Widom-Larsen, Brillouin (and some others) propose that electrons acquire
782 KeV mass/energy and overcome the electroweak barrier to combine with
protons, deuterons or tritons to produce low momentum neutrons.
Storms notes [1] that an electron must reach relativistic speeds to gain
782 KeV in a lattice, - seemingly a very tall order, due to collisions.
Others, e.g. Hagelstein, et al[2], doubt that field strengths in LENR
experiments provide this extra energy ("renormalized" mass).
I think both objections may overlook collective effects.
In an arc, colliding electron-proton(deuteron) wave packet pairs are
strongly squeezed together by equal, opposite magnetic forces.
Even when the composite packet has velocity zero (lab frame), the packets
continue absorbing field energy by becoming more oscillatory, localized and
overlapping as spectra shift to high mass/energy eigenstates. In pictures:
TIME Low resolution ASCII graphic of
| e-p collision with (lab) velocity ~ 0
|
V PROTON ELECTRON
| -----> <----- Decreasing
| _____________ _____________ Magnetic
| / \ / \ Vector Potential
| / PROTON \ / ELECTRON \
| / 'p' \ / 'e' \ A
| -------------------+--------------------- ------------->
|
V |\ 'HEAVIER' |
| | \ ELECTRON |
| _____________ | \ /\ |
| | \| \ / \ V
| | | \/ \ /\ /\ |
| | | \/ \/ \ A |
| -------------------+--------------------\ -------> |
| V
| | A-field
| |\ transfering
| | \ | 'HEAVY' momentum
| | \ |\ ELECTRON to e-p pair
| ___________|___\ | \ | |
| | | |\| \|\ |
| | | | | | | |
| | /\| | \ \ \ A |
| -------------/------+-------\-\---------- ---> V
V significant e-p electron wave packet
wave packet overlap becomes squeezed, more
localized, oscillatory,
- spectrum shift to high
mass/energy eigenstates
Electron velocities in arcs are usually far below relativistic, but the arc
magnetic field stores huge energy and momentum that is transferred to/from
colliding particles when the arc current rises, falls, or is interrupted.
To gain 782Kev in energy, an electron can equivalently acquire (see [6])
momentum = 6.3480 * 10^-22 [N*sec] -- where [N] = newtons
The following example shows that this does not require exotic lab equipment.
Assume the electron is in an arc plasma uniformly distributed in a tube
with radius=R, length=10*R, current=I aligned with the z-axis of 3-space.
We want to compute how much field momentum can be transferred to a electron
'e' in a collision at a radial distance 'r' from the tube center.
=============================== x-axis
^ e \ /
| ^ <----- I[Amps] \ /
| | r \ /
2R -------+------------------- <------x----- z-axis
| / \
| / \
v / y-axis
===============================
|<------ L = 10*R ------->|
The (under-utilized) "magnetic vector potential" field (denoted A(r))
depends only on local currents. Very conveniently [3,4] --
q*A(r) = momentum impulse (as a vector) that a charge 'q' at point 'r'
picks up if currents sourcing vector-field 'A' are shut off
By ref[5], near the outer surface of the electron plasma tube (r = R),
the momentum available to electrons, protons, or deuterons is
[e]*|A(R)| = [e] * (u0/4*pi) * ln(2L/R) * I
= (1.6*10^-19 [C]) * (10^-7 [N/Amp^2]) * ln(20) * I
= 4.8 * 10^-26 [C] * [N/Amp^2] * I
{Note that this only depends on the R and L ratio.}
So, the minimum current which can provide a colliding electron (at a
radial distance R) in this arc with 782 KeV is
I = {6.348 * 10^-22 [N*sec]} / {4.8 * 10^-26 [C*N/Amp^2]}
= 1.33 * 10^4 [Amp]
-- [e] = electron charge = 1.6*10^-19 [C], [C] = coulomb
u0 = permeability of free space = 4*pi*10^-7 [N/Amp^2]
ln = natural log, ln(20) ~ 3
[Amp] = [C]/[sec]
Much greater arc currents are routinely achieved [7].
NOTES -
1) Only electrons can acquire significant relativistic mass from
a momentum "kick" in arcs due to their small mass.
More massive protons, deuterons or tritons will not gain much mass.
2) The equation for |A(r)| is singular at r=0 (see [5]).
This is not "unphysical" since volume integral is still finite.
It shows that much smaller currents still can produce "heavy electrons"
at the center of current flow, but less frequently.
3) It is not obvious whether inner K-shell electrons of an atom in an
arc can be forced into the nucleus - resulting in "electron capture"
4) Perhaps a similar analysis applies to currents in emulsions of metal
particles in dielectric fluids [8].
5) Widom-Larsen also calculate the collective magnetic force using the
"Darwin Lagrangian" which includes pairwise magnetic energy between
electrons.
REFERENCES -
[1] (p. 29) "A Students Guide to Cold Fusion"
http://lenr-canr.org/acrobat/StormsEastudentsg.pdf
[2] "Electron mass shift in nonthermal systems"
http://arxiv.org/pdf/0801.3810.pdf
[3] "Feynman Lectures on Physics" Vol.3, Ch.21 (p.5)
http://www.peaceone.net/basic/Feynman/V3%20Ch21.pdf
[4] "On the Definition of 'Hidden' Momentum" (p.10 - note cgs units)
http://hep.princeton.edu/~mcdonald/examples/hiddendef.pdf
[5] UIUC Physics 435 EM Fields & Sources - LECTURE NOTES 16 (p. 8)
http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/P435_Lect_16.pdf
[6] Accelerating Voltage Calculator
http://www.ou.edu/research/electron/bmz5364/calc-kv.html
[7] "EXPERIMENTAL INVESTIGATION OF THE CURRENT DENSITY AND THE HEAT-FLUX
DENSITY IN THE CATHODE ARC SPOT"
http://www.ifi.unicamp.br/~aruy/publicacoes/PDF/IfZh%20current%20density%20and%20U.pdf
[8] AMPLIFICATION OF ENERGETIC REACTIONS - Brian Ahern
United States Patent Application 20110233061
http://www.freepatentsonline.com/y2011/0233061.html - EXCERPT:
<<Ultrasonic amplification may have usefulness, but it is inferior to
are discharges through nanocomposite solids due to a process called the
inverse skin effect. In ordinary metals, a rapid pulse of current
remains close to an outer surface in a process referred to as the
skin effect. Typically, the electric current pulses flow on the outer
surface of a conductor. Discharges through a dielectric embedded with
metallic particles behave very differently. The nanoparticles act as a
series of short circuit elements that confine the breakdown currents to
very, very small internal discharge pathways. This inverse skin effect
can have great implications for energy densification in composite
materials. Energetic reactions described fully herein are amplified
by an inverse skin effect. These very small discharge pathways are so
narrow that the magnetic fields close to them are amplified to
magnitudes unachievable by other methods >>
Comments/criticisms are welcome.
-- Lou Pagnucco
