaudhary 2 1 Massachusetts Institute of Technology 2 University of Engineering and Technology, Lahore

The problem Donor-receiver type model closely related to experiment Could account for excess heat in F&P experiment if we had a transition strongly coupled to the phonons Could account for Karabut experiment if we had a transition strongly coupled to the phonons Electron-nuclear coupling too weak Electron-electron coupling in atoms too weak Nuclear coupling strong enough… But no known coupling with phonons

Go back to beginning… If nuclear coupling could exchange phonons, then we could understand new experiments But we know that nuclear coupling does not exchange phonons Let’s go back and see how it works… If we understand why there is no coupling, perhaps we might be able to find some fine print

Center of mass separation Consider two-body problem with central field potential Introduce center of mass and relative variables

Clean separation Hamiltonian in center of mass and relative coordinates separates Center of mass and relative degrees of freedom are independent, with no coupling between them Center of mass Relative

Clean separation is powerful and important The separation between the center of mass problem and relative problem in nonrelativistic physics is very powerful! Means that can work relative problem in rest frame Means that can analyze center of mass dynamics independent of relative problem Separation is clean even with many bodies Our intuition about how the world works is strongly connected to this separation

Separation of condensed matter and nuclear physics Suppose that we wanted to describe the coupling between condensed matter and nuclear degrees of freedom. We might start with a model that considers neutrons, protons, and electrons at the outset on equal footing:

Separation of nuclear and condensed matter physics Since the center of mass problems and relative problems separate cleanly, the nuclear and condensed matter problems are independent in this model Nuclear problem Condensed matter problem

Two separate fields Condensed matter physics Nuclear physics

No basic coupling between condensed matter problem and internal nuclear problem in fundamental construction of the nonrelativistic theory. (But there is some weak coupling due to electron-nuclear interactions.) This basic result is a consequence of the clear separation of center of mass and relative problems in the nuclei.

Take away message Donor-receiver model consistent with excess heat and Karabut As long as can find a strongly-coupled transition Electron-nuclear coupling too weak Electron-electron coupling also too weak Only coupling strong enough is internal nuclear But internal nuclear separates cleanly Which is why condensed matter and nuclear physics are separate fields

Center of mass separation in Dirac model

Relativistic two-body problem Two-body Dirac equation: Normally people work with field theory models that are much more complicated… We choose Dirac model here since it is the simplest model that could conceivably be relevant.

Center of mass separation is problematic Problem is very old and very famous Studied since 1930s Mathematical problems associated with center of mass separation in relativistic problem No general separation in literature Mathematical no go theorem about separation

A new model for CMNS

Need for a CMNS Hamiltonian Want to include configuration mass effects on lattice dynamics Recently understood that we can work with Dirac models We can make a new Hamiltonian to describe condensed matter nuclear physics (CMNS) We hope that it clarifies what the strongly-coupled receiver transition is

Need to revisit Dirac Now assume that many states are present: Can we get an equation for the expansion coefficients?

Diagonal matrix elements

Each state now has its own kinematic mass, and we now understand the energy-momentum relation for each basis state.

Off-diagonal matrix elements Mixing within relative part of problem New term P. L. Hagelstein and I. U. Chaudhary, “Including nuclear degrees of freedom in a lattice Hamiltonian,” J. Cond. Mat. Nucl. Sci. (in press)

Unexpected (at least by me) coupling between center of mass motion and internal states

Nonrelativistic limit Nonrelativistic limit of finite basis expansion:

Now have a better description for the dynamics of a composite particle (such as a nucleus. And we have a coupling between the center of mass motion and the internal nuclear states.

New relativistic model for nucleons and electrons Start with relativistic model for nucleons and electrons

Coalesce nucleons into nuclei

A CMNS Hamiltonian

This gives us a new starting point for condensed matter nuclear physics modeling that includes coupling between vibrations and nuclear excitation

Not two separate fields Condensed matter physics Nuclear physics

Lattice-nuclear Hamiltonian Now use Born-Oppenheimer approximation to replace electrons with equivalent potentials between nuclei

Now we have a fundamental Hamiltonian for Fleischmann- Pons excess heat effect and Karabut collimated x-ray emission

If new coupling term goes away… Then we recover usual condensed matter lattice model

Take away message Wanted a model to describe mass shift for excited nuclear configurations Start with many-body Dirac model Develop finite basis approximation Get new relativistic and nonrelativistic finite basis models Use to construct new condensed matter nuclear Hamiltonian Get mass shift effect Also get new coupling term

Deuteron matrix element

New matrix element The new coupling matrix element in the case of the deuteron can be written as

Nonrelativistic approximation

Results from numerical calculations We used the Hamada-Johnston model for strong force interactions, and computed deuteron wavefunctions to evaluate the matrix element:

What does it mean? We had estimated earlier that a coupling matrix element of 1 keV would allow us to be consistent with the model numbers of the Letts 2 laser experiment excess heat The new evaluation of the coupling matrix element corresponds to a maximum of about 100 eV Result is orders of magnitude larger than for electron- nuclear or electron-electron coupling Is sufficiently large to motivate developing a new model for excess heat in Letts experiment and others

D 2 / 4 He calculation in progress We are evaluating the new interaction matrix element now for phonon exchange in connection with the D 2 / 4 He transition Calculation involves lots of terms and some significant numerical integrations. Hope to complete it in a few weeks

Significance of results Donor-receiver model requires estimates for coupling matrix elements New relativistic CMNS Hamiltonian describes coupling between lattice and internal nuclear degrees of freedom Now have receiver coupling matrix element for deuteron Will shortly have donor coupling matrix element Will allow ab initio calculation of excess heat within donor- receiver model