Enhanced low energy fusion rate in palladium (Pd) due to vibrational deuteron dipole-dipole interactions and associated resonant tunneling that over-cancels the Jastrow factor between deuteron pair wavefunctions
aa r X i v : . [ c ond - m a t . o t h e r] N ov Enhanced low energy fusion rate in palladium(Pd) due to vibrational deuteron dipole-dipoleinteractions and associated resonant tunnelingthat over-cancels the Coulomb barrierbetween neighbouring interstitial deuteronicpair wavefunctions
J.S.Brown
Clarendon Laboratory, 1 Parks Rd, Oxford, UK
Abstract
It is observed that interstitial hydrogen nucleii on a metallic lattice are stronglycoupled to their near neighbours by the unscreened electromagnetic field mediatingtransitions between low-lying states. It is shown that the dominant interaction is ofdipole-dipole character. By means of numerical calculations based upon publisheddata, it is then shown that in stoichiometric PdD, in which essentially all interstitialsites are occupied by a deuteron, certain specific superpositions of many-site productstates exist that are lower in energy than the single-site ground state, suggesting theexistence of a new low temperature phase. Finally, the modified behaviour of thetwo-particle wavefunction at small separations is investigated and prelimary resultssuggesting a radical narrowing of the effective Coulomb barrier are presented.
Key words:
RDDI, phase transition, protons, deuterons, metal, interference,entanglement, fusion
PACS:
Several metallic elements, notably palladium, vanadium, niobium and nickel,can reversibly absorb hydrogen up to the point of stoichiometry, in which everyavailable interstitial site - of octahedral (O) or tetrahedral (T) symmetry - isoccupied by a hydrogen nucleus. A single hydrogen nucleus in such an environ-ment exhibits a spectrum of singlet, doublet and triplet state representationsof the local point symmetry group. The ground state is invariably a singlet
Preprint submitted to Elsevier 23 November 2018 ith even [+ + +] parity along each of the symmetry axes. The next level is,in an fcc lattice, a triplet of states with parities [ − + +], [+ − +], [+ + − ] andan excitation energy of the order of 60 meV. The dipole moment between theground state singlet and the first excited triplet is typically of the order of0 . Ae . Since the electronic Fermi gas couples weakly to the electromagneticfield quanta in this part of the spectrum, such a dipole moment gives rise toan essentially unscreened resonant dipole-dipole interaction (RDDI) betweennearest neighbours [1]. Simple geometrical considerations reveal this to be ofthe order of 20 meV per pair. Not only is this typically several times largerthan screened static Coulomb interaction, it is also manifestly an appreciablefraction of the (on-diagonal) excitation energy of the dipole itself. Since in thequasi-stoichiometric loading regime each hydrogen has several nearest neigh-bours, it has previously been speculated [2] that there exist many-site statesfor which the total collective effect of the interaction is a multiple of the pairinteraction. This paper sets out to answer the, in our opinion, intriguing ques-tion as to whether there exists any such collective state of quantum-entangleddipoles whose total energy is lower than the simple product of ground states,and to obtain an upper bound on the lowest possible energy of such an en-semble of coupled oscillators. The single particle states ψ n are the solutions of H ( r ) ψ n ( r ) = " − ¯ h M H ∇ + V ( r ) ψ n ( r ) = ǫ n ψ n ( r ) (1)–where V(r) is the periodic potential experienced by an infinitely heavy posi-tive charge with fixed metal core positions. The reader is directed to [3] for adetailed discussion of the derivation of this potential using the DFT procedure.In view of the relatively large mass M H of hydrogen nucleii, the lowest en-ergy solutions will generally be well-localised about local minima in V. Theseminima will coincide with sites of octahedral or tetrahedral symmetry in cu-bic lattices and hence the levels ǫ n are an assortment of singlets, doubletsand triplet representations of the cubic point symmetry groups. The static(zero frequency) components of the potential disturbance due to the hydrogennucleus is subject to a screening law of the approximate (Thomas-Fermi) form V HH ( r ) = e e − Kr r (2)2here K is proportional to the DOS at the Fermi surface.K is typically much greater than a reciprocal lattice vector. The static CoulombH-H interaction between nearest neighbours is consequently small, typicallynot more than a few meV, and essentially state-independent. By contrast, theattenuation of the electromagnetic field due to a transition between levels isnegligible over the dimensions of a lattice cell.Since the interparticle interaction is so strongly frequency dependent, the fullHamiltonian cannot be written in closed analytical form. However, matrixelements between pairs of two-site states are simply given by: H i ,j ; i ,j = e h j , j | e − K | r − r − R | δ ( ǫ i − ǫ j ) δ ( ǫ i − ǫ j ) | r − r − R | | i , i i (3)where R is the intersite displacement vector. The δ ( ǫ i − ǫ f ) factors express thefact that only transitions between different levels give rise to an unscreenedinteraction.If just the lowest order term in the multipole expansion of the Coulomb op-erator | r − r − R | is retained, there is no need for double integration over bothvolumes. In this approximation, (3) reduces to the familiar expression for adipole-dipole interaction: H i ,j ; i ,j ≈ e R h j | r | i i · h j | r | i i − e R h j | R · r | i ih j | R · r | i i (4)If a classical dipole is located at every interstitial O-site in an fcc lattice, theinteraction energy is lowest with the following orientations over a constant-zplaquette: ← ◦ → ◦ ← ◦ → ◦ ←◦ ↑ ◦ ↓ ◦ ↑ ◦ ↓ ◦→ ◦ ← ◦ → ◦ ← ◦ →◦ ↓ ◦ ↑ ◦ ↓ ◦ ↑ ◦← ◦ → ◦ ← ◦ → ◦ ← (5)- where the open circles represent the sites of the metal cores at locations[ ], [ ],[ ] etc.It can be shown that there is zero net interaction between parallel layers wheneach layer has such an arrangement. Guided by this classical analogue, we willlimit our search for minimum energy states to those constructed from:[+ + +] parity states at all O-sites in a z = 0 plaquette[ − + +] parity states at O-sites of even y and3+ − +] states at O-sites of odd y .[+ + +] ground states at all sites external to the plaquetteFor the rest of this paper we will use the shorthand | s, n i to denote the n thstate of [+ + +] parity, | p x , n i to denote the n th state of [ − + +] parity and | p y , n i to denote the n th state of [+ − +] parity. The singlet ground state isaccordingly written as | s, i . If the site location needs to be made explicit, wewill append this in bold type thus: | s, , i . For clarification, we reproducebelow an example of a pair of five-O-site states that are degenerate in zeroethorder and that are linked by the dipole-dipole interaction of (4): | s, i ◦ | s, i◦ | p y , i ◦| s, i ◦ | s, i , | s, i ◦ | s, i◦ | s, i ◦| p x , i ◦ | s, i or equivalently: | s, , i ⊗ | s, , i ⊗ | p y , , i ⊗ | s, , i ⊗ | s, , i , | p x , , i ⊗ | s, , i ⊗ | s, , i ⊗ | s, , i ⊗ | s, , i (6)The Hamiltonian matrix in the subspace of the two-O-site states | s, , i ⊗ | p y , , i and | p x , , i ⊗ | s, , i is, according to (4): H = ǫ s, + ǫ p, − √ e d a − √ e d a ǫ s, + ǫ p, (7)where the dipole length d ≡ h s, | x | p x , i = h s, | y | p y , i The energy eigenvalues are in this case simply ǫ s, + ǫ p, ± √ e d a More generally, and assuming full hydrogen occupancy and perfect latticesymmetry, the full many-site Hamiltonian matrix depends numerically uponjust the lattice parameter a , the dipole lengths d and the energies ǫ of thesingle-site states. Quasi-stoichiometric PdH and PdD are natural candidates for our model be-cause the adiabatic effective potential experienced by the hydrogen nucleushas been determined from ab initio
DFT calculations [3,4] and the theoretical4pectra found to agree well with IR spectroscopic measurements over a widerange of substoichiometric loading ratios. For this work we checked the resultspublished in [3] by solving (1) on a real-space wedge mesh of pitch 0.03 ˚A, us-ing the published effective adiabatic Pd-H potential and boundary conditionsappropriate to the desired parity. The lattice parameter, corresponding to adisplacement like [ ] in our notation, is 4 . A . The lowest few eigenvectorsof the very large sparse matrix were solved using the dominant-diagonal iter-ation method, as in [5].The lowest single-site energy levels, relative to the O-site potential, were foundto be:level PdH (meV) PdD (meV) ǫ s,
82 52 ǫ p,
151 95 ǫ s,
233 149 ǫ p,
289 186- with the following dipole lengths along each of the three cartesian axes:dipole PdH (˚A) PdD (˚A) h s, | x | p x , i h s, | x | p x , i h s, | x | p x , i -0.003 -0.002 h s, | x | p x , i i ,j ; i ,j PdH (meV) PdD (meV) h s, , | ⊗ h p x , , | H | p y , , i ⊗ | s, , i -27 -21 h s, , | ⊗ h p x , , | H | p y , , i ⊗ | s, , i -20 -15 h s, , | ⊗ h p x , , | H | p y , , i ⊗ | s, , i -15 -11 h s, , | ⊗ h p x , , | H | p y , , i ⊗ | s, , i -30 -23 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -13 -10 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -9 -7 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -7 -5 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -14 -11 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -1 -1 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -1 -1 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i -1 -1 h s, , | ⊗ h p x , , | H | p x , , i ⊗ | s, , i As an illustrative example, we will calculate the lowest 5-site energy achievablefor a given number of | p, i states. There are just five 5-site states having 1 | p, i and 4 | s, i states, namely: | s, , i ⊗ | s, , i ⊗ | s, , i ⊗ | s, , i ⊗ | p x , , i| s, , i ⊗ | s, , i ⊗ | s, , i ⊗ | p x , , i ⊗ | s, , i| s, , i ⊗ | s, , i ⊗ | p y , , i ⊗ | s, , i ⊗ | s, , i| s, , i ⊗ | p x , , i ⊗ | s, , i ⊗ | s, , i ⊗ | s, , i| p x , , i ⊗ | s, , i ⊗ | s, , i ⊗ | s, , i ⊗ | s, , i (8)6he corresponding Hamiltonian matrix is (in meV): H =
43 5 − − −
15 43 21 − − −
21 21 43 21 − − − − − −
21 5 43 + 5 ǫ s, I (9)- where we have separated out the energy of the conventional ground state.The energy eigenvalues in this small subspace are 3,29,37,59 and 88 meVrelative to 5 ǫ s, . For the 80 five-site states that comprise just one | p, i andfour | s, n < i states, the lowest two energy eigenvalues are found to be -2and 26 meV relative to 5 ǫ s, . It is clear from this that the | s, i states make asignificant contribution to the lowest energy eigenvector.In an attempt to find a converged value for the absolute mimimum site energy,Hamiltonian matrices were constructed for a series of plaquettes of increasingsize up to a limit set by the memory capacity of our machine. The 15 sitesincluded were: [000], [020], [110], [200], [220], [130], [310], [330], [420],[1-10], [-110], [240], [130], [040], [150] , in that order.The results obtained for the energies ( E , E ) of the lowest two states relativeto N ǫ s, are summarized in the following table. For the larger plaquettes, anenergy cut-off was applied in order to limit the size of the matrix.7ites p -states Cut-off (meV) States E (meV) E (meV) E / Site (meV)5 1 - 80 -2 26 05 2 - 80 31 43 66 1 - 192 -4 19 -16 2 - 240 12 37 28 1 - 1024 -15 12 -28 2 - 1792 -9 12 -19 1 - 2304 -16 9 -29 2 300 1044 -14 5 -212 1 300 2784 -21 -7 -212 2 300 3696 -33 -9 -312 3 300 2200 -34 -12 -312 4 300 4455 -24 -8 -215 2 300 9660 -40 -23 -315 3 300 5915 -44 -25 -315 4 250 1365 -26 -8 -2 The intrinsic complexity of this exact method and the inapplicablity of a per-turbative approach have so far confounded our attempts to establish a lowerbound on the absolute minimum site energy. It follows from the variationalprinciple that inclusion of higher | s, n i states, as well as further increase in pla-quette size, will result in even lower minimum energies. A mean-field approachis perhaps indicated, but we have as yet to find a sufficiently accurate formula-tion. It is nevertheless already clear from the above data that entangled statesare favoured in the stoichiometric regime. The existence of a low temperaturephase in which all the deuterons cohere in a mesoscopically entangled state ishence strongly indicated. 8 Over-cancellation of Coulomb barrier
At small interparticle distances :- K | r − r | <<
1, the off-diagonal elements ofthe type we have been considering are comparable in magnitude, but oppositein sign, to the static Coulomb pair repulsion term. It is hence reasonable sosuppose that, once coherence has been established, the height and width ofthe effective Coulomb barrier between neighbouring s, p state pairs is reduced,with a concommitant increase in the - normally infinitesimally slow - D-Dfusion rate. In order to investigate the magnitude of this effect, we solvedthe two-particle Hamiltonian for the two states discussed in connection with(7) above. Both the static (2) and dynamic (3) interactions were included.Memory constraints limited us to a grid resolution of 0 . A . In view of (6)and (7), the solution was constrained to be of the form: ψ ( x , y , z , x − a , y − a , z ) − ψ ( x − a , y − a , z , y , x , z ) (10)where ψ is odd in its 5th argument and even in all others.It was found that the lowest energy solution, with ǫ ≈ ǫ s, + ǫ p, − √ e d a (11)- exhibited an increased probability for close encounters of the two hydrogennucleii right down to the limit of our resolution. At | r − r | = 0 . A , the am-plitude was enhanced by about an order of magnitude over the simple productstate that pertains when interaction is neglected. This exciting result impliesthat the dipole-dipole attraction effectively over-cancels the Coulomb repul-sion at least down to this length scale. The region of overlap was concentratedabout the T-site lattice potential minima that are equidistant between thetwo O-sites. A search is currently being undertaken for other metallic lattices with highaffinity for hydrogen and flat effective potentials. A multi-level grid DFT al-gorithm of high accuracy has been developed for this purpose.9 eferences [1] G. Kurizki, A. Kofman, V.Yudson, Phys. Rev.
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R35–R38 (1996).[2] J.Brown, arxiv.org/abs/cond-mat/0608292[3] H.Krimmel, L. Schimmele, C. Els¨asser, M. F¨ahnle, J.Phys. Condens. Matt. , 1711–1715 (2005).[5] M.Puska, R.Nieminen, Phys. Rev. B29 , 5382–5397 (1984)., 5382–5397 (1984).