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Fundamental of Rate Theory for CMNS ICCF19 slides

Presentation · April 2015

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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.

• Free particle motion (plasma)

never makes collision/fusion, due to zero volume/size.

• Borrowing QM results of cross

section, we may use classical mechanics as Boltzmann transport equation. (We need size!)

Technova -6404-NT-22 5

Cross section

The Case of Hot Plasma Fusion7)

• 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 (Ek).

• <Macroscopic Fusion Rate> = < Nd(Ek)2vσdd (Ek)>

Gamow-Teller peak

Nd : deuteron density, v: relative d-d velocity, σdd = (S(Ek)/Ek)exp(-Γdd): fusion cross section, Ek: relative d-d kinetic energy Γdd : Gamow factor

• Free particle motion and collision process:

Nd(Ek) = N∙(Ek/T2)exp(-Ek/T) : Maxwell-Boltzmann distr.

6

AT ICCF17 TSC theory

Technova -6404-NT-22 6

Fusion Rate Formula for Collision Process4)

• f two free particles
• T = <Ψf│Hint│Ψi>

= <Initial State Interaction> x<Intermediate Compound State> x<Final State Interaction>

• Cross Section ~ T2 ρ(E’)
• ρ(E’): final state density
• Reaction-Rate(σv): (4π2/h)vT2 ρ(E’)
• <Initial> = <El. EM Int><Strong Int>
• <Final>=BRs to Irreversible Decays

Technova -6404-NT-22 7

2)Fusion r rate t e theo eory f for t trapped ed D D(H) H) particles es3,

3,4) 4)

Trapped particles in EM-potential make nuclear reactions by QM superposition

• Pair or cluster trapped in Electro-

Magnetic (chemical) potential

• Overlapping weight within strong/weak nuclear

interaction range (1.4fm/2.4am) should be estimated by QM. Overlapping of QM-clouds Rdd: inter-nuclear distance deuteron deuteron Coulombic (EM) trapping potential: Vs1(1,1) potential for instance electron

Technova -6404-NT-22 8

EM- Chemical Trapping Potential

Reaction Surface of Strong Interaction Thomas-Fermi Screening potential

Nuclear Optical Potential for reaction rate formula

• Forward Equation:

(1)

• Adjoint Equation:

(2)

• Ψ*x(1) – Ψx(2):

Ψ       + + ∇ − = ∂ Ψ ∂ iW V M t i

2 2

2  

* 2 *

2 2

Ψ       − + ∇ − = ∂ Ψ ∂ − iW V M t i  

t i t i t t i ∂ ∂ = ∂ ΨΨ ∂ =       ∂ Ψ ∂ Ψ + ∂ Ψ ∂ Ψ ρ    * * *

[ ]

[ ] [ ]

ρ ρ ρ W i j div i W i M t i 2 2 * * 2

2 2 2

+ − = + Ψ ∇ Ψ − Ψ ∇ Ψ − = ∂ ∂    

10

AT ICCF17 TSC theory

Technova -6404-NT-22 10

Fusion Rate Formula by Fermi’s Golden Rule

i f

r W FusionRate Ψ Ψ >= < ) ( 2 

Ψ = Ψ + Ψ + + Ψ ∇ − E r V r iW r V m

c nr

) ( )] ( ) ( [ 2

2 2



) ( ) ( ) ( r r r

c n

Ψ ⋅ Ψ = Ψ

Nuclear Potential Coulomb Potential Inter-nuclear wave function EM Field wave function Born-Oppenheimer Approximation

11

AT ICCF17 TSC theory

Technova -6404-NT-22 11

Adiabatic QM Equations for Initial State Interaction

) ( ) ( )] ( ) ( [ ) ( 2

2 2

r E r r iW r V r m

n n n nr n

Ψ = Ψ + + Ψ ∇ −  ) ( ) ( ) ( ) ( 2

2 2

r E r r V r m

c c c c c

Ψ = Ψ + Ψ ∇ − 

Inter-Nuclear QM Schroedinger Equation: Outer-Nuclear QM Schroedinger Equation for Electro-Magnetic Field:

12

AT ICCF17 TSC theory

Technova -6404-NT-22 12

Fusion Rate Formula by Fermi’s First Golden Rule with Born-Oppenheimer Approximation

Vn ci cf Vn ni nf

r W FusionRate Ψ Ψ ⋅ Ψ Ψ >= < ) ( 2 

π

π 

2

4

n

R Vn ≈

: Effective Volume of Nuclear Strong (Weak) Interaction Domain

π



: Compton wave length of pion (1.4 fm) (weak boson: 2.5 am) Rn : Radius of Interaction surface of strong (weak) force exchange

13

AT ICCF17 TSC theory

Barrier Factor

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

• →

n

: : d ;

; down wn q qua uark

( uud uud ) ( ) ( u*

u*d) d) (udd dd) ) : u* u* ; an anti ti-up q quark rk : : d* d* ; an anti ti-down q quark rk

n p p n

For D + D Fusion; PEF = 2

Technova -6404-NT-22 14

<W> value Estimation

DD 0.008 DT 0.115 3D 1.93 4D 62.0

Inter-nuclear wave functions are governed by the real part of optical potential V(R), having Woods-Saxon type well shape that makes inter-nuclear wave functions very localized near around V(R). Therefore, <W> becomes approximated by the surface sticking at R=R0.

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

16

With time elapsed, potential becomes deeper and moves to left. Fusion Interaction Surface : Elevated KE

AT ICCF17 TSC theory

Technova -6404-NT-22 16

Electron 3) 8Be* formation 15 fm

Deuteron 4He 4He

4re = 4x2.8 fm

p or d Electron

d+ d+ d+ d+

e- e- e- e-

1.4007 fs N-Halo 4) Break up to two 4He’s via complex final states; 0.04- 5MeV α + BOLEP photons (main excited energy damping) 2) Minimum TSC reaches strong interaction range for fusion 1) TSC forms

Electron Center

4D/TSC Condensation Reactions

17 Technova -6404-NT-22

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0.01 0.1 1 10 100 1000 10000 1 2 3 4

Rdd (pm) Edd (eV) Time (fs)

4D/TSC Langevin cal. : Nf = 4, Vs1, BA = 0.85 Collapse Rdd (pm) or Edd (eV)

0.01 0.1 1 10 100 1000 10000 100000 0.00001 0.0001 0.001 0.01 0.1 1 10

4D/TSC condensation-collapse calculation with Vs1(1,1) potential (tc = 3.6247 fs)

Rdd (pm) Edd (eV)

tc – time (fs) Rdd (pm) or Edd (eV)

Rdd=Rgs (pm) P2d ; 2D Barrier Factor P4d; 4D Barrier Factor 0.0206 0.0412 0.103 0.206 0.412 0.805 (μdd) 1.03 2.06 4.12 10.3 21.8 (dde*(2,2) 74.1 (D2 molecule) 4.44E-2 1.06E-2 1.43E-3 3.35E-5 9.40E-7 1.00E-9 9.69E-11 6.89E-15 9.38E-21 2.16E-32 1.30E-46 1.00E-85 1.98E-3 1.12E-4 2.05E-6 1.12E-9 2.16E-13 1.00E-18 9.40E-21 4.75E-29 8.79E-41 4.67E-64 1.69E-92 1.00E-170

20

)) , ( exp( ) , ( Z m n Z m P

dd nd

Γ − =

dR E Z m R V Z m

z m b r d s dd

∫

− = Γ

) , (

) , ; ( 218 . ) , ( µ

Barrier factors were calculated time-dependently Based on the Heavy Mass Electronic Qausi-Particle Expansion Theory (HMEQPET)

Technova -6404-NT-22

m = 9000/Rdd, Z =2 (e-e pair), r0 = 5 fm set [b0(m,Z) = Rdd (fm), for positive Vs] Vtsc is approximated by Vs1(m,Z) EQPET pseudo-potential

• 5.00E+03
• 4.00E+03
• 3.00E+03
• 2.00E+03
• 1.00E+03

0.00E+00 1.00E+03 2.00E+03 3.00E+03 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00

Vs1(m, Z) potential vs. Rgs (nm)

Vs(0.4,2) Vs(4,2) Vs(44,2)

Rgs = 1 pm Rdd (nm) Vs (eV) Rgs = 10 pm Vmin = -394 eV Rgs = 100 pm Vmin = -36 eV

• 5.00E+03
• 4.00E+03
• 3.00E+03
• 2.00E+03
• 1.00E+03

0.00E+00 1.00E+03 2.00E+03 3.00E+03 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 Conversion of potentials for barrier factor calculation

Vs(44,2) Vs(9,2) Rdd (nm) Vs (eV)

Range of Gamow integral Surface of strong interaction Rgs = 1 pm b0 = 1 pm

1.00E-99 1.00E-93 1.00E-87 1.00E-81 1.00E-75 1.00E-69 1.00E-63 1.00E-57 1.00E-51 1.00E-45 1.00E-39 1.00E-33 1.00E-27 1.00E-21 1.00E-15 1.00E-09 1.00E-03 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 Rdd (fm) Barrier Factor

1.00E-23 1.00E-21 1.00E-19 1.00E-17 1.00E-15 1.00E-13 1.00E-11 1.00E-09 1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01 1.00E+03 1.00E+05 1.000E+01 1.000E+02 1.000E+03 1.000E+04 DD Fusion Power Level vs. Rdd and Life-Time

1W/mol 10kW/mol

Rdd (fm) Life Time (sec)

Cluster Rdd = Rgs (pm) Barrier Factor Steady Cluster d-d Fusion Rate (f/s) Steady Cluster 4d Fusion rate (f/s) Fusion Rate for d-d collision formula (f/s) D2 74.1 1.0E-85 2.4E-66 3.6E-86

(4.0E-72)*

dde*(2,2) 21.8 1.3E-46 3.2E-27 1.0E-46

(1.0E-31)*

ddμ 0.805 1.0E-9 2.4E+10 1.5E-9

(1.0E+8)*

4D/TSC- minimum 0.021 1.98E-3 3.7E+20 Collision Rate Formula UNDERESTIMATES fusion rate of steady molecule/cluster

* Frequency of d-d pair oscillation by QM-Langevin calculation was considered.

Technova Inc. AT 26

20 40 60 80 100 120 140

5 10 15 20

Rdd (pm) Edd (eV)

Time (fs) Rdd (pm) or Edd (eV)

D2 Langevin cal.: Vs1(1,1.41) Pot., Rdd(gs) = 74 pm, GSOF=1.1x1014 Hz

To see observable heat level by d-d fusions, inter-nuclear d-d distance should become Less than Rdd of 1 pm with feasible life-time:

at Rdd = 1 pm and life-time =3.0E-18 s (TSC-cal): <Fusion yield> =(barrier-factor)x<Wћ/2>x<life-time> = (1.0E-10)x(1.5E19)x(3.0E-18) = 4.5E-9 (fusion/d-d) Order of Macroscopic Fusion Rate: 6.023E23(d-d/s)x4.5E-9 = ca. 3E15 f/s/mol-dd =ca. 3 kW/mol-dd

The Case of 4D/TSC-min transitory BEC: 1.0 W power = 1.7E11 4D/TSC fusions/s

) ) ( exp( 1

4 4

dt t

c

t d d

∫

− − = λ η

)) ( ; ( 10 88 . 1 )) ( ; ( 10 04 . 3 ) (

4 23 4 21 4

t R r P t R r P W t

dd d dd d d

× = × = λ <fusion rate per 4D/TSC-min> = 3.7x1020 f/s ; for steady state Real yield of 4d fusion : η4d ≈ 1.0 per TSC-cluster Happens in

• Ca. 2x10-20 s

Technova -6404-NT-22 28

10 kW: 1.7E15 4D/TSCs per second = 2.8 nano-mol 4D/TSCs per second

Vtsc (keV) vs. R' at Rdd(t)=25 fm using Vs(2,2)

• 150
• 100
• 50

0.01 0.02 0.03 0.04 0.05 0.06

R' (pm) Vtsc (keV)

Vtsc (keV)

Summary of Results for CCF of CMNR

Technova -6404-NT-22 29

Cluster Type Collapse?

(Rdd-min ≦10 fm)

Rdd (gs) Remarks d-e-d (2D+) N 138 pm d-μ-d N 0.79 pm

DD fusion in 0.1ns

3D+ N 85 pm 4D/TS Y 100% 4D fusion 6D/RD Y

100% 6D fusion

8D/RD Y

100% 8D fusion

• r 4D fusion?

12D/RT N

• ca. 80 pm

20D/RT Y

What kind of fusion?

Summary

• For CMNR rate formulas, we need to take life-time of D(H)s in chemical

trapping potential well into account properly.

• We cannot use formulas for two-body collision theory for CMNR, as it

drastically underestimates fusion rate.

• Fermi’s first golden rule should be used for rate formulas.
• Feasibly enhanced cluster fusions may happen only for collapse states of

dynamic condensation: 4D/TSC, 6D/RDC, 8D/RDC for examples.

• Therefore, we need time-dependent fusion rate calculation to estimate real

fusion yield per D(H)-condensing-cluster formation.

• If inter-nuclear d-d distance for condensate becomes less than 1.0 pm,

we can expect reaction rates with ‘observed heat-power levels’.