“Nuclear Reactions in Micro/Nano-Scale Metal Particles”Yeong E. Kim

Department of Physics, Purdue UniversityWest Lafayette, Indiana 47907, USA

August 22, 2011

BACKGROUD

• The first invited talk on the subject was presented at the First APFB1999 conference, Tokyo, Japan, August 23 – 28, 1999, organized by Professor Shinsho Oryu et al. :Y. E Kim and A. L. Zubarev, “Effective Linear Two-Body Method for Many-Body Problems In Atomic and Nuclear Physics”, Few-Body Systems Supplement 12, edited by S. Oryu, M. Kamimura, and S. Ishikawa, pages 7-14 (2000).

• Since 1999, there have been 9 refereed publications and 7 papers in conference proceedings.

March 23, 1989March 23, 1989

• Pons and Fleischmann announced that electrochemical cells with heavy water are producing more heat than can be accounted for by chemical means and speculated that nuclear reactions must be occurring.

• Thousands of scientists worldwide attempted experiments—most failed

Initial Claim: Radiationless fusion reaction (Electrolysis Exp.) D + D → 4He + 23.8 MeV (heat) (no gamma rays)According to the conventional nuclear theory in free space, the above fusion reaction is not expected to be observable at room temperature, due to (1) the DD Coulomb repulsion (Gamow factor), and (2) the violation of the momentum conservation in free space.

3

Conventional DD Fusion Reactions in Free-Space

[1] D + D→ p + T + 4.033 MeV

[2] D + D→ n + 3He + 3.270 MeV

[3] D + D→ 4He + γ(E2) + 23.847 MeV

Measured branching ratios: (σ [1], σ[2], σ[3]) ≈ (0.5, 0.5, 3.4x10-7)

In free space it is all about the coulomb barrier! GES(E)

E Eexpσ(E)

The three well known “hot” dd fusion reactions

For Elab < 100 keV, the fit is made with σ(E) = GE / EeS

E

Reaction [1] Reaction [2]

SRI Labyrinth(L and M) Calorimeter

and Cell

Brass Heater Support and Fins

Water Outlet Containing Venturi Mixing Tube and Outlet RTD's

Acrylic Flow Separator

Stainless Steel Dewar

Heater

Locating Pin

Acrylic flow restrictor

Gas Tube Exit to Gas-handling

Manifold

Acrylic Top-piece

Water In

Water Out

Hermetic 16-pin Connector

Gasket

Quartz Anode Cage

PTFE Ring

PTFE Ring

PTFE Spray Separator Cone

Recombination Catalyst in Pt Wire Basket

Pt Wire Anode

Catalyst RTD

PTFE Plate

Hermetic 10-pin Connector

Stainless Steel Outer Casing

PTFE Liner

Quartz Cell Body

Gasket

Screws

Pd Cathode

Stand

Inlet RTD's

Over 50,000 hours of calorimetry to investigate the Fleishmann–Pons effect have been performed to date, most of it in calorimeters identical or very similar to this.

SRI FPE Replication

a)a) Current threshold Current threshold IIcc = = 250mA/cm2 and linear slope.and linear slope.

b)b) Loading thresholdLoading threshold D/Pd > 0.88D/Pd > 0.88

5

IIc c = =250mA/cm2

D/Pd = 0.88

The conditions required for positive electrolysis results:(1) Loading ratio D/Pd > 0.88 and (2) Current density Ic > 250 mA/cm2

• Caltech (1989/90): N.S. Lewis, et al., Nature 340, 525(1989)

• Harwell (1989): Williams et al., Nature 342, 375 (1989)

• MIT (1989/90): D. Albagli, et al., J. Fusion Energy 9, 133 (1990)

• Bell Labs (1989/90): J. W. Fleming et al., J. Fusion Energy 9, 517 (1990)

• GE (1992): Wilson, et al. J. Electroanal. Chem. 332, 1 (1992)

6

2/ 0.77 0.05,0.79 0.04,0.80 0.05 (70 140) /cD Pd I mA cm

2/ 0.76 0.06,0.84 0.03 (80 110) /cD Pd I mA cm

2/ 0.62 0.05,0.75 0.05,0.78 0.05 (8 69,512) /cD Pd I mA cm

2/ 0.45 0.75 (64,128,256,600) /cD Pd I mA cm

2/ 0.69 0.05 100 /cD Pd I mA cm

In no single experiment did following samples of NULL results simultaneously have the required D/Pd ratio (D/Pd > 0.88) and critical current density (Ic > 250 mA/cm2 ) !

SEM images from Dardik, et al., Proceedings of ICCF-14 , 2008Micro-craters in palladium, possibly following extreme heat release, when loaded with heavy hydrogen in electrolysis experiments. No micro-craters were observed with hydrogen. There have been many other reports of observing the micro-craters from electrolysis experiments with heavy water.

A2 system for H2 run

Reaction chamber

Pressure gaugeVacuum gauge

A1 systemfor D2 run

H2 gascylinder

Vacuum pump

D2 gascylinder

A2 system for H2 run

Reaction chamber

Pressure gaugeVacuum gauge

A1 systemfor D2 run

H2 gascylinder

Vacuum pump

D2 gascylinder

Tout

Tc

(6 ml/min)

Reaction chamber

Vacuum chamber

Heater

Vacuum pump

Pin

D2 or H2

gas

Cold trap

Pd membrane

Vacuum pump

Vacuum pump

Tin

Pd powder

Heater

Thermocouples

ChillerTout

Tc

(6 ml/min)

Reaction chamber

Vacuum chamber

Heater

Vacuum pump

Pin

D2 or H2

gas

Cold trap

Pd membrane

Vacuum pumpVacuum pump

Vacuum pump

Tin

Pd powder

Heater

Thermocouples

Chiller

A. Kitamura et al./ Physics Letters A 373 (2009) 3109-3112

8

(c) Mixed oxides of PdZr

0 500 1000 1500

0

0.4

0.8

1.2

0

0.4

0.8

1.2

Time [min]

Ou

tpu

t p

ower

[W

]

Pre

ssu

re [

MP

a]

Power (D2) Power (H2) Pressure (D2) Pressure (H2)

9

•Output power of 0.15 W corresponds to Rt ≈ 1 x 109 DD fusions/sec for D+D → 4He + 23.8 MeV

10.7-nmφPd

Fig. 3(c): A. Kitamura et al., Physics Letters A, 373 (2009) 3109-3112.

1MPa = 9.87 Atm

Theory of Bose-Einstein Condensation Nuclear Fusion (BECNF) in Metal

In metal, hydrogen (deuterium) atom is ionized and becomes mobile as proton (deuteron) in metal, as proven experimentally by Coehn 1929! This implies that we can achive a very high density (~1022/cm3 !) of deutron-electron plasma in a metal !!

For BECNF theory, assume one single basic concept that deuterons form Bose-Einstein condensates in metal (“nuclear” BEC), and

Develope a consistent physical theory which will • (1) be capable of explaining experimental

observations, and • (2) have predictive powers, capable of making

theoretical predictions, which can be tested experimentally

Boson-Einstein Condensation (BEC) Mechanism

N-body Schroedinger equation for the system is

2 2N N2 2

iii ji=1 i=1 i<j

1 eH= Δ + mω +r

2m 2 r -r

where m is the rest mass of the nucleus.

[The electron degrees of freedom can be incorporated by using the electron-screened Coulomb potential (Debye screening)].

(1)

(2)

H E

11

Equivalent Linear Two-Body (ELTB) Method [Kim and Zubarev, J. Phys. B: At. Mol. Opt. Phys. 33, 55 (2000); Phys. Rev. A 66, 053602 (2002)]

(3 1)/2

( )( )( ,... ) ,N Nr r

1/ 2

2

1

N

ii

r

(3)

Use of a variational principle with leads to2 2 2

2 22 2

(3 1)(3 3)[ ( )] ,

2 2 2 4

d m N NV E

m d m

2 (3 / 2)( )

3 2 (3 / 2 3/ 2)

N NV

N

* 0H d * 1d

(4)

Eq. (4) can be solved analytically to obtain the solution for ().

Optical Theorem Formulation of Nuclear Reactions

[Y. E. Kim, A. L. Zubarev, J.-H. Yoon, Y. J. Kim, Phys. Rev. C 55, 801 (1997)]

The total elastic nucleus-nucleus amplitude (two potential formula):

( ) ( ) ( )cf f f

where is the Coulomb amplitude.( )cf 2 ( )( ) (2 1) (cos )

cli n el

l

f l e f P

where , is the l-th partial wave S-matrix, and is the Coulomb phase shift.

c( 1) / 2n(el) nf S ik

The Optical Theorem:( ) ( )Im

4n el r n el

l l

kf

where is the partial wave reaction cross-section.r

The elastic scattering amplitude, :( )n elf ( )2 2

2n el c cf tk

where is the Coulomb wave function.

For the s-wave, Eqs. (7) and (8) yield

c

(5)

(6)

(7)

(8)

1212

( )f

ns

0 0 02 2

2Im

4r c ck

tk

(9)

(rigorous)

(valid for low energies)( )Im4

n el rkf

Parameterization of the Short-Range Nuclear Force and Fusion Rates

The reaction cross-section is conventionally parameterized as r2 ,r S

eE

2

2

1, , / 2

2 2BB

r mkr e

S is the astrophysical S-factor and is the Gamow factor.2e

From the previous slide

For the nuclear force , we use the Fermi pseudo-potential to write( )FV r

0Im Im ( ) ( )2

F At V r r

where is determined from Eqs. (9) and (10) .

For our case of N-particles, we obtain the reaction rate from Eq. (9) after replacing by the solution of the N-body Schroedinger Eq. (1):

2 /BA Sr

(10)

(11)

(12)

1313

(9)0 0 02 2

2Im

4r c ck

tk

Im2 i j ij

trap

tR

trapR0c

Fusion Rates for N=2 Case

From the previous slide,

where is given by the Fermi potential ,

For N = 2 case, Eq. (13) reduces to

where is the solution of the Schroedinger Eq. (1) with N=2.

Near is the two-body Coulomb wave function, c(r).

From Eq. (13), we have

The reaction rate for N = 2 case is proportional to the Gamow factor, , and hence is consistent with the conventional formula for fusion rate for the N=2 case !

Im ijt

(12)

(14)

1414

0,r

2 2(0)CtrapR e

2e

0Im Im ( ) ( )2

F At V r r

Im2 i j ij

trap

tR

Im2 ij

trap

tR

(13)

1/22D

t trap trap trap D

N 1 3R N R R B Vn

N 4S

Alternative Derivation of R t , Eq. (16): Use of , obtained from solution of the mean-field equation, in Eq. (12) yields Eq. (16) within a factor of 2 ! [Y.E. Kim and A.L. Zubarev, Italian Physical Society Conference Proceeding, Vol. 70, 375 (May 2000)]

Reaction Rates for Large N

15

3/2 1/22

trap D3trap

1 3 N 1 3R B B n N

2 4S S

D

where S is the S-factor in units of keV-barn, B = 2ħ / (π me2) = 1.4 x 10-18 cm3/sec x (keV-barn)-1, Dtrap is the average diameter of the trap, ND is the total number of deuterons, N is the number of deuterons in a trap, and nD is the deuteron density. S and are only two unkown parameters !

Im2 i j ij

trap

tR

(12)

(15)

(16)

Significances of Theoretical ResultsNuclear fusion rate R for large N does not depend on the Gamow factor in contrast to the reaction rate for nuclear fusion in free space !

This could provide explanations for overcoming the Coulomb barrier.

This is consistent with Dirac’s conjecture* that boson creation and annihilation operators can be treated simply as numbers when the ground state occupation number is large. This implies that for large N each charged boson behaves as an independent particle in a common average background potential and the Coulomb interaction between two charged bosons is suppressed. * Paul A. M. Dirac, “The Principles of Quantum Mechanics” (second edition), Oxford 1935, Chapter IX, Section 63, p. 235

There is a similar classical analogy of uniform charge distribution in a sphere. the electric field is zero at the center.

16

BECNF theory can explain the following experimental observations either qualitatively or quantitatively.

Experimental Observations from both electrolysis and gas loading experiments (as of 2010, not complete) (over several hundreds publications !):

[1] The Coulomb barrier between two deuterons is suppressed

[2] Excess heat production (the amount of exess heat indicates its nuclear origin)

[3] 4He production comensurate with excess heat production, no 23.8 MeV gamma ray

[4] More tritium is produced than neutron R(T) >> R(n)

[5] Production of nuclear ashes with anomalous rates: R{4} << R {6} and R {5} << R{6} i. e. R(T) << R(4He) and R(n) << R(4He)

[6] Production of hot spots and micro-scale crators on metal surface

[7] Detection of radiations

[8] “Heat-after-death”

[9] Requirement of deuteron mobility (D/Pd > 0.9, electric current, pressure gradient, etc.)

[10] Requirement of deuterium purity (H/D << 1)

17

Proposed Experimental TestsI. Experimental tests of the concept of BEC of deuterons in

metals (this concept is new)• Experiment 1: Measure the velocity distribution of

deuterons by low-energy neutron scattering• Experiment 2: Measure the diffusion rate of deuterons

to establish possible superfluidity

II. Experimental tests of theoretical predictions• Experiment 3: Temperature dependence of the reaction

rate mini-ignition at extremely low temperatures

18

Fraction of Deuterons in the BEC State in Metal at Various Temperatures

For BOSE-Einstein distribution, a fraction F(T) of deuterons below the temperature T or Ec satisfying

can be calculated as

where

For T = 300o K with

F (300o K) = 0.084 (8.4%)

F(77.3o K) = ~ 0.44 (~44%) ! (Liquid NitrogenTemp.)F(20.3o K) = ~ 0.94 (~94 %) !! (Liquid Hydrogen Temp.)F(4.2o K) = ~0.99 (~99 %) !!! (Liquid He-4 Temp.)

19

3

4( ) 1/23V

N E dE E dEh

(2m )

/0 0

( )( ) ( )

1BE E kT

N E dEn E N E dE

e eN

/dB dBd h m

2.5

dB d

0

1( ) ( ) ( )

cE

BEF T n E N E dEN

19

• Experiment 1: Measure the velocity distribution of deuterons by low-energy neutron scattering

20

~ 400 nK ~ 200 nK ~ 50 nK In 1995, measurement of the velocity distribution was used to establish the existence of the BEC of atoms in a magnetic trap at extremely low temperatures, for which the Nobel prize was awarded in 2000 to C. Wieman, E. Cornell, and W. Ketterle.

• Experiment 2: Measure the diffusion rates of deuterons to establish possible superfluidity of deuterons in metal Explore a number of experimental methods for observing the superfluidity In 1996, the Nobel prize was awarded for discovery of superfluidity of 3He to D. Lee, D. Osheroff, and R. Richardson.

D-T targets at National Ignition facility

Radiograph of a high-density carbon capsule with a smooth, frozen layer of D-T inside.

Experiment 3: Temperature dependence of the reaction rate - mini-ignition at extremely low temperatures

Left: A 2-mm-diameter polished beryllium ICF capsule with a 10-micron fill tube attached. Right: 2-mm polished high-density carbon ablator capsules with the silicon mandrel inside.

Proposed Experiment 3:D-Pd targets for BECNF

For BECNF, use 1-cm diameter containerfilled with micro/nano- scale metal particles pre-loaded with deuterons

Cryogenic Target System (NIF)Ignition target inserter cyrostat

A NIF target is suspended at the end of its cryogenic cooling system via a copper support beam.Precise temperature control is achieved by subcooling the target to below requirements and then using small electric heaters to precisely raise the temperature to the exact level required.

Proposed Experiment 3:Adopt the NIF’s the cryogenic target systemfor BECNF

Target Chamber at National Ignition Facility

Technicians on a specially-designed target chamber service system lift make adjustments to the target alignment sensor and positioner.

Cyrogenic Target Positioner (cyroTARPOS)

The cryoTARPOS was tested off-site in preparation for installation in the NIF target bay.

Proposed Experiment 3:Use the NIF’s target chamber or a newlybuilt ignition chamber for BECNF

Observation of Hydrogen-Nickel Nuclear Reactions

S. Piantelli, et al. , Department of Physics, University of Siena, ItalyS. Focardi, E. Campari, et al. , Department of Physics, University of Bologna, ItalyObservations of exess heat (2 ~ 4 x input energy) and some gamma rays with nickel metal plate/cylinder in a reactor pressurized with hydrogen gas

Publications:1. F. Piantelli, Atti Acad. Fis. Series XV, Tomo XII, pp 89-96 (1993)2. S. Focardi, R. Habel, and F. Piantelli, Nuovo Cimento A, 107, 163 (1994)3. S. Focardi, et al., Nuovo Cimento A, 111, 1233 (1998); 4. E. Campari, S. Focardi, F. Piantelli, et al., 5th Asti Workshop, Asti, Italy (2004)and ~ six other publications

Patent Application:Silvia Piantelli, “METHOD FOR PRODUCING ENERGY AND APPARATUS THEREFOR”, Internatioanl Application (Pub. No.:WO 2010/058288 A1, Pub. Date:May 27, 2010)

Spatial distribution of Ni and Cu on the sample surface

Experimental cell for hydrogen loading with Ni cylinder (red)

E. Campari, S. Focardi, F. Piantelli, et al.,Proceedingd of ICCF 11, Marseilles, France (2006)

“Experimental test of a mini-Rossi device at the Leonardocorp, Bologna 29 March 2011”reported by Hanno Essen* and Sven Kullander** , 3 April 2011*Associate Professor of Theoretical Physics, Swedish Royal Institute of Technology, Stockholm, Sweden** Professor of Physics Emeritus, University of Upssala, Chair of Energy Committee, Royal Swedish Academy of Sciences• Micro/nano scale Ni particles/powers with hydrogen gas at 25 bars• Electric input power of 0.3 kW (resistance heating) and output power of 4.69 kW as estimated from vaporization of input water (18o C ) at a constant flow rate, during a period of 5 hour 45 minutes exsess heat generation of ~ 14 x input energy. 2-cm Pb shielding

Generalized BECNF Theory for Hydrogen-Nickel SystemHydrogen-Nickel ReactionsAssume (1) mobile Ni nuclei and (2) mobile composite bosons consisting of

two protons with spins coulpled anti-parallel forming singlet states (S=0)This allows us to use the generalized BECNF theory for two species of bosons.Predictions are possibilities of reacations (i) ANi(2p(S=0),

p)A+1Cu, with even A = 58, 60, 62, and 64, and (ii) ANi(2p(S=0), p)A-2Ni, with even A = 58, 60, 62, and 64

For (i), 59Cu(81.5 seconds) and 61Cu(3.333 hrs) are radioactive, both decay to unstable Ni nuclei by the electron capture, both of which subsequently decay to stable Ni isotopes by emitting characteristic gamma-rays. use as experimental tests

For (ii), all of Ni isotopes produced are stable except 56Ni. However, its production reaction rate is expected be substantially lower than those of Ni isotopes.

58Ni(2p(S=0), p)59Cu

60Ni(2p(S=0), p)61Cu

Conclusions and Summary● BECNF theory is based on one single physical assumption of the

new basic concept of BEC of deuterons in metals.

● BECNF theory provides consistent theoretical explanations for experimental observations.

● Experimental tests are proposed for the basic concept of “nuclear” BEC of deuterons in metals.

● Experimental tests are also proposed for BECNF mini-ignition at extremely low temperatures. If successful, it can be used in the target chamber at the National Ignition Facility, or in a newly built ignition chamber.

● Recently, generalized BECNF theory is used to make theoretical predictions for BECNF processes in hydrogen-nickel systems. Theoretical predictions will be compared with experimental data, when more accurate data become available in the near future !

29

Backup Slides

30

“Experimental test of a mini-Rossi device at the Leonardocorp, Bologna 29 March 2011”reported by Hanno Essen* and Sven Kullander** , 3 April 2011*Associate Professor of Theoretical Physics, Swedish Royal Institute of Technology, Stockholm, Sweden** Professor of Physics Emeritus, University of Upssala, Chair of Energy Committee, Royal Swedish Academy of Sciences• Micro/nano scale Ni particles/powers with hydrogen gas at 25 bars• Electric input power of 0.3 kW (resistance heating) and output power of 4.69 kW as estimated from vaporization of input water (18o C ) at a constant flow rate, during a period of 5 hour 45 minutes exsess heat generation of ~ 14 x input energy.

P

E-Cat Hyperion (5 ~ 30 kW) being manufactured by Defkalion Green Technologies (DGT) in Xanthi, Greecehttp://www.defkalion-enrgy.com

A: Reactor(s) container, thermally insulated and lead shieldedB: Hydrogen tankC: Electronic control unitCP: Pump for heat transport (closed circuit)Dimension: 22 x 18 x 14 inchesPin < 0.5 kW is used to ignite and to sustain reactions for generating Pout = 5 ~ 30 kW

Demo for a larger unit (1.15 ~ 3.45 MW) is scheduled in late October 2011.It will contain ~300 units of Hyperion (5 – 30 kW) in parallel configuration, and will fit in a truck container (20 feet long).

Application of BECNF to Hydrogen-Nickel System

Andrea Rossi’s Energy Catalyzer (“E-Cat”)Observations of exess heat (10 ~ 60 x input energy !) with micro/nano-scale nickel metal particles in a reactor pressurized with hydrogen gas

• 2007: His new discovery of the excess heat effect using micro/nano-scale Ni particles with pressuraized hydrogen gas.• 2008: Applications for Italian and international patents. • 2009: The technology licensed to a new start-up company, Defkalion Green Technologies (DGT) in Greece with capitalization of ~200M Euros ! • 2011: Two positive demonstrations of the E-Cat in January and March, 2011;• 2011: In July, Greek Government issued DGT a commercial license for marketing in Greece, after extensive tests and evaluations of the E-Cat.

Rossi’s Energy-Catalyzer (E-Cat) Demo on March 29, 2011

Target Chamber at National Ignition Facility

Exterior of the NIF target chamber under construction. The square openings are for the quads of beamlines; the round openings will accommodatenearly 100 pieces of diagnostic equipment.

Technicians on a specially-designed target chamber service system lift make adjustments to the target alignment sensor and positioner.

Possible Scenarios for Creation of Micro-Craters Observation [6]: Production of hot spots and micro-craters. Episodes of “Melt Down” reported by Fleischmann, and others.• Example of 10 nm diameter PdD particles

38

Exp. Obs. [6]

o3 4

trap D trapD 10nm 100A N n D 3.56 10 deuterons6

3/2 2 216 3

trap 3 3trap trap

1 3 N NR B (0S .36 10 cm / sec)

2 D D

216 3 10

trap 3trap

NR (0.36 10 cm / sec) 4.5 10 / sec

D

11 3

3 18 3 8trap trap trap

trap

V 5.0 10 cm (example)V

V D 0.52 10 cm N 0.96 10 traps6 V

• Explosion time

5reaction

10 5reactiontrap exp losion

trap

N 5.4 10 DD fusions(example)N

R 4.5 10 / sec 1.2 10 sec/R

exp losion

39

Power DD Fusion Rate Fuel Lifetime

100 HP(419 Watt)

2.6x1013 /sec 165 years

1 KW 6.2x1013 /sec 69 years

1 MW 6.2x1016 /sec 25 days

For 1 cm3 Palladium containing 6.8 x 1022 deuterons,Rt = ~ 1029 /sec with =1 and S= 55 KeV-barn, under optimal conditions

1/22D

t trap trap trap D

N 1 3R N R R B Vn

N 4S

Total Fusion Rate for D(m) + D(m) 4He(m) + 23.85 MeV

Is BECNF process scalable for practical applications ? We need further theoretical and experimental research.

(3)

P13/14 Simultaneous Series Operation of Light & Heavy Water Cells;

Excess Power & Current Density vs. Time

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

430 454 478 502 526 550 574 598 622

I (A/cm^2) Pxs D2O (W) Pxs H2O (W)PPInIn = 10 W = 10 W

200mA/cm2

Coulomb potential and nuclear square well potential

EEWKBRG

GeTET 0)(

2

)2( 2221 cZZ

EG

ar

R

WKBR drE

r

eZZET

21

221

2

22exp)(

E

U

(E+U)

-V0

B

V(r)

≈ ≈

R ra rbr

U = Escreening

(Electron Screening Energy)

Gamow Factor – WKB approximation for Transmission Coefficient

B

E

B

E

B

E

E

EET GWKB

R 1cos2

exp)( 1

R

eZZB

221

ar

eZZE

221

Equivalent Linear Two-Body (ELTB) Method (Kim and Zubarev, Physical Review A 66, 053602 (2002))

For the ground-state wave function , we use the following approximation

(3 1) / 2

( )( )( ,... )N Nr r

where1/ 2

2

1

N

ii

r

It has been shown that approximation (3) yields good results for the case of large N (Kim and Zubarev, J. Phys. B: At. Mol. Opt. Phys. 33, 55 (2000))

By requiring that must satisfy a variational principle with a subsidiary condition , we obtain the following Schrödinger equation for the ground state wave function ()

* 0H d * 1d

2 2 22 2

2 2

(3 1)(3 3)[ ( )]

2 2 2 4

d m N NV E

m d m

(3)

(4)

2 (3 / 2)( )

3 2 (3 / 2 3/ 2)

N NV

N

where (5)

4242

Optical Theorem Formulation of Nuclear Fusion Reactions (Kim, et al. Physical Review C 55, 801 (1997))

In order to parameterize the short-range nuclear force, we use the optical theorem formulation of nuclear fusion reactions. The total elastic nucleus-nucleus amplitude can be written as

( ) ( ) ( )cf f f

where is the Coulomb amplitude, and can be expanded in partial waves( )cf ( )f

2 ( )( ) (2 1) (cos )cli n el

l

f l e f P

In Eq. (7), is the Coulomb phase shift, , and is the l-th partial wave S-matrix for the nuclear part.

c ( 1) / 2n(el) nf S ik ns

For low energy, we can write (optical theorem)( )Im

4n el rk

f

where is the partial wave reaction cross section.r

In terms of the partial wave t-matrix, the elastic scattering amplitude, can be written as

( )n elf

( )2 2

2n el c cf tk

where is the Coulomb wave function. c

(6)

(7)

(8)

(9)

4343

Parameterization of the Short-Range Nuclear Force

For the dominant contribution of only s-wave, we have( )

0Im4

n el rkf

( )0 0 0 02 2

2n el c cf tk

Where is conventionally parameterized as r

2r Se

E

2

2

1, , / 2

2 2BB

r mkr e

, is the “Gamow” factor,2e

From the above relations, Eqs. (10), (11), and (12), we have

0 0 02 2

2Im

4r c ck

tk

For the case of N Bose nuclei, to account for a short range nuclear force between two nuclei, we introduce the following Fermi pseudo-potential ( )FV r

0Im Im ( ) ( )2

F At V r r

where the short-range nuclear-force constant A is determined from Eqs. (12) and (13) to be . 2 /BA Sr

For deuteron-deuteron (DD) fusion via reactions D(d,p)T and D(d,n)3He, the S-factor is S = 110 KeV-barn.

and

(10)

(11)

(12)

(13)

(14)

and S is the S- factor for the nuclear fusion reaction between two nuclei.

4444

Derivation of Fusion Probability and Rates

For N identical Bose nuclei confined in an ion trap, the nucleus-nucleus fusion rate is determined from the trapped ground state wave function as

Im2 i j ij

t

tR

where is given by the Fermi potential Eq. (14), . Im ijt

From Eq. (15), we obtain for a single trap

3

4t B

cR B Nn

m

where is the probability of the ground state occupation, 2 3/ , /Be c n N r

is Bose nuclei density in a trap, and with

For the case of multiple ion traps (atomic clusters or bubbles), the total ion-trap nuclear fusion rate R per unit time and per unit volume, can be written as

3

4t B

cR n B Nn

m

where nt is a trap number density (number of traps per unit volume) and N is the average number of Bose nuclei in a trap.

Im ( ) / 2ijt A r

(15)

(16)

(17)

3 / 8B Am c 2 /BA Sr

4545

Possible Scenarios for Creation of Micro-Craters Observation [6]: Production of hot spots and micro-craters. Episodes of “Melt Down” reported by Fleischmann, and others.• Example of 10 nm diameter PdD particles

49

Exp. Obs. [6]

o3 4

trap D trapD 10nm 100A N n D 3.56 10 deuterons6

3/2 2 216 3

trap 3 3trap trap

1 3 N NR B (0S .36 10 cm / sec)

2 D D

216 3 10

trap 3trap

NR (0.36 10 cm / sec) 4.5 10 / sec

D

11 3

3 18 3 8trap trap trap

trap

V 5.0 10 cm (example)V

V D 0.52 10 cm N 0.96 10 traps6 V

• Explosion time

5reaction

10 5reactiontrap exp losion

trap

N 5.4 10 DD fusions(example)N

R 4.5 10 / sec 1.2 10 sec/R

exp losion

Requirement forBose-EinsteinCondensation (BEC):

λDB > d

where d is theaverage distance between neighboringtwo Bosons.

5050

51

Created in 1995 by C. Wieman, E. Cornell, W. Ketterle, et al. Nobel Prize in 2000

• Experiment 1: Measure the velocity distribution of deuterons by low-energy neutron scattering

52

~ 400 nK ~ 200 nK ~ 50 nK

In 1995, measurement of the velocitydistribution was used to establish the existence of the BEC of atomsin a magnetic trap at extremely low temperatures, for which the Nobel prizewas awarded in 2000 to C. Wieman, E. Cornell, and W. Ketterle.

• Experiment 2: Measure the diffusion rate of deuterons to establish possible superconductivity.

In 1996, the Nobel prize was awarded for discovery of superfluidity of 3He.

Atomic BEC vs. Nuclear BECλDB > d , λDB =

Atomic BEC: d ≈ 7 x 103 Å = 0.7 μm (for nRb = 2.6 x 1012/cm3)

υc ≈ 0.6 cm/sec near T ≈ 170 n Kelvin

( ~ 2000 atoms in BEC out of ~ 2 x 104 atoms 10 % in BEC)

(1) Increase λDB by slowing down neutral atoms

using laser cooling and evaporation cooling

Nuclear BEC: d ≈ 2.5 Å (for nD = 6.8 x 1022/cm3 in metal)

υc ≈ 0.78 x 105 cm/sec (υkT ≈ 1.6 x 105 cm/sec at T= 300 Kelvin)

(1) Increase λDB by cooling deuterons or by applying EM fields

(2) Decrease d further by increasing density, using ultrahigh pressure device such as Diamond Anvil Cell (DAC), etc.

h

mυ

53

[Kim, 2009]