See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/320345294 Fundamental of Rate Theory for CMNS ICCF19 slides Presentation · April 2015 CITATIONS READS 0 24 1 author: Akito Takahashi Osaka University 341 PUBLICATIONS 1,664 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Leading the Japanese Gvt NEDO project on anomalous heat effect of nano-metal and hydrogen gas interaction View project Neutron tranport and moderation View project All content following this page was uploaded by Akito Takahashi on 12 October 2017. The user has requested enhancement of the downloaded file.
Fundamental of Rate Theory for CMNS Akito Takahashi Technova Inc. and Osaka University To be presented at ICCF19, Padua Italy, April 2015
No good to use the rate theory of free particle collision • Despite the importance of reaction rate estimation in modeled CMNS theories, only a few authors have treated nuclear reaction rates properly. Some theories have borrowed rate formulas of two body collision process which is the case of nuclear reactions for the random free particle motion as in plasma and gas phase or beam-target interactions.
CMNR (condensed matter nuclear reactions) needs to use rate estimation for trapped state H(D)s • Intrinsically the CMNS nuclear reactions should happen between trapped particles of proton and deuteron (H or D) in negative potential wells organized by the ordering of condensed matter such as periodic lattice, mesoscopic nano- particle and surface fractal conditions. Such trapped H(D)- particles should have finite lifetime or existing time in the negative potential well and are keeping mutual inter-nuclear distances for finite time-intervals before fusion reactions.
Rate Formula by Fermi’s First Golden Rule should be used for CMNR • We need therefore to use formulas based on the Fermi’s first golden rule for rate estimation, due to the finite life time of trapped particles. This paper recalls the procedure and formulas for the fundamental of rate theory for CMNS, as has been used in the TSC theory development.
1) C Classic ical me al mechanic ics and f free p partic icle fusion ion Classical mechanics has no intrinsic capability to treat nuclear fusion, because of zero-size (point) particle assumption. • Borrowing QM results of cross • Free particle motion (plasma) section, we may use classical never makes collision/fusion, mechanics as Boltzmann due to zero volume/size. transport equation. (We need size!) Cross section Technova -6404-NT-22 5
The Case of Hot Plasma Fusion 7) • Confinement of high kinetic energy deuterons (plasma) in a very large scale (torus) room , like Tokamak magnetic field confinement. • Average kinetic energy of d-d (or d-t) reaction for ITER is aimed to be about 10keV (E k ). • <Macroscopic Fusion Rate> = < N d (E k ) 2 v σ dd (E k )> Gamow-Teller peak N d : deuteron density , v: relative d-d velocity, σ dd = (S(E k )/E k )exp(- Γ dd ): fusion cross section , E k : relative d-d kinetic energy Γ dd : Gamow factor • Free particle motion and collision process: N d (E k ) = N ∙ (E k /T 2 )exp(-E k /T) : Maxwell-Boltzmann distr. 6 AT ICCF17 TSC theory Technova -6404-NT-22 6
Fusion Rate Formula for Collision Process 4) of two free particles • T = < Ψf│Hint│Ψ i> = <Initial State Interaction> x<Intermediate Compound State> x<Final State Interaction> • Cross Section ~ T 2 ρ (E’) • ρ (E’): final state density • Reaction-Rate( σ v): (4 π 2 /h)vT 2 ρ (E’) • <Initial> = <El. EM Int><Strong Int> • <Final>=BRs to Irreversible Decays Technova -6404-NT-22 7
es 3, 3,4) 4) 2)Fusion r rate t e theo eory f for t trapped ed D D(H) H) particles Trapped particles in EM-potential make nuclear reactions by QM superposition • Pair or cluster trapped in Electro- • Overlapping weight within strong/weak nuclear Magnetic (chemical) potential interaction range (1.4fm/2.4am) should be estimated by QM. electron Rdd: inter-nuclear distance deuteron deuteron Coulombic (EM) trapping potential: Overlapping of QM-clouds Vs1(1,1) potential for instance Technova -6404-NT-22 8
Reaction Surface of Thomas-Fermi Screening potential Strong Interaction EM- Chemical Trapping Potential
Nuclear Optical Potential for reaction rate formula • Forward Equation: ∂ Ψ 2 = − ∇ + + Ψ 2 i V iW (1) ∂ 2 t M • Adjoint Equation: ∂ Ψ 2 * − = − ∇ + − Ψ 2 i V iW * (2) ∂ t 2 M • Ψ *x(1) – Ψ x(2): ∂ Ψ ∂ Ψ ∂ ΨΨ ∂ ρ * * Ψ + Ψ = = i * i i ∂ ∂ ∂ ∂ t t t t [ ] ∂ ρ 2 [ ] [ ] = − Ψ ∇ Ψ − Ψ ∇ Ψ + ρ = − + ρ 2 2 i * * i 2 W i div j i 2 W ∂ t 2 M 10 AT ICCF17 TSC theory Technova -6404-NT-22 10
Fusion Rate Formula by Fermi’s Golden Rule 2 < >= Ψ Ψ FusionRate W ( r ) f i 2 − ∇ Ψ + + Ψ + Ψ = Ψ 2 [ V ( r ) iW ( r )] V ( r ) E nr c 2 m Nuclear Potential Coulomb Potential Ψ = Ψ ⋅ Ψ ( r ) ( r ) ( r ) n c EM Field wave function Inter-nuclear wave function Born-Oppenheimer Approximation 11 AT ICCF17 TSC theory Technova -6404-NT-22 11
Adiabatic QM Equations for Initial State Interaction Inter-Nuclear QM Schroedinger Equation: 2 − ∇ Ψ + + Ψ = Ψ 2 ( r ) [ V ( r ) iW ( r )] ( r ) E ( r ) n nr n n n 2 m Outer-Nuclear QM Schroedinger Equation for Electro-Magnetic Field: 2 − ∇ Ψ + Ψ = Ψ 2 ( r ) V ( r ) ( r ) E ( r ) c c c c c 2 m 12 AT ICCF17 TSC theory Technova -6404-NT-22 12
Fusion Rate Formula by Fermi’s First Golden Rule with Born-Oppenheimer Approximation Barrier Factor 2 < >= Ψ Ψ ⋅ Ψ Ψ FusionRate W ( r ) nf ni cf ci Vn Vn Vn ≈ π : Effective Volume of Nuclear Strong (Weak) 2 4 R π Interaction Domain n : Compton wave length of pion (1.4 fm) (weak boson: 2.5 am) π R n : Radius of Interaction surface of strong (weak) force exchange 13 AT ICCF17 TSC theory
Scaling of PEF ( Pion Exchange Force) for Nuclear Fusion by Strong I nteraction Two B Body y Intera ract ction: PEF EF = = 1 n + + π + → p (udd dd) ) (ud*) ) (uud uud) : u : u ; up ; up q qua uark p p + π : d ; ; down wn q qua uark n - → : (udd dd) ) ( u* u*d) d) ) : u* ( uud uud ) ( u* ; an anti ti-up q quark rk : : d* d* ; an anti ti-down q quark rk For D + D Fusion; PEF = 2 n p n p Technova -6404-NT-22 14
<W> value Estimation Inter-nuclear wave functions are governed by DD 0.008 the real part of optical potential V(R), having Woods-Saxon type well shape DT 0.115 that makes inter-nuclear wave functions very localized near around V(R). 3D 1.93 Therefore, <W> becomes approximated by the surface sticking at R=R0. 4D 62.0 Technova -6404-NT-22 15
Fusion rate should be estimated time- dependently, e.g. for TSC Condensation: No Stable State, but into sub-pm entity Fusion Interaction Surface With time elapsed, potential becomes deeper and moves to left. : Elevated KE 16 AT ICCF17 TSC theory Technova -6404-NT-22 16
Electron Center 4D/TSC e- Condensation Electron d + Reactions p or d d + e- 1.4007 fs d + e- 4r e = 4x2.8 fm e- 2) Minimum TSC d + reaches strong 1) TSC forms interaction range for fusion Electron N-Halo 4 He 4 He 4) Break up to two 4 He’s via Deuteron complex final states; 0.04- 15 fm 5MeV α + BOLEP photons (main 3) 8 Be* formation excited energy damping) Technova -6404-NT-22 17
4D/TSC Langevin cal. : Nf = 4, Vs1, BA = 0.85 Rdd 10000 (pm) Edd 1000 (eV) Rdd (pm) or Edd (eV ) 100 Collapse 10 1 0.1 0.01 0 1 2 3 4 Time (fs) Technova -6404-NT-22 18
4D/TSC condensation-collapse calculation with Vs1(1,1) potential (t c = 3.6247 fs) 100000 Rdd (pm) Edd (eV) 10000 Rdd (pm) or Edd (eV) 1000 100 10 1 0.1 0.01 0.00001 0.0001 0.001 0.01 0.1 1 10 t c – time (fs)
Barrier factors were calculated time-dependently Based on the Heavy Mass Electronic Qausi-Particle Expansion Theory (HMEQPET) R dd =R gs (pm) P 2d ; 2D Barrier P 4d ; 4D Barrier Factor Factor 0.0206 4.44E-2 1.98E-3 Vtsc is approximated by 0.0412 1.06E-2 1.12E-4 Vs1(m,Z) EQPET pseudo-potential 0.103 1.43E-3 2.05E-6 0.206 3.35E-5 1.12E-9 0.412 9.40E-7 2.16E-13 m = 9000/R dd , Z =2 (e-e pair), r 0 = 5 fm 0.805 ( μ dd) 1.00E-9 1.00E-18 set [b 0 (m,Z) = R dd (fm), for positive Vs] 1.03 9.69E-11 9.40E-21 2.06 6.89E-15 4.75E-29 4.12 9.38E-21 8.79E-41 ∫ b ( m , z ) Γ = µ − 0 ( m , Z ) 0 . 218 V ( R ; m , Z ) E dR 10.3 2.16E-32 4.67E-64 dd s d r 0 21.8 (dde*(2,2) 1.30E-46 1.69E-92 74.1 (D 2 1.00E-85 1.00E-170 = − Γ molecule) P ( m , Z ) exp( n ( m , Z )) nd dd Technova -6404-NT-22 20
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