Dear Colleagues!
I have a question about the penetrability of the Coulomb barrier.
In some books (Blatt&Weisskopf, for example), it is defined as 1/(F^2+G^2). Here F and G are the regular and irregular Coulomb functions. However, in a number of works and textbooks, penetrability is called a quantity kR/(F^2+G^2). (as we usually have for calculating the reduced width).
This value, as it is easy to see, can be more than 1 and is not a penetrability from the point of view the ratio of the transmitted flow to the falling one. Where did this name come from?
With respect,
Sergey Torilov
In quantum-nuclear physics, the Coulomb barrier penetration factor or penetrability rigorously defines as the ratio of the squared modulus of the wave function at the target nucleus radius, Rn, corresponding to the Coulomb barrier energy, Ec = (Z1* Z2) e2/ Rn, over the same quantity evaluated at the classical turning point, Rc, i. e., the distance at which an incoming ion of charge Z1e and energy E lower than Ec is repulsed back. In these conditions, the penetrability is expressed as
P(E, Rn) = K R/(F2 +G2) with K being the internal region wave number, and is always lower than unity. It tends to 1 while R approaches Rn or E approaches Ec.
Dear professor Saad Ouichaoui,
Thank you very much for the answer!
However, I would like to clarify one point. The Coulomb functions take into account not only Coulomb interaction, but angular momentum as well.
We have the same equation for neutron too. Let us to consider the partial wave expansion for neutron wave. For s-wave we will have kR/(sin()^2+cos()^2).
Way we have the condition for kR1 for E>1 MeV.
Best regards,
Sergey Torilov
Just a last comment on this important question, very well treated in Illiadis book "Nuclear Physics of Stars" for nuclear astrophysics applications where the penetrability is approximated by the Gamow factor.
The quantum definitions from classical books you have cited are obviously correct. One must just have in mind that they relate to sub-Coulomb barrier nuclear reactions. For charged particle incident énergies higher than the barrier, the penetration factor is indeed higher than unity, as well accounted for in the definitions.
Best regards,
Saad.
Dear professor Saad Ouichaoui,,
Thank you for your comment!
I would like to clarify my question. For the simplest square potential well, the solution for the Coulomb potential in the case of a purely outgoing wave is well known. It is 1/(F^2+G^2).
And of course there is no kR multiplier. This value is
Hi, Sergey!
You are quite well informed regarding the various formulations of this point in the literature. Exploring again the latter, I found no peculiar model associated with the definition involving kR, i.e., energy E of the incoming particle towards the Coulomb barrier (in case of a charge particle ), whatever can be the shape of the potential (Woods-Saxon in Vogt's treatment). The expression kR/(F2 + G2) is not an a priori set definition but a result of computation. Then the only effective quantal-nuclear model is the barrier penetration by tunnel effect, meaning that part of the incident waves go inside the potential well, that is the transmission probability or "penetrability". And as you surely know, the transmission coefficient has different expressions depending on the interaction forces (potentials) in presence.
Best regards,
Saad.
Dear professor Saad Ouichaoui,
Thanks for the interesting discussion. I think that from time to time it will be useful for us to recall the simple models that appeared at the very beginning of nuclear physics.
Best regards,
Sergey.
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