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*From*: Ludwik Kowalski <kowalskil@MAIL.MONTCLAIR.EDU>*Date*: Sun, 15 Feb 2004 12:01:07 -0500

1) In electrostatic you might ask students

to estimate the coulomb barrier preventing

positive D ions to fuse at low temperatures.

Here is my suggestion. First tell students

that in addition to repulsive forces the

ions attract each other by very strong

nuclear forces. But these forces do not

obey the 1/r^2 law. They are negligibly

small when r is above a distance R (called

range). For x<R the attractive nuclear

forces are much larger than repulsive

electric forces. That would be a good

justification for defining the coulomb

barrier, CB, as the value of the electric

potential at r=R.

The rest is trivial. Assume that R=3 F,

for example, and calculate CB. Note that F

is the unit of length (femtometer or 10^-15 m).

My answers were 478 kV and 358 kV, at R=3 F

and 4 F, respectively. You may or may not link

this problem with the cold fusion controversy.

2) If you do then consider addressing the

screening effect. Some scientists say that

screening is possible

when D ions are embedded

in metals, such as Pd or Ti. Simply stated,

and without trying to argue about what causes

screening (local clouds of electrons at crystal’s

boundaries?) one can simply declare: screening

consists of lowering of the coulomb barrier by

nearby electrons.

To illustrate screening do the following.

Place one deuteron at x=0 at treat the other

deuteron, at x>0, as a probe charge. That is what

I did to calculate coulomb barriers. Then place

an electron at some negative value of x, for

example, -2 F. The CB is now V=V1 + V2 (where

V1 is the positive part due to the deuteron

and negative V2 is the negative part due the

electron). You will see that CB approaches

zero when electron is approaching the origin,

as it should be. My answer, for R=4 F, was

CB=120 kV for the electron placed at x=-2 F.

And nothing prevents you from introducing more

than one screening electron.

4) Yes, I know that three or more particles would

usually not be at rest

on the x axis. And I know

that the nuclear potential is not a rectangular

well. My goal is to estimate the orders of

magnitude of CB, and to illustrate the idea

of screening.

5) By the way, we usually think that the

so-called “free electrons” in metals are

uniformly distributed, like in ionized

gases. What evidence do we have for this?

6) No, I am not trying to poison your mind

with poisonous and heretical pseudo science.

Ludwik Kowalski

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