On the Nuclear Saturation

0 Institute of Theoretical Physics, Nago..'Va University TammDancoff method is applied to the bound system of many particles, on the view point of the meson theory. The obtained results seriously modify the usual method in which the many body problem is treated with the simple addition of the two body potential. It is shown that this modification has a possibility to give the nuclear satqration. The; usual attempts to resolve the difficulty on the nuclear s~turation have been developed in the following two ways: (i) the modification of the two-body potential (or, two body interaction), or (ii) taking account of the interaction other than (i). On the point (i), it is well known that Majorana force leads to the nuclear saturation. However, Majorana force is not strong enoughl ), as indicated in the experiment of the low energy nucleon-nucleon scattering. The hard core2) suggested in the high energy proton-proton scattering experiment cannot resolve the problem3), because it is far less effective to the neutron-proton force due to its short range. The situation remains unchanged, even if the tensor force is included4). These attempts have been all based on the assumption that the nuclear force effective in the nucleus is the simple addition of the two-body force. In this paper, a criticism against these usual methods will be given on the view point of the method developed by Tamm and Dancoff5) in the meson theory. Before entering this. problem, we shall discuss briefly on the point (ii). It has already suggested6l that the many body force** will play an important roll in the nucleus, while the concrete analysis has been left unchallenged. Another interesting trial is the introduction of the non-linear interaction suggested from the meson theory7). This interaction makes the nuclear force less effective in higher nuclear density, and gives the binding energy proportional to the mass number A. This view point, however, has the defectS) as follows; if the nuclear force is damped in the region of high nuclear density, the force acting on the nucleon which traverse through the heavy nucleus would be quite different from the one given in the lighter nucleus (the most simple case is two-body Introduction *) The author is indebted to Yukawa.Yomiuri Fellowship for the financial aid. The derivation of the equation Here, If, is the wave vector of the meson. The total Hamiltonian becomes as H}Jf=ElfT. Through our discussion, the nucleon field In order to rewrite eq. (6), we make use of the relation ~p/ .~iJk(t)=~iJk(t)~P/ -(k2/M)~iJk(t) j 2M I I j 2M I (,~ p/ -E )1Jf(O)=2J2J(l/W,,) Hk (i)Hk (t) .1Jf(O) 'j=12M i,lk -~2J(l/Wk)lfk(Z)(l/~P/ +Wi,.-E)Hk(t)( p/ -E).IJf(O) i,lk 2 M 3=12M S. Ogawa and S. Marumori In eq. (10) and the [ 1 + '2..2:, (l/(J)k) Hk(i)i-i k(l) /(~ p/ +wl<.-E)](2:., p/ -E)Ip"( 0). ki,1 zM 2M =~2:,(l/(I)l,,)Hk(i) . Hk(l) IF (0). k i 1 we can obtain the equation which coincide with the umal Schrodinger equation derived in the adiabatic approximation. In the deuteron problem, the expectation value of each term in eq. (11) would become as ~ lEI ~Z-3 Mev. and we obtain the validity of the usual treatment ill the resultant. From our view point we can see that the usual treatment or the derivation of eq. (lZ) is based on the follow ing two assumptions; Now, returning to the eq. (5), we find that the quantity (2J pf/zM- E ) <Z.Wk (V) <z'Wk. l/(2J-;~+wk-E) Accordingly, we get G,Z the Nuclear Saturation Binding energy and saturation *) It should be noted E<O in the bound system. [ 1+(V) (E ) 1 _ _ E][(Ek".>-EJ=(V), kin +w,. ~ If we make use of the approximation (ii) in the preceding paragraph, we find eq. (16) takes a usual form of the binding. energy as E= (Ekln) - ( V), Nucleon-nucleus collision ( 102 +KA-E)'fT= ~ V(ij) .(tJ( P02 +KA+iotr.-E) ./jr. zM i' 3~0 zM into the following -two terms, and assume A A V=:E V(ij) + ~ V(O,j) = VA + Vo' i, .1=1 j=.1 Oil the Nuclear Saturation Let X(Eo) .ep(E}) be the state of the system in the case where the incident nucleon is infinitely apart from the nucleus, then we get [(pN2M) -h~J2jX(Ei)ep(E}) +[wV/(KA-E}+ (tJ)]. ~X(Ei)ep(E}) - (KA-E}) IJx(i:!.:)ep(E) - [ {(p,,2/2M) -Eo }wV/ (KA-E} + (,j)2J 2:; X(Ei) ep (E}) . *) We are not concerned with the problem of the nuclear disintegration. 2,J[ (Po2/2M- E'][1 + wVI (KA-E}+ (0) 2JX(E;) (Ej) = [(1- (KA-E}) wi (KA- E} + (0) 2] Vo[x (Bo) (Eo) + ~X(E;) (El)J. - (25) In the derivation of this equation, we assume the excitation energy of nucleus to be smaU compared with the binding energy itself, that is, IB}-Ejl <tIE}I, IE}I. Taking account of eq. (5), we can rewrite eq. (27) as *) x'(Eo) is x(Eo) plus the scattered wave with the energy Eo. (Po2/2M-.E:-vii) X(h~) =X(EO)Vio+bX(Ej)Vij, i4'j Vij= ((B~i), 'lI(El. tion from the usual' treatment of the nucleon-nucleus collision, though the conc,rete character of the interaction is too complicated to analyse. Concluding remarks ('2:: p/ +Wl.-E)'-1 -l/F(l!,~ k .p,L, V, V,"",w, (U,',) - 2M 2M References (...truncated)