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Homework answers / question archive / When we developed Gamow's theory of alpha emission, we assumed the alpha particle was preformed in the nucleus
When we developed Gamow's theory of alpha emission, we assumed the alpha particle was preformed in the nucleus. there is some evidence that nucleon clusters continually form and breakup in the nucleus and that alpha emission requires the formation of an alpha cluster at the nuclear surface prior to emission. Using classical (i.e. not quantum mechanics) arguments, come up with an estimate for the probability of an alpha particle being formed at the surface of a nucleus. Calculate your probability for a nucleus with A=238
First, in the classical fashion we take a nucleus to be a sphere filled with A nucleons, each of volume V1 so that the whole sphere has volume
V = A*V1
and radius
(3V/4pi)^{1/3}
In this model the 238 nucleus has radius
R = (3*238*V1/4pi)^{1/3}
and the alpha particle has radius
r = (3*4*V1/4pi)^{1/3}.
Now we say that alpha particle is at the surface of the larger nucleus if its center is within its radius r from the surface of the larger sphere of radius R.
The volume of the shell of thickness r at the surface of the larger sphere is the difference between the volume of the whole sphere and the volume of the sphere below the shell:
(4pi/3) [R^3-(R-r)^3]
Since the volume of the sphere is 4piR^3/3, the probability of the alpha particle to form at the surface is the ratio of the volumes of the surface shell and the whole volume:
p = { (4pi/3) [R^3-(R-r)^3] } / { (4pi/3) R^3 } = 1 - (1-r/R)^3
= 1 - [ 1 - (4/238)^(1/3) ]^3 = 0.6
