Quantum Plasma Nuclear Fusion Theory for Anomalous Enhancement of Nuclear Reaction Rates Observed at Low Energies with Metal Targets

Yeong E. Kim and Alexander L. ZubarevPurdue Nuclear and Many-Body Theory Group

(PNMBTG)

Department of Physics, Purdue UniversityWest Lafayette, IN 47907 USA

June 5, 2007

Fig. 1. Periodic table showing the studied elements. Low Ue values (Ue < 80 eV, small effect) in light-shade (or yellow), are found for insulators and semiconductors. Metals in dark shade (or green) display high Ue values (Ue ≥ 80 eV, large effect). Elements of group 3 and 4 and the lanthanides show a small effect despite being metals probably because of the low density for quasi-free mobile deuterons.

TG(E) = exp Table I. Summary of results.

Ue (eV) Ue (eV) Ue (eV) Ue (eV)

Metals Metals Semiconductors lanthanides

Be 180±40 Pd 800±90 C ≤60 La ≤60

Mg 440±40 Ag 330±40 Si ≤60 Ce ≤30

Al 520±50 Cd 360±40 Ge ≤80 Pr ≤70

V 480±60 In 520±50 Insulators Nd ≤30

Cr 320±70 Sn 130±20 BeO ≤30 Sm ≤30

Mn 390±50 Sb 720±70 B ≤30 Eu ≤50

Fe 460±60 Ba 490±70 Al2O3 ≤30 Gd ≤50

Co 640±70 Ta 270±30 CaO2 ≤ 50 Tb ≤30

Ni 380±40 W 250±30 Groups 3 & 4 Dy ≤30

Cu 470±50 Re 230±30 Sc ≤30 Ho ≤70

Zn 480±50 Ir 200±40 Ti ≤30 Er ≤50

Sr 210±30 Pt 670±50 Y ≤70 Tm ≤70

Nb 470±60 Au 280±50 Zr ≤40 Yb ≤40

Mo 420±50 Tl 550±90 Lu ≤40

Ru 215±30 Pb 480±50 Hf ≤30

Rh 230±40 Bi 540±60

C. Rolfs et al., Progress of Theoretical Physics No. 154, 373 (2004);F. Raiola et al., Eur. Phys. J. A 19, 283 (2004).

eG- E / E+U QPNF Mechanism

Generalized Momentum Distribution Function(Leo P. Kadanoff and Gordon Baym, “Quantum Statistical Mechanics”, W.A.

Benjamin, New York (1962), Chapter 4.)

The Second-Quantized Formalism in Heisenberger Representation for Identical Particles (bosons or fermions).

Particle Creation Operator

Particle Annihilation Operator

The Fourier Transform:

One-particle Green’s function:

The correlation functions:

† ( , )r t

( , )r t

†

1 1 1 1

1G ( , ; , ) (1,1 ) ( (1 ))r t r t G Tr

i

> †1 1

†1 1

1G (1,1 ) (1 ) ,

1(1,1 ) (1 ) ,

t ti

G t ti

† †1 1

†1 1

( (1 )) (1 ), (1 ), (+ for bosons, - for fermions)

Tr for t tfor t t

3( , ) , )ip r i tp d r dt e r t

QPNF Mechanism

Introduce the Fourier Transforms of G> and G<

3( , ) ( , )p r i t

G p i d r dt e G r t

3( , ) ( , )ip r i tG p i d r dt e G r t

( , ) ( , ) ( , )A p G p G p

( , ) 1 ( ) ( , )( , ) ( ) ( , ) ( , )

G p f A pG p f A p n p

( )

1 1( ) ,

1f

e kT

( , ) :A p Spectral Function

The boundary condition on G can then be represented by writing

where

QPNF Mechanism

( ) ( , ) ( ) ( , )2 2

d dn p n p A p

A(p,ω) is shown to have a general form:

2

2

( , )( , )

( , )( ) Re ( , )

2c

pA p

pE p p

For a system of free particles,

and

in the classical limit, βμ→∞.

Galitskii and Yakimets (GY) use.

(Kadanoff-Baym (KB))

( , ) ( , ) ; ( ) ( ) ; ( , ) ( , )KB p GY KB GY KB GYA p E n E p E p , ( , ) ( , )GYKB

n p E p

(Kadanoff-Baym (KB))

QPNF Mechanism

2

oA (p, ω) = 2πδ(ω - p /2m)

2

2

( / 2 )00 ( / 2 )

1( ) ( ) ( , )

2 1p m

p m

dn p A p e

e

________________V.M. Galitskii and V.V. Yakimets, Zh.Eksp,Teor,Fiz. 51, 957 (1966)[Sov.Phys.JETP 24,637 (1967)]. (GY)

Kadanoff-Baym Equation

Theory of Quantum Plasma Nuclear Fusion (QPNF)

Quantum Correction for Non-Ideal Plasma Distribution Function• Generalized Distribution Function (V.M. Galitskiy and V.V. Yakimets, J. Exptl.

Theoret. Phys. (U.S.S.R) 51, 957 (1966))

where n(E) is Maxwell-Boltzmann (MB), Fermi-Dirac (FD), or Bose-Einstein (BE) distribution, modificed by the quantum broadening of the momentum-energy dispersion relation, δγ(E-εp), due to particle interactions

This Lorentzian distribution reduces to the δ-function in the limit of Δ→0 and γ →0,

δγ(E, εp) = δ(E, εp)

pf ( , ) ( ) ( , )E p n E E

p 2 2

(E, )(E, )

[(Ε (E,p)) (E,p)]p

p

QPNF Mechanism

The Momentum-Energy Dispersion Relation:

: kinetic energy in the center of mass coordinate of

interacting pair of particles, µ is the reduced mass.

: energy shift due to interaction (screening energy, etc.)

: line width of the momentum-energy dispersion

where ρc is the density of Coulomb scattering centers (nuclei) and

is the Coulomb scattering cross section.

is an effective charge which depends on εp. For small values of εp, is expected to be For larger values of εp, it is expected that

eiZ

p 2 2(E, )(E, )

[(Ε (E,p)) (E,p)]p

p

2p / 2p

( , )E p

( , )E p

( , ) 2 / c cE p E

e e 2 2 2c i j(Z Z e ) / p

eiZ e

i<<Z .eiZ

.ei iZ Z

QPNF Mechanism

The Nuclear Fusion Rate:

where the normalization N is given by

For a high energy region, εp>>kT, γ, and Δ,

compared with the MB case

p

02

p

v ε v ( )f ),

f ( ) ( ) ( ),ε / 2 ( is the reduced mass)

cmrel rel

p

N d E p

p dE n E Ep

f ( ) 1pN d p

e e 2 2i j

2 4 8

(Z Z e )8 1f ( ) ( ) ( , ) c

p p

N kTp n E E p dE N

p

p

p

- /p

- /p

- /p

(ε ) f ( ) (ε ) f ( ) , (ε ) f ( ) ,

p

p

p

kTMB MB

kTFD FD

kTBE BE

kT

kT

n p en p en p e

QPNF Mechanism

Deviation from Maxwell-Boltzmann(MB)Distribution Function for thermal meta-equilibrium

Nonextensive Statistical Mechanics (Tsallis)• Generalization of Boltzmann-Gibbs (BG) Entropy S

→ (Power Law)

(BG)

• Generalization of MB distribution fuction (Power Law) → (MB)

C. Tsallis, “Possible Generalization of Boltzmann-Gibbs Statistics.” J. Stat. Phys. 52 (1988): 479-487.

M. Gell-Mann and C. Tsallis (eds.), “Nonextensive entropy - Interdisciplinary applications”, Oxford University Press, Oxford (2004).

(Applications to physics, chemistry, biology, economics, linguistics, medicine, geophysics, cognitive sciences, computer sciences, and social sciences)

1

1

1(1 )

1

wq

q i ii

S p pq

log , as q 1i ii

S p p

/(1 )

1 (1 )q q

Eq

kT

/ , as q 1E kTe

Fusion Reaction RatesThe total nuclear fusion rate Rij, per unit volume (cm-3) and per unit time (s-1)

between a pair of nuclei, i and j is given by

(9)• between a beam particle and a target particle (bt)

• between a beam particle and a plasma particle (bp)

• between a plasma particle and a plasma particle (pp)

• between a plasma particle and a target particle (pt) , are reaction rates due to new processes involving the quasi-free

mobile deuterons in a plasma state with a momentum distribution of GY type. and are expected to be much smaller than based onconsideration on different densities involved. The recent calculationy Coraddu et al. indicates that is negligible.

bt bp pp ptij ij ij ij ijR R R R R

btijRbpijR

ppijRpt

ijR

, ,bp pp ptij ij ijR R and R

bpijR pp

ijR ptijR

bpijR

is given by

(10)

• Φi is the incident beam particle flux (# per cm2), • ρt is the stationary target particle density, • E0 is the incident kinetic energy in the laboratory

system, • is the stopping power with the

laboratory kinetic energy Ei,• σij(E) is the cross-section for reaction between

particles i and j with the relative kinetic energy E in the center of mass (CM) system.

btijR

0

0 0( , ) ( , )/

Ebt iij e i t ij e

i

dER E U E U

dE dx

/idE dx

σij(E) is conventionally parameterized as

(11)

• EG is the Gamow energy, EG=(2παZiZj)2 μc2/2 with the reduced mass μ

• S(E) is the astrophysical S-factor. • To accommodate the effect of electron

screening for the target nuclei, σij(E) is modified to include the electron screening energy,Ue , and parameterized as

( )( ) exp[ / ]ij G

S EE E E

E

( )( , ) exp[ /( )]e

ij e G ee

S E UE U E E U

E U

(12)

Dominant contribution from quantum plasma nuclear fusion

can be written as (13)

(1)

• is the conventional fusion rate calculated with the MB

distribution is the QPNF contribution given by

(14)

• where EG is the Gamow energy, EG =(2παZiZj)2 μc2/2, • ρi is the number density of nuclei • Sij(0) is the S-factor at zero energy for a fusion reaction

between i and j nuclei, assuming Sij(E) ≈ Sij(0).

ptijR

int 0( )( )pt C Qij ij ijR V E R R

3i j c i jQ 2 e e 2

ij rel A ij i j 3ij ij G

ρ ρ ρ ρ ρ1 ( c)R v N (4(5!)) S (0)(Z Z )

1+ 1+ c EAN

2int 0 0( ) ( ) / 4bV E x E D

QijR

CijR

Parameterization of experimental data for low-energy reaction rates

For comparison of our theoretical estimates with experimental data, we use the parameterization of the experimental data for anomalous enhancement based on the following equation,

(15) where is given (12)

with Ue replaced by experimentally extracted value of obtained by fitting the experimental data.

0exp exp exp0 0( , ) ( , )

/E i

ij e i t ij ei

dER E U E U

dE dx

exp( , )ij eE U( )

( , ) exp[ /( )]eij e G e

e

S E UE U E E U

E U

Applications to Experimental Results of

Low-Energy Nuclear Reactions For our theoretical estimates for the reaction rate, we

approximate it using eq. (13), as (16)

where is given by eq. (10) and is given by eq. (14). Define the enhancement factor F(E) as

(17)

and (18)

where are given by eqs. (1), (10), (14), and (15), respectively. UA is the adiabatic value for the screening energy.

int 0( )bt pt bt Qij ij ij ij ijR R R R V E R

btijR Q

ijR

exp exp

exp

( , )( ) ,

( , )ij e

btij A

R E UF E

R E U

int( , ) ( )( ) ,

( , )

bt Qij A ij

theo btij A

R E U V E RF E

R E U

exp expint ( ), ( , ), , and ( , )bt Q

ij A ij ij eV E R E U R R E U

Fig. 1. Enhancement factors Fexp(E) [Eq. (17)] and Ftheo [eq. (18)] for D(d,p)T reaction with Ta target as a function of the deuteron laboratory kinetic energy E. F=1 is the expected conventional value.

… … Ftheo(E) with ρi= 2.5 x 1016cm-3

- - - - Ftheo(E) with ρi= 1.7 x 1017cm-3

-·- -·- Ftheo(E) with ρi= 4.5 x 1017cm-3

—Fexp(E) (Raiola et al., Eur. Phys. J. A 13 (2002) 377)

Using exp 309 eVeU

Fig. 2. Enhancement factors Fexp(E) [Eq. (17)] and Ftheo [eq. (18)] for reaction with Pd6Lix (x=1%) target as a function of the deuteron laboratory kinetic energy E. F=1 is the expected conventional value.

… … Ftheo(E) with ρi= 3.9 x 1017cm-3

- - - - Ftheo(E) with ρi= 1.3 x 1018cm-3

-·- -·- Ftheo(E) with ρi= 6.0 x 1018cm-3

—Fexp(E) (Kasigi et al., J. Phys. Soc. Jpn 73 ( 2004) 608)

Using

6 4( , )Li d He

exp 1500 eVeU

Fig. 3 … … Ftheo(E) with ρi= 1.2 x 1018cm-3

- - - - Ftheo(E) with ρi= 2.8 x 1018cm-3

-·- -·- Ftheo(E) with ρi= 6.8 x 1018cm-3

—Fexp(E) (Cruz et al., Phys. Lett. B 624 (2005) 181)

Using

6 3( , )Li p HeFig. 3. Enhancement factors Fexp(E) [Eq. (17)] and Ftheo [eq. (18)] for reaction with Pd6Lix (x=1%) target as a function of the deuteron laboratory kinetic energy E. F=1 is the expected conventional value.

exp 3760 eVeU

Proposed Experimental Tests of Theoretical Predictions

1. New Density Dependence

(Kasagi et al., J. Phys. Soc. Jpn. 71 (2002) 2881.)

Rij(PdLi) > Rij(Li)

(Cruz et al., Phys. Lett. B 624 (2005)81.)

2. Increase Quasi-free Deuteron/Proton Densities

to increase the reaction rates, by applying

(1) electric current to target metal

(2) laser beams to reaction zone in metal target.

cij i jQij c i j

RR

6 3 7 4Li( d,α) He and Li( p,α) Hereactions:

ijR ( ) ( )ijPdO R Pdd(d,p) t reaction :

Proposed Experimental Tests (Koltick and Kim)

n p n p

Use thermal neutrons E~0.025 eV

Thermal energy needed to breakProton interstitial binding E ~0.1 eV

Probability to observe scattered neutron at

1 eV Maxwell-Boltzmann 10-5

10 eV Maxwell-Boltzmann 10-44

MUST INSURE THAT BACKGROUND NEUTRONS ARE NEAR ZERO

3. Search for 1/p8 using Neutron As a Probe

Neutron Scattering

The neutron can be consider at rest and are scattered by the fast protons.

The slower the neutrons the longer they remain in the target to be scattered to high energy

The neutron is scattered uniformly in energy

Spallation Neutron Source• SNS is an accelerator-based neutron source in Oak Ridge,

Tennessee, USA.

•The first neutrons were produced Friday, April 29, 2006

•Partnership of 6 US Department of Energy Laboratories

•February 19th Beam Acceleration 1 GeV

Spallation Neutron Source

Accelerator system includes, Ion source, Linear accelerators, Accumulator ring, Mercury target, 18 Beam Lines

Time of Flight SpectrometersModerator

Shutter

Guides

Choppers

Sample

Detector Array

The ARCS, a wide angle chopper spectrometer at SNS, utilizes high speed choppers, a super-mirror neutron guide and large angle array of He3 neutron detectors to allow precise neutron time of flight measurements to |-q| ~< 30 Å-1.

Concluding Remarks• Theory of quantum plasma nuclear fusion (QPNF) is

based on conventional quantum statistical physics and nuclear physics.

• QPNF theory may provide a consistent explanation for anomalous enhancement of low-energy nuclear reaction rates with metal targets.

• QPNF theory makes a set of many theoretical predictions which can be tested experimentally.

• QPNF theory has many potential and important applications in clean fusion energy generation and astrophysics.

Back up Slide

ARCS Detector Development

• 3He Detectors,109 1m 8-pack module• 2 short modules above and below beam• Absorbing baffles designed to block cross-talk within 90°

Table 1. Parameters for the bare S-factor used in this paper.

6 4Li(d,α) He [37]

6 3Li(p,α) He [14]

Reaction S(0) (keV-b) S1 (barns) S2 (b/keV)

D(d,p)T [8] 53 0.48 -

16.9 x 103 -41.6 28.2 x 10-3

3.0 x 103 -3.02 1.93 x 10-3

Table 2. Values of at different values of the beam deuteron laboratory kinetic energy Eb for the D(d,t)T reaction calculated from Eqs. (19),

(1b-3b), (1), (14), (10), and (16) respectively.

( )bE keV( )bE ( )bx E

int ( )bV Eint ( ) Q

b ijV E R ( , )btij b AR E U

ijR

Eq. (19) (cm) (cm3) (s-1) (s-1) (s-1)

4 4.6 x 10-7 2.1 x 10-5 3.7 x 10-5 5.9 x 10-2 1.7 x 10-2 7.6 x 10-2

7 3.1 x 10-6 2.8 x 10-5 4.9 x 10-5 3.6 3.7 7.3

10 8.1 x 10-6 3.3 x 10-5 5.8 x 10-5 2.9 x 10 5.6 x 10 8.5 x 10

20 3.1 x10-5 4.7 x10-5 8.2 x10-5 5.9 x102 3.5 x103 4.1 x103

30 5.2 x 10-5 5.7 x 10-5 1.0 x 10-4 2.1 x 103 2.2 x 104 2.4 x 104

Table 3. Values of at different values of the beam deuteron kinetic energy Eb for the reaction calculated from Eqs. (19), (1b-3b), (1), (14),

(10), and (16), respectively.

( )bE keV

( )bE ( )bx E int ( )bV E int ( ) Qb ijV E R ( , )bt

ij b AR E UijR

Eq. (19) (cm) (cm3) (s-1) (s-1) (s-1)

30 5.8 x 10-6 4.3 x 10-5 5.5 x 10-6 0.1 0.1 0.2

40 2.0 x 10-5 5.0 x 10-5 6.3 x 10-6 1.5 2.5 4.0

50 4.8 x 10-5 5.7 x 10-5 7.1 x 10-6 9.2 2.3 x 10 3.2 x 10

60 8.8 x 10-5 6.2 x10-5 7.8 x10-6 3.4 x10 1.2 x102 1.5 x102

70 1.4 x 10-4 6.8 x 10-5 8.5 x 10-6 9.5 x 10 4.1 x 102 5.1 x 102

Table 4. Values of at different values of the beam proton kinetic energy Eb for the reaction calculated from Eqs. (19), (1b-3b), (1), (14),

(10), and (16), respectively.

( )bE keV( )bE ( )bx E int ( )bV E

int ( ) Qb ijV E R ( , )bt

ij b AR E U ijR

Eq. (19) (cm) (cm3) (s-1) (s-1) (s-1)

30 1.9 x 10-5 3.1 x 10-5 2.5 x 10-5 1.4 0.7 2.1

40 4.1 x 10-5 3.6 x 10-5 2.7 x 10-5 7.7 6.7 14.4

50 6.9 x 10-5 4.1 x 10-5 3.2 x 10-5 2.5 x 10 3.2 x 10 5.7 x 10

60 1.0 x 10-4 4.6 x10-5 3.6 x10-5 5.7 x10 1.0 x102 1.6 x102

70 1.3 x 10-4 5.0 x 10-5 3.9 x 10-5 1.1 x 102 2.6 x 102 3.7 x 102