Mathematical Principles of Theoretical Physics

4.5. STRONG INTERACTION POTENTIALS 231

where

(4.5.63) ρn= 10 −^16 cm, kn= 1013 cm−^1.

4.Parameters Anand g^2 s.First, we use the experimental data in (4.5.56) to determine the
parameterAnin (4.5.62). By (4.5.56) we know that

Fexp=0 atr=

1

2

× 10 −^13 cm.

Hence, let

Fn=0 atr=

1

2

× 10 −^13 cm.

Then, it follows from (4.5.62) and (4.5.63) that

(4.5.64) An=ρneknr/k^2 nr^3 = 8 e^1 /^2 × 10 −^3.

Next, we assume that

(4.5.65) Fn=FY atr 1 = 10 −^13 cm.

Then, by (4.5.59) and (4.5.62), we deduce from (4.5.65) that

9

(

ρw

ρn

) 6

g^2 s

[

1

r^21

−

An

ρn

k^2 nr 1 e−^1

]

=−g^2

2

r^21

e−^1 ,

which leads to

g^2 s=

2

9

ρn

(Anknr^21 −eρn)

(

ρn

ρw

) 6

g^2.

In view of (4.5.63)-(4.5.64) andr 1 = 1 /kn, we derive

(4.5.66) g^2 s=

2

9

e−^1 /^2

8 −e^1 /^2

(

ρn

ρw

) 6

g^2 ,

wheree= 2 .718 andgis as in (4.5.58).
Finally, fromFn=0 we can also deduce that

Fn>0 asr>r 2 ⋍ 9 × 10 −^13 cm.

4.5.6 Physical conclusions for nucleon force

The discussion above leads to the following physical conclusions:

1. The modified Yukawa potential based on PID and PRI is

(4.5.67) Φn=βg

[

1

r

−

8 e^1 /^2

r 1

( 1 +

r

r 1

)e−r/r^1

]

,