Mathematical Principles of Theoretical Physics

230 CHAPTER 4. UNIFIED FIELD THEORY

wheregis the Yukawa strong charge, and

(4.5.58) g^2 = 1 ∼ 10 hc ̄.

By (4.5.57) we can deduce the classical nucleon force as

(4.5.59) FY=−g
dΦY
dr

=−g^2

(

1

r^2

+

1

r 1 r

)

e−knr.

It is clear that the nucleon force (4.5.59) is alway attractive, i.e.

(4.5.60)

FY< 0 for anyr> 0 ,

FY→ −∞ forr→ 0 ,

FY/F 1 ≃0 forr> 10 × 10 −^13 cm,

whereF 1 = 2 g^2 /er 12 , andeis the base of the natural logarithm.
Comparing (4.5.60) with (4.5.56), we find that the Yukawa theory has a large error in
0 <r<^12 × 10 −^13 cm. In particular, by (4.5.57) the Yukawa potential can be shown in Figure
4.3.

Yukawa potential r

Figure 4.3: Theoretic curve of Yukawa potential energy

3. Modified Yukawa potential. The nucleon potentialΦnderived by the unified field
   model based on PID and PRI is given by

Φn= 3

(

ρw

ρn

) 3

gs

[

1

r

−

An

ρn

( 1 +knr)e−knr

]

,

and the potential energyVnof two nucleons is

(4.5.61) Vn= 3

(

ρw

ρn

) 3

gsΦn= 9

(

ρw

ρn

) 6

g^2 s

[

1

r

−

An

ρn

( 1 +knr)e−knr

]

.

The nucleon force is given by

(4.5.62) Fn=−
dVn
dr

= 9

(

ρw

ρn

) 6

g^2 s

[

1

r^2

−

An

ρn

k^2 nre−knr

]

,