Mathematical Principles of Theoretical Physics

4.5. STRONG INTERACTION POTENTIALS 233

4.5.7 Short-range nature of strong interaction

By the layered potentials (4.5.41) for the strong interaction, we see that when the nucleons
form an atom, the nucleon potential is no longer valid, and the correct potential becomes
the strong interaction potential for atoms given by the fourth formula in (4.5.41). The corre-
sponding force formula is given by

(4.5.73) Fa=N^2

(

ρw

ρa

) 6

g^2 s

[

1

r^2

−

Aa

ρa

k^2 are−kar

]

,

where

(4.5.74)

for atom : ka= 10 −^9 ∼ 10 −^10 cm, ρa= 10 −^8 cm,

for molecule: ka= 10 −^7 cm, ρa= 10 −^7 cm.

It is clear that the attractive force in (4.5.73) is of short-ranged. The repulsive force in
(4.5.73) looks as if it is long-ranged. However by (4.5.74) the factors(ρw/ρa)^6 for the atoms
and molecules are very small. Hence, the strong repulsive forces for atoms and molecules
almost vanishes.
In fact, the bound force between atoms and molecules is the electromagnetic force with
strength given by

(4.5.75)

e^2

hc ̄

=

1

137

, ethe electric charge.

Hence at the atomic and molecular scale, the ratio between strong repulsive force and the
electromagnetic force is

(4.5.76)

Fa

Fe

=Ns^2 g^2 s

(

ρw

ρa

) 6

/Ne^2 e^2 ,

whereNsis the number of strong charge, andNeis the number of the electric charge. Note
that each nucleon has three strong charges, and the protons are almost the same as neutrons.
Therefore, we assume that
Ns= 6 Ne.

In view of (4.5.70), the ratio (4.5.76) becomes

Fa

Fe

= 4 β^2

(

ρn

ρa

) 6

g^2 /e^2 , β^2 =

2

8

√

e−e

⋍ 0. 2.

By (4.5.58) and (4.5.75), we have

g^2

e^2

is in the range of

1

137

∼

1

13. 7

.

Then, byρn= 10 −^16 cm and (4.5.74), we derive that

Fa

Fe

∼

{

10 −^50 at the atom level,

10 −^56 at the molecular level.

This clearly demonstrates the short-range nature of the strong interaction.