# SOM Alpha-X Dumbbell Models

Author: Phil Fraley

Copyright © 2009 by Phillip E. Fraley

Three models with an alpha as one component and either a deuteron (Li-6), triton (Li-7) or helion (Be-7) as the other are stable or semi-stable (Be-7 has a half-life of 52 days).

## Li-6 Model

The Li-6 dumbbell model is shown in Fig. 8.
Figure 8

Li-6 ModelThe radius of the black circle

is the Li-6 radius.

larger image

r = 0.853 6^{1/3} = 1.550 fm

4 M 2r 1/3 v_{1} + 2 M 2r 2/3 2v_{1} = √2ћ

Dividing by M r_{p} c = ћ: 8 r/r_{p} v_{1}/c = √2

v_{1} = 0.0240 c. v_{2} = 2 v_{1} = 0.0480 c

kinectic energy: 1/2 4 M v_{1}^{2} + 1/2 2 M v_{2}^{2}

Dividing by M c^{2} = 938.259 MeV yields 3.229 MeV total

Figure 9

Li-7 or Be-7 ModelThe radius of the black circle

is the Li-7 or Be-7 radius.

larger image

## Li-7 and Be-7 Models

The Li-7 dumbbell model is shown in Fig. 9 which also applies to Be-7.
r = 0.853 6^{1/3} = 1.632 fm

4 M 2r 3/7 v_{1} + 3 M 2r 4/7 4/3v_{1} = √2ћ

Dividing by M r_{p} c = ћ: 8 r/r_{p} v_{1}/c = √2

v_{1} = 0.02275 c. v_{2} = 4/3 v_{1} = 0.03033 c

kinectic energy: 1/2 4 M v_{1}^{2} + 1/2 3 M v_{2}^{2}

Dividing as above yields 2.265 MeV total

There are five 'dumbbell' models: deuteron, Li-6, Li-7, Be-7 and possibly Be-9, Be-9 to be discussed later. Li-7 is the only one without an soe+ and also is the only one where the magnitude of the binding energy (2.47 MeV) is approximately equal to the kinectic energy (expected if the potential is 1/r and is central).

The soe+ radius is 0.7 times the green deuteron radius of Fig. 8 and is only slightly smaller than the green helion radius of Fig. 9.