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 1015 m 2.576 1014 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 1012 m 5.76 1011m 13.076 5.76 10 m 7.53 10 m 11 6 13 10 13.076 7.53 1010 m 9.85 109 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 1013 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 1.52 105 eV r 11 E7 8.91 10 eV r 2 11 5 6 5.76 10 m 7.53 1010 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 1011 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 1011 m 6 7.53 1010 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
© Copyright 2024 ExpyDoc