Longview,
Below is my calculation of the energy/momentum absorbed from the magnetic
vector potential in a collision between an electron and a positively charged
particle of nearly equal and opposite momenta in an intense current arc
- based on the Feyman reference below ([3] p.21-4 "Two kinds of momentum).
It is just copy of a past posting to Vortex-L. I want to check it again and
revise it with some new data, but time does permit. For simplicity, I
assume spinless particles. Perhaps such collisions occur in arcs, or near
electrodes. The crude Ascii-graphics may have to be viewed with fixed
font-size to appear correctly. It is a bit tedious to read and check:
'Slow' arcing electrons can gain relativistic mass
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).
In an arc, colliding electron-proton(deuteron) wave packets 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
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].
REFERENCES -
[1] (p. 29) "A Student’s 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 Ch21.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.
http://web.hep.uiuc.edu/home/s…re_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 current density and U.pdf
[8] AMPLIFICATION OF ENERGETIC REACTIONS - Brian Ahern
United States Patent Application 20110233061
http://www.freepatentsonline.com/y2011/0233061.html - EXCERPT:
Comments/criticisms are welcome.