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244) Coulomb barriers etc.

Ludwik Kowalski (7/30/05)
Department of Mathematical Sciences
Montclair State University, Upper Montclair, NJ, 07043

A discussion list, CANA, has been created for members of ISCMNS (International Society of Condensed Matter Nuclear Science) four days ago. The purpose is to discuss scientific and technical aspects of cold fusion. Several interesting messages have already been posted. What follows is my own message. I hope that other nuclear physicists will comment on it. Will this generate some food for thoughts? I hope so. And I will write about this here, accordingly.

I read the paper recommended by Dennis with great interest. The phrase -- “the effective target area” -- made me think about the coulomb barrier. Why about the coulomb barrier when the bombarding particles are neutral? I will explain this shortly. But before going ahead let us look at something else. The 1 / v law, mentioned by the authors, implies that the apparent size of a nucleus depends on the projectiles with which a nucleus is bombarded. This fact has been known since the beginning of nuclear era. The unit of cross section “barn” was introduced to recognize unexpectedly large cross sections. In the first nuclear reactor (Fermi’s pile, 1942) blocks of pure graphite were used to slow down neutrons to benefit from large cross sections at small v. Yes, most CANA members know this; but being a retired teacher I can not miss an opportunity for lecturing.

If the term “effective target area” is accepted then the term “effective coulomb barrier” should also be accepted. I am saying this because the height of the coulomb barrier and the size of a nucleus are intimately related. To justify this one must go back to the well known definition of the coulomb barrier, CB. Coulomb barrier is the value of potential energy (of two interacting positive particles) at a certain distance between their centers. The potential energy, V, consists of the sum of two terms, V1 and V2. The positive term, V1, called coulomb potential, is due to electric repulsion, the negative term V2, called nuclear potential, is due to attractive nuclear forces. The dependence of V1 on d (distance between the centers of two particles) is of the 1/d type, according to Coulomb’s law. The dependence of V2 on d is usually much more complicated. In textbooks that dependence is often represented by deep potential wells.

Let me take a crude rectangular well (rather than a more realistic well introduced by Yukawa, by Saxon and Wood, etc.). In that model V2=0 everywhere outside a volume of radius R. At d<R the absolute value of V2 suddenly becomes much larger than the absolute value of V1. In other words, CB is the value of V1 when D=R. For many of you this hint is sufficient to see why CB would go down rapidly if R (the effective radius) were allowed to become larger. But let me illustrate this numerically for a two body system -- a Gd nucleus and an alpha particle. For a gadolinium R is close to 7.5 F. The potential energy of the system, at d>R, can be calculated on the basis of Coulomb’s law (where k stands the familiar 1/4*Pi*epsilon-zero):

V = V1 = k*Z1*Z2*e2 / d =9*109 *2*64*2.56*10-28 / d

where d is in meters and V is in joules. In practical units (F and MeV ) one has:

V (in MeV) = 1.44*Z1*Z2 / d (in F)

Replacing d by R=7.5 F, and knowing that Z1*Z2 is 128, one finds that CB = 24.5 MeV. A more realistic potential would give a lower CB, perhaps slightly below 20 MeV, but that is not important in the context of this message.

What would happen to CB if R were doubled? In that case the coulomb barrier would be reduced by the factor of two. The paper mentioned by Dennis refers to much larger changes in the “effective R” (several orders of magnitude). That, according to the above speculation, would nearly eliminate the coulomb barrier. Yes, I know that the speculation is silly; the value of R (beyond the region occupied by neutrons and protons) is determined by the range of nuclear forces. The large effective R, as far as I know, does not correspond to unusually large range of strong nuclear forces. Cross sections of nuclear reactions induced by charged particles, even for gadolinium, are never as large as for reactions induced by slow neutrons.

So what is the point of all this? In my mind this speculation is connected with the 1/ v law. That law describes cross sections induced by slow neutrons in the non-resonance regions of v. The common explanation -- “effects of interactions become stronger at small v because times of interactions become longer” -- does not satisfy me. Why not? Because i know that alpha particles approaching a nucleus, in central collisions, also have very small v. In fact, v becomes zero when the distance of minimum approach is reached (alpha particle starts bouncing back). That kind of slowing down does not result in high probabilities of fusion. Why do slower neutrons fuse much more frequently than faster neutrons? Why slowing down of charged particles (for example, at d<10*R) does not increase probabilities of fusion with charged particles? These two questions must be related. “Spending more time near the nucleus” should have the same effect on all projectiles, charged and uncharged.

I would be happy to know how to solve the puzzle without leaning on theoretical concepts beyond classical physics. Please, be aware, that I am not addressing the issue of tunneling effect; nobody ever tried to explain this effect, as far as I know, by using classical physics. But the explanation of the 1/v law, for example in the article chosen by Dennis, appears to be classical. That is my main point. Let me end with a quote extracted from the <> article. “A slow-moving neutron spends more time near a target nucleus than a fast-moving neutron, so it is more likely to interact. That makes the cross section -- the nucleus's effective ‘target area’--larger for slower neutrons.” Thanks for providing this URL, Dennis.

A saying “one picture is often worth more than many words” is applicable. But CANA messages cannot contain pictures; they are for text only. To create a picture follow these steps:

a) Using the last formula above (and several values of d between 2 F and 20 F), plot positive V1 (in MeV) versus d.

b) On the same graph plot negative V2 versus d, according to a rectangular well potential. In doing this assume that R=7.5 F and that for d<R the absolute value of V2 is “moving toward infinity.”

c) Calculate V = V1 + V2, for several values of d (also between 2 and 20 F), and plot V versus d, for example, in red.

The red curve will have a peak at d=R. The height of that peak is called coulomb barrier, CB. It is not difficult to see what effect a change in R should have on CB. For much smaller R, the CB would be much higher; for much larger R, the CB would be much smaller. Coulomb barriers are often compared with gravitational barriers associated with mountains and hills (as illustrated in Figure 1 of unit # 40, on this website).

Appended on 8/xx/2005:

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