A model for enhanced fusion reaction in a solid matrix of metal deuterides

International Conference on Condensed Matter Nuclear Science. 2008 ^ | July 2008 | K P Sinha

[Kevmo]

A model for enhanced fusion reaction in a solid matrix of metal deuterides

A model for enhanced fusion reaction in a solid matrix of metal deuterides

Sinha, K.P. and A. Meulenberg. A model for enhanced fusion reaction in a solid matrix of metal deuterides. in

ICCF-14 International Conference on Condensed Matter Nuclear Science. 2008. Washington, DC.

A model for enhanced fusion reaction in a solid matrix of

metal deuterides

K. P. Sinhaa and A. Meulenbergb

a Department of Physics, IISc, Bangalore 560012, India (kpsinha@gmail.com)

b HiPi Consulting, Frederick, MD, USA (mules333@gmail.com)

Abstract

Our study shows that the cross-section for fusion improves considerably if d-d

pairs are located in linear (one-dimensional) chainlets or line defects. Such nonequilibrium

defects can exist only in a solid matrix. Further, solids harbor lattice

vibrational modes (quanta, phonons) whose longitudinal-optical modes interact

strongly with electrons and ions. One such interaction, resulting in potential

inversion, causes localization of electron pairs on deuterons. Thus, we have

attraction of D+ – D- pairs and strong screening of the nuclear repulsion due to

these local electron pairs (local charged bosons: acronym, lochons). This

attraction and strong coupling permits low-energy deuterons to approach close

enough to alter the standard equations used to define nuclear-interaction crosssections.

These altered equations not only predict that low-energy-nuclear

reactions (LENR) of D+ – D- (and H+ – H-) pairs are possible, they predict that

they are probable.

Introduction

The central idea of this paper is that the solid matrix in which LENR takes place provides

conditions such as: (1) confinement of deuterons (or deuterium atoms) in linear chains or in line

defects, (2) dynamical solid-state modes (phonons), which can store and exchange energy, and

(3) the strong interaction between appropriate phonon modes (particularly longitudinal-optical

modes) with electrons and ions. The resultant resonant D+-D- pairs in this environment permit

attractive forces and/or strongly-screened repulsive forces, rather than the normally-expected

strong-repulsive Coulomb forces between positively-charged nuclei. These options are not

available in a gaseous plasma, or in materials that lack (at least) short-range order.

It is a goal of this paper to provide an understandable, standard-physics basis (under special

conditions) for the extensive body of results presently available from LENR.1

Model

Let us consider a linear chain of deuterons surrounded by an equal number of electrons. The

Hamiltonian for such a system is 2

H = He + HL + HeL + A, (1)

where the electron contribution is

2

He =  Em C+

mCm +  tmn (C+

mCm + h.c). (2)

Here the C+

m (Cm) denote the electron creation (annihilation) operators in the Wanneir state

|m>, at site “m” with spin . Em is the onsite single-particle energy of the electron and tmn = -

|t| is the nearest-neighbor hopping integral. The lattice Hamiltonian is

HL = ħD m (d+

m dm + ½ ), (3)

where D is the vibration frequency of the deuterium atom D (taken as an Einstein oscillator) and

with d+

m (dm) denoting the phonon creation (annihilation) operators.

The interaction of electrons with the above phonon modes is described by

HeL = għD C+

mCm (d+

m + dm), (4)

where g is a dimensionless coupling constant. The last term in Equation 1, A, is a constant

negative energy due to negative space charge in the channel. Note that in a low-dimension (one

or two) structure, the potential energy between two deuterium atoms is much deeper and

negative, relative to that of atoms in a 3-D lattice.3 A suitable unitary transformation4,5 leads to a

displaced harmonic oscillator [d'

m  (dm + )] and, in the transformed total Hamiltonian, the onsite

single-electron energy E*

m = Em – Ed, with Ed = g2 ħD; the hopping integral (in Equation 2)

t*

mn = tmn exp(-g)2; (5)

and the electron effective mass2

m* = me exp[Ed /ħD] = me exp[g2]. (6)

Even for a very conservative value of g2 = 1.6, this will give m* = 5me (see below Equation 9).

Let us now consider the situation of two deuterons and two electrons in a chain. This

introduces Coulomb repulsion (Ue) between two electrons about an atom at site “m” in the same

orbital state |m>, but having opposite spin. The displacement transformation

(C+

m)* = C+

m exp [-g (dm – d+

m)], (7)

gives the effective Hamiltonian and the various parameters are obtained as

E*

m = Em - Ed; Ue

* = Ue – 2 Ed ; and t* = |t| exp [- g2] (8)

For U < 2Ed, U* becomes negative. Thus, there is potential inversion for the 2 electrons in the

singlet state and they will form a small on-site localized pair, a sort of composite boson

(lochon)2,6. Under this condition, the D- state will be more stable (has lower energy) than the

neutral atom D (though not necessarily more stable than the D+ = d state). This would lead to

the existence of D+-D- pairs. They would exist in the resonating state, D--D+  D+-D-, further

reducing their energy and inter-nuclear distance.

Strong Screening

Bound electrons reduce the effective charge of nuclei. An occasional transfer of one such

electron between two deuterium atoms forms a transient electron pair within a D+-D- pair. At

separations larger than the orbital radius of the electrons, this transfer changes a neutral

relationship to an attractive one. At separations smaller than a fraction of the orbital radius of the

electrons, it still gives a significant reduction in “effective” Coulomb repulsion between the

nuclei. This effective potential will be represented by

3

Ud* = ((e*)2/r ) = (e2/r) (1- exp [-as/L]) , (9)

where as is the strong-electron-screening length, L = (ħ/mL

* L) is the rationalized deBroglie

wavelength1 of the lochon, mL

* being the effective mass of the lochon, and L its speed (as

determined from its energy in the atomic and molecular potential wells of the two deuterons) .

This screening by lochons is a short-range effect and reduces the repulsive potential between

reacting nuclei (deuterons here). Screening by itinerant electrons is weak in this range (relative

to that of the bound electrons) and hence not considered here.7

The coupling into an optical-phonon mode, along with the attractive potential of the D+-Dpair,

briefly produces a nearly 1-D encounter that greatly increases the potential-well depth of

this short-lived “molecule.”

Penetration Factor and Cross Section

Next, we discuss the fusion reaction of a screened d-d reaction in 1-D. For an incident

particle of effective charge e*, the penetration factor P(l, Ea) decreases rapidly with its decreasing

total energy, Ea, where l is its orbital angular-momentum state. In this low-energy situation,

particles in an l = 0 state contribute most.8 We have:

  r0

R

2 2 1/2

d

1/2 2

o P(0, Ea ) (V (R)/Ea ) exp[-2 (| k - (2M / )(e*) /r) |) dr]

  r0

R

1/2

o

1/2

o (V (R)/E ) exp[-2 k (|1- (r /r) |) dr] a , (10)

where k is the wave vector of particle a, Md is the reduced mass of two deuterons, and with

Vo(R) = (e*2)/R:

ro = (e*)2 2Md/ħ2 k2, R << ro . (11)

The integral requires careful treatment since, as r  0, it has a singular term. Hence, resorting

to an asymptotic expansion (a better approximation for expression of the integral as r 0), we

get,

P(0,Ea) = (Vo(R)/Ea)1/2 exp [- (e*)2/ħ r] , (12)

where r is the relative velocity. The cross-section (a,b) of this reaction is

(a,b) = (constant / Ea) exp[-(e*)2/ħ r]

= (/k2) exp[-(e2/ħ r) (1 – exp (-as/ L))] , (13)

where k2 = (2 Md Ea / ħ) = Md

2 r

2 / ħ and Md, Ea, as , and L are as above. The critical

difference between this development9 and the prior work (standard model) is a factor of 2 in the

exponent that exists in the regular solution and is gone here (valid at least for r => R).

The key values in the present model are those calculated for the deBroglie wavelengths for the

lochon and the deuterons, as a function of d-d gap, and the value taken for as. This value is the

screening provided by the bound electrons/lochon and is given in Ichimaru7 (page 9) based on

the ion-sphere model. Normally this is ½ the sum of the atomic-orbital radii for the charge state

of the two atoms [aij = (ai + aj)/2]. In our model, aij = 0.53A for the D-D case and, since we can

1 Several of the references herein use rather than  (the deBroglie wavelength), in their equations. We follow suit.

4

ignore the radius of the bare deuteron, aij = ~ 0.3A for the D+ - D- case. So, we assume a range of

values between the initial value as = aij /2 (the Bohr radius divided by 2 since we are using L

in the equation) and that of the 1-D case (as described above and under the appropriate

circumstances), which will reduce as by up to an order of magnitude.

The value of L varies from ~10-9 down to ~10-10 cm, while the lochons accelerate between the

deuterons and the Coulomb field grows as the gap shrinks. This large lochon size, relative to the

nuclear-interaction distances, is a major limitation for strong screening. However, being bound,

the electron/lochon screening of the deuterons increases with kinetic energy (i.e., as orbits

shrink), rather than decreasing as is the case for free electrons. The 1-D nature of the problem

affects the electron s-orbital orientation, in that the electron/lochon direction of motion is along

the d-d axis, and therefore this localization and the velocity-induced shrinkage of L (along the dd

axis) aids in the screening.

Reaction Rate

The reaction rate (per cm3 per second) for d-d- fusion with lochon screening is given by

Rdd = r*

dd KBTL

2 Nd

2/ħ [(e2/ħ r) (1 – exp (-as/ L)]

x exp [-(e2/ħ r )(1 – exp (-as/ L))] ; (14)

where r*

dd = ħ2/2 MN e2 is the nuclear Bohr radius for a pair of deuterons; MN is the average mass

per nucleon  1.66 x 10-24 gm; KB is the Boltzmann constant; T is the temperature (with KBTL

2

having dimensions of energy x area); and Nd is the concentration of deuterons per unit volume.

The model presented in the foregoing section is more appropriate for reaction on the surface or

defect-plane in the lattice. The reaction rate is the number of effective collisions of deuterons

per unit area per sec. To convert to this picture, set KBTL

2 => KBTL = energy x length and set

Nd => NS = number of deuterons per unit area (rather than per unit volume).

The results of Equation 14,

modified for surfaces, are plotted

in Figure 1, taking some acceptable

values of the parameters involved.

However, the figure plots the

reaction rate assuming that 100%

of the available lattice sites are

actively involved. It is likely that

only a percentage of the sites can

be made to contribute to LENR.

Three values of as/L (1, 0.2, and

0.1) have been selected for the

lochon case.

Figure 1 The D+ - D- reaction

rate (for a surface), as a function

of D+ - D- separation distance, for

three ratios of as/L (1, 0.2, & 0.1).

5

Conclusion

In the foregoing sections, we have presented a model incorporating conditions in the

condensed matter state that can facilitate fusion of deuterons aided by interaction of electrons

with phonon modes of the system. The cross-section of the reaction improves considerably

owing to the presence of d-d pairs in line defects and with strong screening provided by bound

electron pairs (lochons). However, only a mechanism, such as D+ and D- pairing can bring the

deuterons close enough to permit a modified standard nuclear model to predict LENR.

Recent experiments by several workers, in which the material (e.g., powder or particles), is

taken to be in the nanometer range, suggest that the creation of large surface area plays an

important role.10 These surfaces may provide the required active sites, in the 2-D geometry that

can harbor lochons and D+ + D- ion pairs. Our computed reaction rate is found to be > 1014 per

cm2 per second (Figure 1) for two-dimensional surfaces, in agreement with the estimate of some

workers.

The role of optical-phonon modes is important for their bringing the D+ + D- pairs together, for

coupling of ions to electrons, and as a source of resonant coupling to provide the required

surface-mode excitation (surface plasmon or phonon) that can lead to enhanced-optical

potentials. Recent work, on excitation of surface plasmon/polaritons with “tuned” lasers,11,12

indicates the importance of this mechanism, where the induced-optical potential aids the fusion

reaction by several orders of magnitude. The known presence of resonant D+ + D- ion pairs in the

solid state (coupled via optical phonons) greatly increases the d-d interaction cross-section by

altering the shape of the Coulomb barrier to the extent of requiring a change in the equations

normally used in nuclear physics.

Acknowledgements

This work is supported in part by HiPi Consulting, New Market, MD, USA, by the Science for

Humanity Trust, Bangalore, 560094, India, by the Science for Humanity Trust, Inc, Tucker, GA,

USA, and by the Indian National Science Academy.

References

1. P. L Hagelstein, M. McKubre, D. J. Nagel, T. A. Chubb and R. J. Jekman, Report to DOE,

USA (2004).

2. K. P. Sinha, Infinite Energy 29, 54 (2000).

3. S.H. Patil, J. Chem. Phys. 118, 2197 (2003).

4. A. P. B. Sinha and K.P. Sinha, Ind. J. Pure and Appl. Phys. 1, 286 (1963).

5. I. G. Lang and Yu. A. Firsov, Sov. Phys. JETP, 16, 1301 (1963).

6. P.W. Anderson, Phys. Rev. Lett. 34, 953 (1975).

7. S. Ichimaru, Statistical Plasma Physics, Vol II, Condensed Plasmas (Addison – Wesley Pub.

Comp. 1994).

8. J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, (Wiley & Sons, N.Y., ‘52).

9. K. P. Sinha and A. Meulenberg, “Lochon Catalyzed D-D Fusion in Deuterated Pd in the

Solid State,” Nat. Acad. of Sc. (India) Letters, Vol.30, No. 7&8, 2007, (arXiv:0705.0595v1)

10. D. J. Nagel, “Proc. of the 13th International Conf. on Cold Fusion, Sochi, Russia (2007).

11. K.P. Sinha and A. Meulenberg, Current Science, 91, 907 (2006), (arXiv:cond-mat/0603213 ).

12. D. Letts and P. L. Hagelstein, “Stimulation of Optical Phonons in Deuterated Palladium,”

this proceedings.

[Kevmo]

A lot of the characters from the PDF do not translate well into HTML. The PDF is a far better read.

[Kevmo]

The Cold Fusion Ping List

[Kevmo]

html link

http://lenr-canr.org/acrobat/SinhaKPamodelfore.pdf

[REDWOOD99]

You mean the off is better than THAT?!

[Kevmo]

Grappling with Whether the E-CAT is a fraud
Wednesday, June 08, 2011 3:55:51 PM · 80 of 98
Kevmo to mad_as_he$$
someone made the comment that all of this is “normal quantum theory”.
***Yup, Widom Larsen theory, as well as KP Sinha’s theory.

[babygene]

What a bunch of crap!!!

[Soul Citizen]

Please add me to the Fusion List ** Thanks **

[El Sordo]

Mah brain hurt...

[JRandomFreeper]

strong screening provided by bound electron pairs (lochons).

Is that a cousin of yours?

/johnny

[garjog]

[JRandomFreeper]

Yep, the special characters are rendered as garbage on my screen.

I followed it well enough, making some assumptions about the non-printable characters.I guess I should look at the PDF and see how far off I was. The answer I got was 42.

/johnny

[garjog]

This is great. Hopefully this reactor will easily interfacee with Entabulator systems.

http://www.youtube.com/watch?v=oIS5n9Oyzsc&feature=player_embedded

[babygene]

Be it known that I have no problem with cold fusion. I have my own experimental apparatus that shows a lot of promise... At this point it is 700% over unity.

However this article is bunk...

[Kevmo]

However this article is bunk...

***This guy is a world class scientist with several articles published in peer reviewed journals, and it pulls only from classical physics. Explain how it is bunk.

[onedoug]

Do it. Then publish the paper.

[grey_whiskers]

What is the energy released per d-d fusion event?

Are there any competing chemical or nuclear reactions? (E.g off the wall suggestion of the type of thing I'm considering, oxygen within the reaction vessel might react with the d to produce heavy water which would segregate the reactants instead of the desired LENR.)

Can we combine this with the predicted reaction rates and make a sanity check against the observed energy output?

Did it say the reaction cross section increased with increasing energy, as it has been suggested that Rossi's reactor got more efficient as it got going?

Is is possible for two d's to enter a quasi-bound state and then separate, thus lowering the true reaction rate?

Can one compare the observed energy output and infer a reaction rate, or (given an estimate of surface area) develop a series of curves, analogous to a phase diagram, of the relative proportions of lattice sites of various surface spacings? (This might help in engineering the metal powder in the future.)

Cheers!

[black_diamond]

Seems to be the same basic concept as the Illudium Q-36 Explosive Space Modulator.

[Kevmo]

Do it. Then publish the paper.
***That appears to be what Rossi’s doing. Perversely enough, he’s being criticized for not publishing the paper.

Matthew 11:16
“To what can I compare this generation? They are like children sitting in the marketplaces and calling out to others:
Matthew 11:15-17 (in Context) Matthew 11 (Whole Chapter)

Luke 7:32
They are like children sitting in the marketplace and calling out to each other: “‘We played the pipe for you, and you did not dance; we sang a dirge, and you did not cry.’
Luke 7:31-33 (in Context) Luke 7 (Whole Chapter)

[Kevmo]

repost

The End of Snide Remarks Against Cold Fusion
http://www.freerepublic.com/focus/f-bloggers/2265914/posts
Friday, June 05, 2009 5:56:08 PM · by Kevmo · 95 replies · 1,770+ views
Free Republic, Gravitronics.net and Intrade ^ | 6/5/09 | kevmo, et al