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. Zubarev Purdue Nuclear and Many-Body Theory Group (PNMBTG) Department of Physics, Purdue University West Lafayette, IN 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.

T G (E) = exp Table I. Summary of results. U e (eV) Metals Semiconductorslanthanides 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 Al 2 O3 ≤30 Gd ≤50 Co 640±70 Ta 270±30 CaO 2 ≤ 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). 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: QPNF Mechanism

Introduce the Fourier Transforms of G > and G < Spectral Function The boundary condition on G can then be represented by writing where QPNF Mechanism

A(p, ω) is shown to have a general form: For a system of free particles, and in the classical limit, βμ→∞. Galitskii and Yakimets (GY) use. (Kadanoff-Baym (KB)) QPNF Mechanism ________________ 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, δ γ (E-ε p ), due to particle interactions This Lorentzian distribution reduces to the δ-function in the limit of Δ→0 and γ →0, δ γ (E, ε p ) = δ(E, ε 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 ρ c is the density of Coulomb scattering centers (nuclei) and is the Coulomb scattering cross section. is an effective charge which depends on ε p. For small values of ε p, is expected to be For larger values of ε p, it is expected that QPNF Mechanism

The Nuclear Fusion Rate: where the normalization N is given by For a high energy region, ε p >>kT, γ, and Δ, compared with the MB case 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): 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)

Fusion Reaction Rates The total nuclear fusion rate R ij, 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 on consideration on different densities involved. The recent calculation y Coraddu et al. indicates that is negligible.

is given by (10) Φ i is the incident beam particle flux (# per cm 2 ), ρ t is the stationary target particle density, E 0 is the incident kinetic energy in the laboratory system, is the stopping power with the laboratory kinetic energy E i, σ ij (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.

σ ij (E) is conventionally parameterized as (11) E G is the Gamow energy, E G =(2 παZ i Z j ) 2 μc 2 /2 with the reduced mass μ S(E) is the astrophysical S-factor. To accommodate the effect of electron screening for the target nuclei, σ ij (E) is modified to include the electron screening energy,U e, and parameterized as (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 E G is the Gamow energy, E G =(2 παZ i Z j ) 2 μc 2 /2, ρ i is the number density of nuclei S ij (0) is the S-factor at zero energy for a fusion reaction between i and j nuclei, assuming S ij (E) ≈ S ij (0).

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 U e replaced by experimentally extracted value of obtained by fitting the experimental data.

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. U A is the adiabatic value for the screening energy.

Fig. 1. Enhancement factors F exp (E) [Eq. (17)] and F theo [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. … … F theo (E) with ρ i = 2.5 x cm F theo (E) with ρ i = 1.7 x cm -3 -·- -·- F theo (E) with ρ i = 4.5 x cm -3 — F exp (E) ( Raiola et al., Eur. Phys. J. A 13 (2002) 377) Using

Fig. 2. Enhancement factors F exp (E) [Eq. (17)] and F theo [eq. (18)] for reaction with Pd 6 Li x (x=1%) target as a function of the deuteron laboratory kinetic energy E. F=1 is the expected conventional value. … … F theo (E) with ρ i = 3.9 x cm F theo (E) with ρ i = 1.3 x cm -3 -·- -·- F theo (E) with ρ i = 6.0 x cm -3 — F exp (E) ( Kasigi et al., J. Phys. Soc. Jpn 73 ( 2004) 608) Using

Fig. 3 … … F theo (E) with ρ i = 1.2 x cm F theo (E) with ρ i = 2.8 x cm -3 -·- -·- F theo (E) with ρ i = 6.8 x cm -3 — F exp (E) ( Cruz et al., Phys. Lett. B 624 (2005) 181) Using Fig. 3. Enhancement factors F exp (E) [Eq. (17)] and F theo [eq. (18)] for reaction with Pd 6 Li x (x=1%) target as a function of the deuteron laboratory kinetic energy E. F=1 is the expected conventional value.

Proposed Experimental Tests of Theoretical Predictions 1.New Density Dependence (Kasagi et al., J. Phys. Soc. Jpn. 71 (2002) 2881.) R ij (PdLi) > R ij (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.

Proposed Experimental Tests (Koltick and Kim) Use thermal neutrons E~0.025 eV Thermal energy needed to break Proton interstitial binding E ~0.1 eV Probability to observe scattered neutron at 1 eV Maxwell-Boltzmann eV Maxwell-Boltzmann MUST INSURE THAT BACKGROUND NEUTRONS ARE NEAR ZERO 3. Search for 1/p 8 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 Spectrometers Moderator 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 He 3 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.

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ARCS Detector Development 3 He 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. ReactionS(0) (keV-b)S 1 (barns)S 2 (b/keV) D(d,p)T [8] x x x 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. Eq. (19)(cm)(cm 3 )(s -1 ) 44.6 x x x x x x x x x x x x x x x x x x x x x x x x x x x 10 4

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. Eq. (19)(cm)(cm 3 )(s -1 ) x x x x x x x x x x x x x x x101.2 x x x x x x x x 10 2

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. Eq. (19)(cm)(cm 3 )(s -1 ) x x x x x x x x x x x x x x x x101.0 x x x x x x x x 10 2