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the Triteron, the H3 Nuclide |
The triteron, consisting of two neutrons and one proton, will be presumed to be a neutron-neutron spin pair and a proton revolving about their center of mass. In order to keep matters simple and yet capture the essentials of the physics of the model it is assumed that the masses of the neutron and proton are equal. This mass is denoted as m. The axis of rotation is presumed to be parallel to the axis of the neutron-neutron spin pair. The rate of rotation is denoted as ω. The separation distance between the centers of the neutrons is a crucial parameter, but it shows up only in the form of one half of its value. This half distance will be denoted as u. The separation distance of the neutrons is established by the force of repulsion between them at the limit of the formation of spin pairs.
Let s denote the distance between the center of the proton and the midpoint of the line between the centers of the neutrons. This distance can be characterized as the width of the triteron. The radius of rotation of the proton is then (2/3)s and the radius of rotation of the neutron pair is (1/3)s. Therefore the angular momentum L of the triteron is
Angular momentum is quantized so
Thus
The centrifugal force on the proton is
The distance between the proton and one of the neutrons, z, is given by
The force F on the proton due to one neutron is
The radial component of the force on the proton due to the two neutrons is therefore
Dynamic balance requires that
Therefore
Thus two expressions for ω² have been derived. When they are equated the result is
Since z²=s²+u², s²=z²−u² and thus
Thus
This is the quantization condition for z. In terms of ζ=z/z_{0} it is
where ζ_{0}=u/z_{0}.
The form of the function on LHS of the quantization condition is:
In the above graph ζ_{0} is set at 1.5.
Let the above function be denoted as Γ(ζ) and thus the quantization condition for ζ is
where φ is equal to (9/8)h²/(mz_{0}H).
The quantization condition for the width s of the triteron can be found from the relation s²=z²−u². Better yet let σ=s/z_{0} and hence
In the function Γ(ζ), ζ may be replaced by the above expression to give Π(σ) so the quantization condition for the width of the triteron is
The shape of Π(σ) is shown below.
Once the quantized values of s are known the values for ω are found through the relation
The kinetic energy K of the triteron is given by
Since ω² is equal to (9/4)n²h²/(m²s^{4})
(To be continued.)
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