In most cases the quantization factor (lambda) is usually multiplied by the nuclear potential, what happens when the factor is "in" the nuclear potential.
The normalization factor is obtained by solving, numerically, the Bohr-Sommerfeld quantization and Wildermuth rule. It is a non-linear problem that can easily be solved using regular algorithms like bisection method or any non-linear equation solver in modern programming languages. The value is usually between 0.1 and 1. If you use a bisection method, you can search for the solution in that range [0.1,1]
Prof. Ibukun David Olusola
In addition to the previous interesting answer, I tried to understand your question, even I am not a nuclear physicist, it is about the use of BS quantization for nuclear models (using the old quantum theory), I would say that it was popular during the time of the quarks model in particle physics (the 60s-70s).
Please look at the following review paper, it is in the Russian language but for sure was translated, and in addition, it gives a very nice review of the problem, I read only a short part of the review :
Resonances in subatomic physics and the principles of similarity and dimension by F. A. Gareev, G. S. Kazacha, and Y. L. Ratis. Physics of elementary particles and nuclei, 1996, Vol 27(1), pp. 97
http://www1.jinr.ru/Archive/Pepan/1996-v27/v-27-1/3.htm
УДК 539.126; 539.125; 539.164.2 Резонансы в субатомной физике и принципы подобия и размерности. Гареев Ф. А., Казача Г. С., Ратис Ю. Л. Физика элементарных частиц и атомного ядра. 1996, том 27, вып. 1, с. 97
http://www1.jinr.ru/Archive/Pepan/1996-v27/v-27-1/synopses.htm
I unquote & summarize part of the abstract: "the authors stress the physical analogy between alpha-decay in nuclear physics and general decay of resonances in physics. On the basis of this analogy, the authors reproduce, some widths for a number of hadronic resonances, in addition, perspectives of the further investigations are also discussed".
Best Regards.
There are several ambiguities in the question.
When we talk about nuclear potential, there are generally two types of meanings:
1) Nuclear forces between nucleons (many-body forces)
2) Average nuclear potential or average field of a nucleon which is the "average" sum of the nuclear forces created by the rest of all the other nucleons (one body force).
3) Bohr Sommerfield's quantfication is a pseudo quantum mechanics. These quantization rules are not based on "real" quantum mechanics.
I have difficulty in placing the context of your question
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