Cold fusion in a dense electron gas

HAL Id: jpa-00211061

https://hal.archives-ouvertes.fr/jpa-00211061

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cold fusion in a dense electron gas

R. Balian, J.-R. Blaizot, P. Bonche

To cite this version:

R. Balian, J.-R. Blaizot, P. Bonche. Cold fusion in a dense electron gas. Journal de Physique, 1989,

50 (17), pp.2307-2311. �10.1051/jphys:0198900500170230700�. �jpa-00211061�

2307

LE JOURNAL DE PHYSIQUE

Short Communication

Cold fusion in a dense electron gas

R. Balian, J.-R Blaizot and P. Bonche

Service de Physique Théorique (*) de Saclay, 91191 Gif-sur-Yvette cedex, ^France (Reçu ^le 2 juin 1989, accepté ^le 7 juillet 1989)

Résumé. 2014 Le facteur de pénétration de la barrière coulombienne est calcule pour ^{deux deutériums}

plongds ^dans

gaz d’électrons dense. Les densitds électroniques nécessaires pour obtenir des taux de fusion compatibles

des observations récentes de "fusion froide" sont de plusieurs ^{ordres de} grandeur

^{delà de celles} auxquelles

peut raisonnablement s’attendre.

Abstract. ²⁰¹⁴ We calculate the Coulomb penetration factor for two deuterons immersed in

dense electron gas. We find that electronic densities orders of magnitude larger than those which could be

expected in metallic palladiun

required ^{in order to} bring the cold fusion rate to

observable value.

J. Phys. ^{France 50} (1989) ^{2307-2311 1er SEPTEMBRE} 1989,

Classification

Physics ^Abstracts

25.88

The major ^factor inhibiting ^the fusion of nuclei at moderate energies ^{is the} Coulomb barrier

penetration ^{factor P}

^{e-B. For the case of s-wave} tunneling, ^{the WKB} approximation ^{leads to}

the expression

where M ^is the reduced mass of the deuteron-deuteron system (2J.L

mD), Ro ^{is the classical} turning point (V(Ro) = E), ^{R, which is} of the order of the deuteron diameter, ^{can be} safely

taken equal ^{to zero. We shall work in atomic units, and set}

^{x a o, E}

( e 2 / a a ) , ^where

au - = h2 /me 2 -- ^{0.529 Â is the Bohr} radius and e 2 /ao - ^{27.2 eV is twice the} binding energy ^{of the}

hydrogen ^atom. Equation (1) ^then gives ^{for the} pure ^Coulomb potential V (r)

e2/r :

(*) Laboratoire de l’Institut de Recherche Fondamentale du Commissariat à l’Energie Atomique.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500170230700

2308

where _me is the electron mass. We have gathered in table I several values of B corresponding

to some typical physical situations. For deuterons at room temperature, ^the Coulomb barrier is

overwhelming (P - 10-2600!) ^and still is for ordinary D2 molecules for which one expects ^{zo - 1}

[1-4]. ^{In a D2 molecule bound} by ^{a muon instead of an} electron, ^{from the} simple scaling ^me

m03BC- 200me, ^one expects ^{zo -} 1/200, and B - 13. These values for B

known to yield

observable rates for "cold fusion" in ddp-molecules [1-3]. ^One may also notice that such values of B are quite comparable ^{to those} expected ^{in hot} plasmas, ^{e.g. in stars,} where fusion of light ^nuclei

occurs preferably ^{for kinetic} energies of the order of

few tens of keV [5].

Table 1. - 7%pical Coulomb barrier penetration factors in various physical situations. In the

of molecules, ^the deuton-deuton potential ^{is not} purely Coulombic ; ^{the estimates} presented in this table

are

obtained from ^{the Coulomb} formula (2) using

input ^the expected ^order of magnitude of ^the turning point ^xo in such molecules. The numbers thus obtained

in rough agreement ^{with those}

quoted e.g. ⁱⁿ reference [3].

The recent announcements [6,7] ^of possible observations of cold fusion of deuteron embed- ded in palladium ^or titanium metallic samples raise the obvious question ^{of whether, in}

^solid

state environment, ^{and in} particular ^{in the} presence ^{of a} dense electron gas, the Coulomb barrier

penetration ^{factor can be} significantly ^{enhanced to} allow the fusion rates to reach observable val-

ues. Let us recall that the fusion rate can be written as A

AnP, ^{where A is the rate} for the fusion reaction D + D

3He +

or 3H + p (A - ^11.5

^10-16 cm3s-1), ^P is the Coulomb barrier

penetration factor and n is an average ^number density of deuterons. ’Paking n - ^6.5

¹⁰²² cm-3,

which corresponds ^to about 1 deuteron per palladium ^{atom, or to an} average distance between deuterons of 2.8 Â, ^one gets ^{A -} 107P _per sec. Th achieve a rate A - lO-23 S-1@ ^{one needs}

therefore B - 70; allowing the deuterons to be within 0.3 Â of each other

gets only ^a slightly

weaker constraint, ^{B - 76.}

The effective potential ^{of two deuterons} immersed in an electron gas may ^be obtained from the Born-Oppenheimer approximation. ^{We shall} actually ^{use here a further} approximation ^based

on the linear response theory. This allows semi-analytical estimates and is accurate enough ^for

our purpose. The calculation is standard [9] ^{and leads to the} following formula for the interaction energy ^{of two} deuterons located at a distances from each other :

where g(z) ^{is the} following ^{function :}

and y - kFao, kF being ^{the Fermi momentum} of the electron gas which is treated as

uniform gas. ^{As a reference} density ^for palladium (a c.f.c. lattice with

3.9 A), ^{we assume} that all the 18 electrons of the s-p-d band contribute to the electron gas, which will obviously ^{lead to} overestimate the screening effect. One obtains thus kF N ^3.25 Â-1, ie. y - ^{1.7. The} shape ^{of the} potential (3) ^is displayed ⁱⁿ figure 1 for several values of _y. One can see that the screened potential ^allows

Fig.1.- The screened potential calculated from equation (3). ^The

labelled by the values of the parameter y

kfao; y

⁰ corresponds ^to the Coulomb potential. ^{For y}

^{0.3, it}

^be

^{that the} potential ^becomes negative around x N 5 and starts to oscillate for larger ^values of x; ^this corresponds ^{to the}

well-known Friedel oscillations which control the behaviour of V (x) ^{at very} large

deuterons of moderate energies ^to approach ^{each other to} closer distances than the Coulomb

potential. This leads to a substantial increase of the penetration ^{factor, as can be seen in table II.}

However, ^for any reasonable values of the electronic density, acceptable ^{values for B} (ire. B 76)

are obtained only ^for large ^{values of}

(E

¹⁰ corresponding ^to energies of the order of 300 eV).

Increasing ^{the value of y} beyond y

^{3 has} only ^{a moderate effect on the} resulting ^{values of xo}

and B; ^for example, ^for

^{10-2 and y}

^{5 one finds xo} = 1.6, ^{B = 180 ; for the same}

^{= 10-2}

and _y

7, ^one gets ^{x4 N 1.4 and B}

^{166. Note that y}

⁷ corresponds ^{to an} electronic density roughly ^{two orders of} magnitude larger ^{than our reference} density ⁱⁿ palladium.

As a further illustration of the difficulty the deuterons may encounter to overcome the Cou-

2310

lomb barrier, ^we have assumed that ^some yet ^unknown collective effect has succeeded in bringing together ^{the two deuterons at a distance xo, smaller than the classical} turning point ^{This lead}

us to calculate the integral (1) ^{for the screened} potential (3) ^{as a} function of _xo chosen to be an

arbitrary parameter. ^{For f}

^{0 and y = 1, one finds B} > 76 unless xo 0.15 ^which corresponds ^to

a distance of 0.1 ^Â.

1’able IL- Some values for ^the exponent ^{B in} the penetration factor, calculated from the potential (3) for ^{various values} of

^{and y. The numbers} corresponding ^{to y =1 and}

⁼¹⁰ coincide with that of

pure Coulomb potential

Finally ^{we note that the} potential (3) ^{is, in the} region ^of interest, fairly ^well approximated by

the familiar Thomas-Fermi expression :

VTF(z) ^{can be} deduced from V(x), equation (3), by setting g(z)

g(O)

^{1 in the} denominator.

Furthermore, ^{we can} underestimate B by noting ^that

With this potential, ^the penetration ^{factor is} easily obtained from equation (2) by replacing

by

f

+ _li. Thus to get B - 76, ^{one needs}

+ 03BC ~ 6 ; ^{for slow} deuterons, ^this implies p - ^{6, which} corresponds ^{to a} screening length ^{à 0.2 and y - 30, that is an electron} density 6000 times larger

than our reference density.

Note that the inequality (6) ^is reminiscent of the exact lower bound fçund in reference [9] ^for

the Born-Oppenheimer potential ^{between two deuterons} embedded in ^a solid. In the latter bound,

03BC denotes the average binding energy ^{of a helium atom in} this solid. Although ^we have evaluated the potential in the linear response approximation ^{within a} uniform electron gas, ^{our numerical} estimates

are compatible with those of reference [9].

In conclusion, ^our rough approach precludes ^the possibility ^of getting ^a sufhcient enhance- ment of cold fusion rate from static screening by electrons. Indeed, this would require ^{either a} screening length of the order of 0.1 À, _meaning ^a local electron density ^{at least 103 times} larger

than the average, ^or

^{deuteron kinetic} energy ^{of at} least 100 eV. To be efhcient, _any dynamical

mechanism would have to bring ^deuteron pairs ^{as close as 0.1 Â} during ^{a time} long enough ^to

accommodate for tunnelling.

References

[1] ^J.D. Jackson, Phys. ^{Rev 106} (1957) ³³⁰

[2] Ya. B. Zel’dovich and S.S. Gershtein, ^Sov. Phys. Uspekhi ³ (1961) ⁵⁹³ [3] ^{C.D. Vm} Siclen and S.E. Jones, ^J. Phys. ^G12 (1986) ²¹³

[4] ^S.E. Koonin and M. Nauenberg, Santa Barbara ITP preprint, April 1989, ^{submitted to Nature} [5] ^D.D. Clayton, Principles of Stellar Evolution and Nucleosynthesis, (Mc Graw-Hill, New-York, 1968),

ch. 4

[6] ^M. Fleischmann, S. Pons and M. Hawkins, J. Electroanal. Chem. 261 (1989) ³⁰¹

[7] ^{S.E. Jones, E.P. Palmer, J.B. Czirr, D.L Decker, G.L Jensen, J.M. Thorne, S.F.} Taylor ^{and J.} Rafelski, University ^of Arizona, preprint AZPH-TH/89-18, ^March 1989, ^{submitted to Nature}

[8] A.L. Fetter and J.D. Walecka, Quantum Theory ^of Many ^Particle Systems, (Mc Graw-Hill, New-York, 1971) chap. 5

[9] ^A.J. Legget ^{and G.} Baym, University of Illinois preprint, April 1989, ^{submitted to Nature.}