Document

An accompanied Long Range
Potential in NN interaction
~Efimov-like structure~
Shinsho Oryu
Department of Physics, Tokyo University of Science
APFB11, August 22, 2011 Sungkyunkwan University (SKKU) , Seoul Korea
Introduction
1)Our aim is to investigate the NN - interaction
by the three - body NNπ system.
-q
q
q'
-q'
2) The three - body AGS equation is reduced to
the coupled channel two - body Lippmann Schwinger ( LS ) equations.
I. Reduction of AGS - Born to two - body potential
Z n , m ( q, q'; E ) 
with   1 
m
 g n (p)m g  m (p')  ,
qq '(  ' x )
 1    1  0.147
M
2m E  q 2  q '2
 2  q 2  q '2
'

  
2 qq '
2 qq '
  q  q'
2m E
; 
;  
qq '
2 qq '

Since the form factors are monotonic functions
x  qq'
2
2
2
2
comparing to Green's function at the low energy,
Since the Green's function with   1  
(   m / M  0.147) ;
(  '  x )  1  (    x )  1  (     x )  1
then  expansion of Green's function leads,

(   ) j ( 2  q 2  q '2 ) j
Z n , m (-q, q'; E )  2C n , m 
2
2 j 1
[


(
q

q
'
)
]
j 0
This is a two - body like potential with energy
dependence.
Introduce Fourier tranceform;
 (R )C n , m
F { Z n , m (-q, q'; E )  
U ( ,  ; r )
4 (2   )
1  r / 2    ( 1)  r /2
U ( ,  ; r )  e


2 e
r
 2    1!2
2

(

1)
( r / 2  1)  r /2



e

3
2!2
2
2
3
2 2

(

1)
(

r / 2  3 r  6)  r / 2



e

5
3!2
2
3
 .....
 U (0) ( ,  ; r )  U (1) ( ,  ; r )  U (2) ( ,  ; r )  ....
The two-body potential reduction has the
energy dependence.
For the standerdization, we adopt
a statistical average with a weight :
 2 1e  a
P


  
2 1  a
e
0
L U (0) ( ,  ; r ) 
1

(2  2)
d 
2  2
a

2 1  a

 e
0
2  2
e  r /2
d
r
a

2  2
r(r / 2  a)
2 1  a

e
Weight function
denotes

nucleon structure (or form factor) effects.
1) Van der Waals type :  e
by   3
and for 2a  a0
2
5
a
0
L U (0) ( ,  ; r ) 
5
r ( r  a0 )
2 1  a
a05
 6
r
e 5r / a0

r
4  a
 e
for r  a0
for r  a0
2) Monotonic : 
2 1  a
e
 1e
 a
a0
L U ( ,  ; r ) 
r ( r  a0 )

e
2
( with 2a  a0 )
(0)
 0 r
by   1
( for a0  r with 0  1 / a0 )
r
a0
 2
( for a0  r )
r
2 1  a
3) Yukawa potential :  e   (  2 0 )
L U ( ,  ; r ) 
(0)
e
 0 r
r
Therefore the energy independent r - space
potential by AGS  Born term is obtained
by Fourier - Laplace transform;
LF
Z
 n , m
(-q,q'; E )  V0L U ( ,  ; r )
(0)
 V ( ; r ) 

V
(k )
( ; r )
k 0
where the lowest order potential is
V0a0
(0)
V ( ; r ) 
r ( r  a0 )
II. Solution of the long range potential
at large distance :
The Schrodinger equation is
 d 2  2  2  1 / 4 
 2  
 l (r)  0

2
r

 dr 
with  2   2  2mE / 2
2m V0 a0
1 2
   
 (l  )
2
2
The solution is the modified Bessel function;
2
2
 (r )   rZ (i r )   rK i ( r )
Boundary condition leads
2  C  n  / 
 e
 n
a
2
 n  2 2 C /   2 n / 
En  

e
e
2
2m  ma

then
En 1  En e
 2 / 
Energy sequence
rms radius is
 /
rn 1  rn e
sequence reversed
where
2m V0 a0
1 2
 
 (l  )
2
2
1 2
 1.7435  (l  )  0
2
only l  0 satisfies, and we obtain
  1.2221
( while   1.0 is the Efimov ' s condition )
Then
2
e
2 / 
 170.98
 /
e
 13.076
En
 e 2
En 1

 170.98
En
En 1 
 5.85  10 3 En
170.98
E1  2.226 MeV
E2  5.85  10 3  2.226 MeV  13.0 keV
E3  5.85  10 3  13.0 keV  76.14 eV
E4  5.85  10 3  76.14 eV  0.44533 eV
E5  5.85  10  0.44533 eV  0.002605 eV
3
E6  5.85  10 3  0.002605 eV  1.5236  10 5 eV
E7  5.85  10 3  1.5236  10 5 eV  8.9108  10 11 eV
r
r
n
n 1
e
 
1

13.076
r
n 1
 13.076 r
n
r 1  1.97  10 m
15
r
r
r
r
r
r
2
 13.076  1.97  1015 m  2.576  1014 m
 13.076  25.76  10 m  3.37  10 m
15
3
 13.076  3.37  10 m  4.40  10 m
13
4
5
7
12
 13.076  4.40  1012 m  5.76  1011m
 13.076  5.76  10 m  7.53  10 m
11
6
13
10
 13.076  7.53  1010 m  9.85  109 m
Bohr
radius
Numerical calculation by Schroedinger equation:
MeV
fm
Our analytic prediction fits to the numerical solution.
q  q'
 q  q
q
q
 q  q'
It is known that a pion-transfer creates an attractive
short range Yukawa potential, however, it missed
out the long range potential accompanied with.
When   0, no multi - bound states exists,
2
however, 1 / r 2 potential is accompanied.
In pp - system, the strength of accompanied
potential is 137 times of the Coulomb,
2
e
1
V0 1
C
NN
V (r )  
;
V r  2  2
r 137r
r
r
15
13
 r  137  10 m  1.37  10 m
Both potentials are coparable at 1.37  1013 m
which is 100 times of normal nuclear range.
It may contribute to the fusion problem.
Conclusion and discussion
1) In the present calculation, only the pion-transfer
AGS-Born term is adopted.
The pion-nucleon form
factors are taken as a constant, but leading terms
are taken into account. Therefore, the results are
qualitatively correct.
However, the detailed
calculation will be presented before long.
2) Our predictions are not observed yet,
for the binding energy lesser than 50keV,
for the phase shift lesser than 50keV.
Proton-proton phase shift

Phase shift of 3 S 1 -state

0
E

Levinson's theorem :  l (0)   l ()  n
  50 keV ?
3) The present theory will complement the stereotype
idea that the NN-interaction is a short range.
4) The long range force is created by the particle
transfer structure. This phenomena will appear in
the other few-body systems.
5) The results will contribute to the study for the neutron
rich nucleus and halo problem in the light nucleus.
6) In the zero energy bound state, the n-p probability
will distribute up to the atomic region.
7) The phenomena will contribute to the nuclear fusion.
N  N ;
( N  ...)  ( N  ...)  
Three-body
Efimov states
Three-body break up threshold
N  ( N  );
( N  ...)  ( N  ...)
Two-body
Present case
D ( NN  );
D ( NN  ...)
Two-body threshold
One-body
8) Zero energy bound states in NN systems could
produce a special nuclear material which has a
different phase from the well known nuclear matter.
9) If the nuclear material will be equilibrium in the
universe, it could take a typical aspect with some
heat radiation. (0.0eV  E  0.0025eV [~250K])
10) Classical three-body problem predicted new planets,
however what we learn from the quantum three-body
problem?
These phenomena could be one of them.
q  q'
 q  q
q
q
 q  q'
Yukawa predicted that a pion-transfer creates an
attractive force, however, he missed out the fact that
the long range potential accompanied with the
Yukawa potential.
When  2  0, no multi - bound states exists,
however, 1 / r 2 potential is accompanied.
For pp - system, strength of NN - force is
137 times of the Coulomb,
2
e
1
V0 1
N
V (r )  
;
V r  2  2
r 137r
r
r
15
13
 137  10 m  1.37  10 m
This range is 100 times of usual nuclear
force where NN force and Coulomb force
are comparable.
It may contribute to the fusion problem.
C
Our NN   system
En
2
e
En 1
r
r
n
n 1
e

 170.98
 
1

13.076
Hydrogen atom
2
1
En  
2
2
2mr0 n
11
r0  5.29  10 m
En  n  1 


En 1  n 
3
E5  2.61  10 eV
r
E6  1.52  105 eV
r
11
E7  8.91  10 eV
r
2
11
5
6
 5.76  10 m
 7.53  1010 m
9
7
 9.85  10 m
Accompanied potential in the many-body
(three-body) system.
The three-body point of view of the ionic bond.
Since   0 , then no
r
2
accompanied potential.





Zero mass particle transfer
Covalent
bond：
H
+
H
H2
e
e
H
H
+
H
H
e
H
-
H
e
H
-
H
Accompanied potential exists, because   0
Nuclear
force
N
q'
q
N

N
 q'
q
N
2
N
E  q 2m  q'
2

N
2
f
2
2m  (q  q' ) 2m
1 / r 2 -type accompanied potential
exists by   0.


C


L F Z   q , q' ;E  


 4  2   
 a


1


2
a





0
0


.....



2 
2
 r  r  a0  1!2  2     r  a0 

C
a0
C
a0

4  2    r  r  a0 
4  2    r 2
 for
a0  r   
C
a0
C
e  r a0

 for r  a0 
4  2    r  r  a0 
4  2    r
Yukawa-type potential accompanies
1 r 2 -type potential.

Z   q , q' ;E   2C 
j 0
     q  q'
j
2
2
  (q  q' ) 
2
2
2

j
j 1
m
2m E
2

,  
m
1 
Fourier tranceform to obtain ｒ-space rep.


F Z   q , q' ;E 
U ( ,  ; r )
C
4 (2  )
Energy dependent potential !!
Laplace transform with weight gives an energy average.
Our NNπ-system
Case of Hydrogen atom
2
En
 e 2
En 1

r n
 e 
r n 1
1
En  
2mr0 2 n 2
 170.98

r0  5.29  1011 m
1

13.076
En  n  1 


En 1  n 
E5  2.61  10 eV
r
E6  1.52  10 eV
r
E7  8.91  10 eV
r
3
5
11
2
5
 5.76  1011 m
6
 7.53  1010 m
7
 9.85  10 m
9
The solution and the binding energy
of 1 r 2 -type potential
The solution is the modified Bessel function. By the
boundary condition, uncountable energy levels are
obtained
En
2
e
En 1
Uncountable energy levels are
concentrated near at zero
energy.

r n

e
r n 1

Rms-radius incleases as far as the
atomic-moleculer region，and
affects the chemical reaction.
Z n , m (  q, q '; E ) 
'
 g n ( p )m g  m ( p ')  ,
2m E   q 2    q '2
2qq '
qq '(  ' x )

 2  q 2  q '2
2qq '
 2  q 2  q '2
2qq '
  

  1  m m  1   ;    1  m m  1   
with
m  m  M ; m  m
then       1   ;   m M
3) Thus the three-body calculation needs
a special treatment.
4) Therefore we used the Fourier transformation of
AGS-Born term to obtain the r-space three-body
effective potential.
5) We obtained the r-space potential with the energy
dependent range.
6) We introduced a typical average method for the
energy dependence.
The method is a kind of Laplace tranceformation.
7) We found that the Yukawa type potential plus the
long range potential.
Possibility of new fusion
１）confinement by laser
  D2  D2    4 He
cooling D2 molecule
２）excitement of deuteron
to zero energy states

D
which have long life time.
D
3）constituent nucleons
D
extend to the molecular


D
orbit, and make singlet
deuteron D*-molecule.
4）D*2 molecule loses
energy and down to the
helium ground state.
D2
( a )(b()b()c()c)

D
(a ) (b) D2 - molecule
D

*
2
D
D
*
D
*
2
*
(b) D - molecule formation
4

He

(c)
D2* -molecule  4 He* -fusion and come down  4 He
尾立先生
ご質問の件、2HのLevelについては、中性子分離エネルギーだけで、構造については
調べられていないのではないかと思います。
もしやと思い、ポリエチレン試料を、冷中性子で照射したスペクトルを見てみました。
添付では、小さなGe結晶で測ったスペクトルです。ノイズを切るために、50kevぐらいで
Discriminationしています。
これ以上下げると、検出器のDead Timeが多くなってしまいますので、過去のデータでは、そ
こまで気にしていませんでした。
先生のご質問の問題を言い換えると、 1Hターゲットを照射して、得られたガンマ線スペクトル
で、 2.2MeVガンマ線と、12.7keVガンマ線は、同時事象かどうか、を調べることだと思います。
2.2MeVガンマ線にゲートをかけて、Coincidenceするガンマ線が何か見えるだろうか、また、
構造があれば、第一励起にいくパスがありますから、2224.57keV-12.7keV=2211.87keVに
ピークが現れるか、ということになります。
検出器の分解能のせいで、2.2MeVのピークが2山やら、テールにコブでもあれば、おやっ？
と思うのですが、手持ちのデータでは、はっきりしませんでした。
Ｄ
Ａ
Ｂ
Ｃ

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