THÔNG TIN TÀI LIỆU
236 A.M. Shirokov et al. Fig. 13 3 d 2 np scattering phase shifts. See Fig. 1 for details substitutes the indexes s and d by the indexes p and f in the above expressions and in other formulas in this section. Within the inverse scattering J -matrix approach, the potential in the coupled partial waves is fitted with the form: V = ⌫, ⌫ N ⌫ n=0 N ⌫ n =0 |n⌫ V ⌫⌫ nn n ⌫ |. (34) Here V ⌫⌫ nn ≡n⌫|V |n ⌫ is the potential energy matrix element in the oscillator basis Fig. 14 3 d 2 np scattering wave functions at the laboratory energies E lab = 2, 10, 50, 150, 250 MeV. See Fig. 2 for details Nucleon–Nucleon Interaction 237 Fig. 15 3 f 3 np scattering phase shifts. See Fig. 3 for details |n⌫=R nl ⌫ (r)|⌫, (35) where the radial oscillator function R nl ⌫ (r) is given by Eq. (4) and |⌫ is the spin- angle function. Different truncation boundaries N ⌫ can be used in different partial waves ⌫. The multi-channel J-matrix formalism is well known (see, e.g., [23, 37]) and we will not discuss it here in detail. The formalism provides exact solutions for the continuum spectrum wave functions in the case when the finite-rank potential V of the type (34) is employed. In the case of the discrete spectrum states, the exact solutions are obtained by the calculation of the corresponding S-matrix poles as is discussed in Refs. [17, 18, 34]. In particular, the deuteron ground state energy E d Fig. 16 3 f 3 np scattering wave functions at the laboratory energies E lab = 2, 10, 50, 150, 250 MeV. See Fig. 4 for details 238 A.M. Shirokov et al. should be associated with the S-matrix pole and its wave function is calculated by means of the J-matrix formalism applied to the negative energy E = E d . Within the J-matrix formalism, the radial wave function u ⌫(⌫ i ) (E, r) is expanded in the oscillator function series u ⌫(⌫ i ) (E, r) = ∞ n=0 a n ⌫(⌫ i ) (E) R nl ⌫ (r). (36) In the external part of the model space spanned by the functions (35) with n ≥ N ⌫ , the oscillator representation wave function a n ⌫(⌫ i ) (E) fits the three-term recurrence relation (8). Its solutions corresponding to the asymptotics (32) are a n⌫(⌫ i ) (E) = δ ⌫⌫ i S nl ⌫ (E) + K ⌫⌫ i (E) C nl ⌫ (E). (37) Equation (37) can be used for the calculation of a n⌫(⌫ i ) (E) with n ≥ N ⌫ if the coupled wave phase shifts δ ⌫ and δ ⌫ i and the mixing parameter ε are known. The oscillator representation wave function a n ⌫(⌫ i ) (E) in the internal part of the model space spanned by the functions (35) with n ≤ N ⌫ , can be expressed through the external oscillator representation wave functions a N ⌫ +1,⌫(⌫ i ) (E)as a n ⌫(⌫ i ) (E) = ⌫ G ⌫⌫ nN ⌫ T l ⌫ N ⌫ , N ⌫ +1 a N ⌫ +1,⌫ (⌫ i ) (E). (38) The matrix elements, G ⌫⌫ nn =− N λ =0 n⌫|λ λ |n ⌫ E λ − E , (39) where N = N ⌫ + N ⌫ + 1, are expressed within the direct J-matrix formalism through the eigenvalues E λ and eigenvectors n⌫|λ of the truncated Hamiltonian matrix, i. e. E λ and n⌫|λ are obtained by solving the algebraic problem ⌫ N ⌫ n =0 H ⌫⌫ nn n ⌫ |λ=E λ n⌫|λ, n ≤ N ⌫ . (40) Here H ⌫⌫ nn ≡n⌫|H |n ⌫ are the Hamiltonian matrix elements. Within the inverse J-matrix approach, we start with assigning some values to the potential truncation boundaries N ⌫ [see Eq. (34)] in each of the partial waves ⌫. As a next step, we calculate the sets of eigenvalues E λ and respective eigenvector components N ⌫ ⌫|λ. This can be done using the set of the J-matrix matching con- ditions which are obtained from Eq. (38) supposing n = N ⌫ . In more detail, these matching conditions are (to be specific, we again take the case of the coupled sd waves so the channel indexes ⌫ and ⌫ i take the values s or d) Nucleon–Nucleon Interaction 239 a N s s(s) (E) = ⌫ =s,d G s⌫ T ⌫ N ⌫ ,N ⌫ +1 a N ⌫ +1,⌫ (s) (E), (41a) a N d d(s) (E) = ⌫ =s,d G d⌫ T ⌫ N ⌫ ,N ⌫ +1 a N ⌫ +1,⌫ (s) (E), (41b) a N s s(d) (E) = ⌫ =s,d G s⌫ T ⌫ N ⌫ ,N ⌫ +1 a N ⌫ +1,⌫ (d) (E), (41c) and a N d d(d) (E) = ⌫ = s,d G d⌫ T ⌫ N ⌫ ,N ⌫ +1 a N ⌫ +1,⌫ (d) (E), (41d) where we introduced the shortened notation G ⌫⌫ ≡ G ⌫⌫ N ⌫ N ⌫ =− N λ = 0 N ⌫ ⌫|λ λ |N ⌫ ⌫ E λ − E . (42) To calculate a N ⌫ ⌫(⌫ i ) (E)anda N ⌫ +1, ⌫(⌫ i ) (E) entering Eqs. (41), we can use Eq. (37) with the K -matrix elements expressed through the experimental data by Eqs. (33). Therefore G ss , G sd , G ds and G dd are the only unknown quantities in Eqs. (41) and they can be obtained as the solutions of the algebraic problem (41) at any positive energy E. These solutions may be expressed as G ss = ⌬ ss (E) T s N s ,N s +1 ⌬(E) , (43a) G dd = ⌬ dd (E) T d N d ,N d +1 ⌬(E) , (43b) and G sd = G ds =− r 0 √ 2EK sd 2 T s N s ,N s +1 T d N d ,N d +1 ⌬(E) , (43c) where ⌬ ss (E) = S N s s (E) + K ss (E) C N s s (E) S N d +1,d (E) + K dd (E) C N d +1,d (E) −K 2 sd (E) C N s s (E) C N d +1,d (E), (44a) ⌬ dd (E) = S N s +1,s (E) + K ss (E) C N s +1,s (E) S N d d (E) + K dd (E) C N d d (E) −K 2 sd (E) C N s +1,s (E) C N d d (E), (44b) 240 A.M. Shirokov et al. and ⌬(E) = S N s +1,s (E) + K ss (E) C N s +1,s (E) S N d +1,d (E) + K dd (E) C N d +1,d (E) −K 2 sd (E) C N s +1,s (E) C N d +1,d (E). (44c) To derive Eq. (43c), we used the following expression for the Casoratian determinant [34,37]: K l n (C, S) ≡ C n+1,l (E) S nl (E) − S n+1,l (E) C nl (E) = r 0 √ 2E 2 T l n,n+1 . (45) It is obvious from Eqs. (42) and (43) that the eigenvalues E λ can be found by solving the following equation: ⌬(E λ ) = 0. (46) The eigenvector components N ⌫ ⌫|λ can be obtained from Eqs. (43a) and (43b) in the limit E → E λ in the same manner as Eq. (20) in the single-channel case: |N s s|λ| 2 = ⌬ ss (E λ ) T s N s ,N s +1 ⌬ λ (47) and |N d d|λ| 2 = ⌬ dd (E λ ) T d N d ,N d +1 ⌬ λ , (48) where ⌬ λ = d⌬(E) dE E=E λ . (49) Equations (47) and (48) make it possible to calculate the absolute values of N s s|λ and N d d|λ only. However the relative sign of these eigenvector components is important. This relative sign can be established using the relation N s s|λ T s N s ,N s +1 N d d|λ T d N d ,N d +1 =− a N d +1,d(s) (E λ ) a N s +1,s(s) (E λ ) =− a N d +1,d(d) (E λ ) a N s +1,s(d) (E λ ) (50) that can be easily obtained from Eqs. (41). Using Eqs. (46)–(50) we obtain all eigenvalues E λ > 0 and corresponding eigen- vector components N ⌫ ⌫|λ. For example, in the case of the coupled pf waves when the NN system does not have a bound state, all eigenvalues E λ are positive and by means of Eqs. (46)–(50) we obtain a complete set of eigenvalues E λ = 0, 1, ,N and the complete set of the eigenvector’s last components N ⌫ ⌫|λ providing the Nucleon–Nucleon Interaction 241 best description of the ‘experimental’ (obtained by means of phase shift analysis) phase shifts δ 1 (E)andδ 3 (E) and mixing parameter ε. However, as in the case of the uncoupled waves, we should take care of fitting the completeness relation for the eigenvectors n⌫|λ that in the coupled wave case takes the form N λ=0 n⌫|λλ|n ⌫ =δ nn δ ⌫⌫ . (51) Due to Eq. (51), in the two-channel case, we should perform variation of the com- ponents N ⌫ ⌫|λ associated with the two largest eigenenergies E λ=N and E λ=N−1 to fit three relations N λ=0 N ⌫ 1 ⌫ 1 |λλ|N ⌫ 1 ⌫ 1 =1, (52a) N λ=0 N ⌫ 1 ⌫ 1 |λλ|N ⌫ 2 ⌫ 2 =0, (52b) and N λ=0 N ⌫ 2 ⌫ 2 |λλ|N ⌫ 2 ⌫ 2 =1. (52c) This immediately spoils the description of the scattering data that can be restored by the additional variation of the eigenenergies E λ=N and E λ=N−1 . As a result, in the case of the coupled pf waves, we perform a standard fit to the data by min- imizing χ 2 by the variation of N p p|λ = N, N p p|λ = N − 1, N f f |λ = N, N f f |λ = N − 1, E λ=N and E λ=N−1 . These six parameters should fit three rela- tions (52), hence we face a simple problem of a three-parameter fit. In the case of the coupled sd waves, the np system has a bound state (the deuteron) at the energy E d (E d < 0) and one of the eigenvalues E λ is negative: E 0 < 0. We should extend the above theory to the case of a system with bound states. For the coupled sd waves case when the np system has only one bound state, we need three additional equations to calculate E 0 and the components N s s|λ = 0 and N d d|λ = 0. The deuteron energy E d should be associated with the S-matrix pole. As it was already noted, the technique of the S-matrix pole calculation within the J-matrix formalism is discussed together with some applications in Refs. [17, 18]. In the case of the finite-rank potentials of the type (34), one can obtain the exact value of the bound state energy E d and the exact bound state wave function by the S- matrix pole calculation within the J-matrix formalism. To calculate the S-matrix, we use the standard outgoing-ingoingspherical wave asymptotics and the respective expression for the J-matrix oscillator space wave function in the external part of the model space discussed, e.g., in Refs. [17, 18, 34, 37] instead of the standing 242 A.M. Shirokov et al. wave asymptotics (32) and respectively modified expression (37) for the J-matrix oscillator space wave function. Using the expressions for the multi-channel S-matrix within the J-matrix formalism presented in Refs. [17,18,34,37], it is easy to obtain the following expressions [14] for the two-channel S-matrix elements: S ss = 1 D(E) C (−) N s s (E) − G ss T s N s ,N s +1 C (−) N s +1,s (E) × C (+) N d d (E) − G dd T d N d ,N d +1 C (+) N d +1,d (E) − G 2 sd T s N s ,N s +1 T d N d ,N d +1 C (−) N s +1,s (E) C (+) N d +1,d (E) , (53a) S dd = 1 D(E) C (+) N s s (E) − G ss T s N s ,N s +1 C (+) N s +1,s (E) × C (−) N d d (E) − G dd T d N d ,N d +1 C (−) N d +1,d (E) − G 2 sd T s N s ,N s +1 T d N d ,N d +1 C (+) N s +1,s (E) C (−) N d +1,d (E) , (53b) and S sd = S ds =− ir 0 √ 2E G sd D(E) , (53c) where D(E) = C (+) N s s (E) − G ss T s N s ,N s +1 C (+) N s +1,s (E) × C (+) N d d (E) − G dd T d N d ,N d +1 C (+) N d +1,d (E) − G 2 sd T s N s ,N s +1 T d N d ,N d +1 C (+) N s +1,s (E) C (+) N d +1,d (E) (54) and C (±) nl (E) = C nl (E) ±iS nl (E). (55) We need to calculate C (±) nl (E)atnegativeenergyE = E d which can be done us- ing Eqs. (55), (10) and (11) where imaginary values of q = q d = i √ 2|E d | are employed. Extension of these expressions to the complex q plane is discussed in Ref. [34]. Since we associate the deuteron energy E d with the S-matrix pole, from Eqs. (53) we have D(E d ) = 0. (56) Nucleon–Nucleon Interaction 243 Assigning the experimental deuteron ground state energy to E d in Eq. (56) and sub- stituting D(E d ) in this formula by its expression (54), we obtain one of the equations needed to calculate E 0 , N s s|λ = 0 and N d d|λ = 0. Two other equations utilize information about the asymptotic normalization con- stants of the deuteron bound state A s and A d .IftheS-matrix is treated as a function of the complex momentum q, then its residue can be expressed through A s and A d [41,42]: i Res q=iq d S l ⌫ l ⌫ = r 0 e i π 2 ( l ⌫ +l ⌫ ) A l ⌫ A l ⌫ . (57) (the factor r 0 in the right-hand-side originates from the use of the dimensionless momentum q). A s and η = A d A s are determined experimentally. Therefore it is useful to rewrite Eq. (57) as i lim q→iq d (q − iq d )S ss = r 0 A 2 s (58a) and i lim q→iq d (q − iq d )S sd =−r 0 ηA 2 s . (58b) Substituting S ss and S sd by its expressions (53)–(54),we obtain two additional equa- tions for the calculation of E 0 , N s s|λ = 0 and N d d|λ = 0. Clearly, in the case of coupled sd waves, we should also fit the completeness rela- tion (51). We employthe followingmethod of calculation of the sets ofthe eigenvalues E λ andthecomponentsN s s|λand N d d|λ.TheE λ valueswithλ = 1, 2, ,N −2 are obtained by solving Eq. (46) while the respective eigenvector’s last compo- nents N s s|λ and N d d|λ are calculated using Eqs. (47)–(50). Next we perform a χ 2 fit to the scattering data of the parameters E 0 , E λ=N−1 , E λ=N , N s s|λ = 0, N s s|λ = N −1, N d d|λ = N,N d d|λ = 0,N d d|λ = N −1,andN d d|λ = N. These nine parameters fit six relations (52a), (52b), (52c), (56), (58a) and (58b), i.e. we should perform a three-parameter fit as in the case of coupled pf waves. Now we turn to the calculation of the remaining eigenvector components n⌫|λ with n < N ⌫ and the Hamiltonian matrix elements H ⌫⌫ nn with n ≤ N ⌫ and n ≤ N ⌫ entering Eq. (40). The coupled waves Hamiltonian matrix obtained by the general J-matrix inverse scattering method is ambiguous; the ambiguity originates from the multi-channel generalization of the phase equivalent transformation mentioned in the single channel case. As in the single channel case, we eliminate the ambiguity by adopting a particular form of the potential energy matrix. As in the case of uncoupled partial waves, we construct 8ω ISTP in the coupled sd waves. Therefore 2N ⌫ + l ⌫ = 8, or 2N s + 0 = 8and2N d + 2 = 8; hence N s = N d +1. In the coupled pf waves, we construct 7ω and 9ω ISTP; clearly we again have N p = N f +1. Thus the potential matrix V ⌫⌫ nn has the following structure: 244 A.M. Shirokov et al. the submatrices V ⌫⌫ nn coupling the oscillator components of the same partial wave are quadratic [e.g., (N p + 1) × (N p + 1) submatrix V pp nn in the 3 p 2 wave] while the submatrices V ⌫⌫ nn with ⌫ = ⌫ coupling the oscillator components of different partial waves are (N ⌫ +1) × N ⌫ or N ⌫ ×(N ⌫ +1) matrices [e.g., (N p + 1) ×(N p ) submatrix V pf nn coupling the 3 p 2 and 3 f 2 waves]. Our assumptions are: we adopt (i) the tridiagonal form of the quadratic submatrices V ⌫⌫ nn and (ii) the simplest two- diagonal form of the non-quadratic submatrices V ⌫⌫ nn with ⌫ = ⌫ coupling the oscillator components of different partial waves. The structure of the ISTP matrices in coupled partial waves is illustrated by Fig. 17. Due to these assumptions, the algebraic problem (40) takes the following form: H ss 00 0 s|λ+H ss 01 1s|λ+H sd 00 0d|λ=E λ 0s|λ, (59a) H ds 00 0s|λ+H ds 01 1s|λ+H dd 00 0d|λ+H dd 01 1d|λ=E λ 0d|λ, (59b) H ss n,n−1 n − 1, s|λ+H ss nn ns|λ+H ss n,n+1 n + 1, s|λ+H sd n,n−1 n − 1, d|λ + H sd nn nd|λ=E λ ns|λ (n = 1, 2, ,N s − 1), (59c) H ds nn ns|λ+H ds n,n+1 n + 1, s|λ+H dd n,n−1 n − 1, d|λ+H dd nn nd|λ + H dd n,n+1 n + 1, d|λ=E λ nd|λ (n = 1, 2, ,N d − 1), (59d) H ss N s ,N s −1 N s − 1, s|λ+H ss N s N s N s s|λ+H sd N s N d N d d|λ=E λ N s s|λ, (59e) and H ds N d ,N s −1 N s − 1, s|λ+H ds N d N s N s s|λ+H dd N d ,N d −1 N d − 1, d|λ + H dd N d N d N d d|λ=E λ N d d|λ. (59f) Even though this set of equations is more complicated than the set (23) discussed in the uncoupled waves case, it can be solved in the same manner. Fig. 17 Structure of the ISTP matrix in the coupled pf waves and of the Version 0 ISTP in the coupled sd waves. The location of non-zero matrix elements is schematically illustrated by solid lines P F PF FP Nucleon–Nucleon Interaction 245 Multiplying Eqs. (59e)–(59f) by N s s|λ and N d d|λ, summing the results over λ and using the completeness relation (51) we obtain H ss N s N s = N λ =0 E λ N s s|λ 2 , (60a) H dd N d N d = N λ =0 E λ N d d|λ 2 , (60b) and H sd N s N d = N λ =0 E λ N s s|λλ|N d d. (60c) Now we multiply each of the Eqs. (59e) and (59f) by its Hermitian conjugate and one of these equations by the Hermitian conjugate of the other, sum the results over λ and use (51) to obtain H ss N s ,N s −1 =− N λ =0 E 2 λ N s s|λ 2 − H ss N s N s 2 − H sd N s N d 2 , (61a) H ds N d ,N s −1 = 1 H ss N s ,N s −1 N λ =0 E 2 λ N s s|λλ|N d d−H sd N s N d H ss N s N s + H dd N d N d , (61b) and H dd N d ,N d −1 =− N λ =0 E 2 λ N d d|λ 2 − H dd N d N d 2 − H sd N s N d 2 − H ds N d ,N s −1 2 . (61c) As in the case of uncoupled waves, we take the off-diagonal matrix elements- H ss N s ,N s ±1 and H dd N d ,N d ±1 to be dominated by the respective kinetic energy matrix elements T s N s ,N s ±1 and T d N d ,N d ±1 and therefore choose the minus sign in the right- hand-sides of Eqs. (61a) and (61c). By means of Eqs. (60) and (61) we obtain all matrix elements H ⌫⌫ nn entering Eqs. (59e) and (59f). Using this information, the eigenvector components N s − 1, s|λ and N d − 1, d|λ can be extracted directly from Eqs. (59e) and (59f): N s − 1, s|λ= 1 H ss N s ,N s −1 E λ N s s|λ−H ss N s N s N s s|λ−H sd N s N d N d d|λ (62a) [...]... −2.224575 −2.224575 (9) 0.4271 5.620 5. 696 5.635 5.67(11) 0.8845 0.8845 0.86 29 0.8845 0.8845(8) 0.0252 0.0252 0.0252 0.0252 0.0253(2) Compilation [44] −2.2245 89 — 1 .98 77 1 .99 97 1 .96 8 1 .96 8 1 .96 76(10) ⎧ ⎨ 1 .96 35 1 .95 60 ⎩ 1 .95 0 0.8781 0.0272 We should improve the tensor component of the N N interaction to increase the d state probability in the deuteron and reduce the rms radius Therefore the only nontrivial... perform the same manipulations with Eqs (59a)–(59d) We take n = Ns − 1, Ns − 2, , 1 in Eq (59c) and n = Nd − 1, Nd − 2, , 1 in Eq (59d) Equations (59c) and (59d) are a bit more complicated than Eqs (59e) and (59f), however the additional terms in Eqs (59c) and (59d) include only the quantities calculated on the previous step As a result, we obtain the following relations for the ⌫⌫ calculation of the. .. 0.21688 394 8836 0.08 090 7735 691 −0.051881443108 −0.055 193 5 898 42 dd Vnn matrix elements n 0 1 2 3 dd Vnn 0.0086674546 59 0.322126471805 0.308516673061 0.061200037 193 dd dd Vn,n+1 = Vn+1,n −0.0833 393 74560 −0.178808 793 641 −0. 093 012604766 sd Vnn = Vnds matrix elements n n 0 1 2 3 4 sd ds Vn,n−1 = Vn−1,n −0.067221025404 0.068044 496 963 0.0 494 00578816 −0.00150 399 81 39 sd ds Vnn = Vnn −0.483308500313 −0.060476585 693 ... 0.0 491 387324 49 −0.001715 094 993 sd ds Vnn = Vnn −0.4824076 895 87 −0.06136 692 8740 −0.08068524 598 7 −0.02041 291 26 39 sd ds Vn,n+1 = Vn+1,n 0.2540123500 19 We attempted the phase equivalent transformation (67) with a more complicated matrix [U ] than Eq ( 69) However, we did not manage to obtain a completely satisfactory interaction It is possible to obtain the potential providing the required values of the. .. −0.17338747 897 5 −0.1630 792 53268 −0.025144 490 505 0 1 2 3 pp Vn,n+1 = Vn+1,n Vnn 0.068281300876 0. 097 104660674 0.047370054433 ff Vnn matrix elements ff n ff −0.018607311 796 −0.012301122585 −0.002274165032 0 1 2 ff Vn,n+1 = Vn+1,n Vnn 0.0081465 294 81 0.0028786684 09 pf Vnn matrix elements pf fp n Vn,n−1 = Vn−1,n 0 1 2 3 −0.02731 096 5160 −0.005320 397 951 0.0 099 068 396 70 pf fp Vnn = Vnn 0.031138374332 0.026548 899 815... −0.278324060 593 −0.011531530086 0.1514476 294 16 0.036322781738 ss ss Vn,n+1 = Vn+1,n 0.211126251530 0.078168834003 −0.0534670718 79 −0.05 592 8268627 dd Vnn matrix elements n 0 1 2 3 dd Vnn 0.0084566 395 92 0.32204 390 7371 0.308 493 158866 0.061181660346 dd dd Vn,n+1 = Vn+1,n −0.083373543646 −0.1788388 098 60 −0. 093 044 099 373 sd Vnn = Vnds matrix elements n n 0 1 2 3 4 sd ds Vn,n−1 = Vn−1,n −0.06 899 75 295 58 0.067744180124... −0.020205646231 sd ds Vn,n+1 = Vn+1,n 0.2540038307 09 for 4 He The sequence of levels in the 4 He spectrum provided by the odd wave 7 ω-ISTP and by the odd wave 9 ω-ISTP is the same but the energies of excited 4 He states are shifted down in the case of the odd wave 7 ω-ISTP by approximately 100 keV or less Therefore the deviations of the 7 ω-ISTP predictions from the experimental odd wave scattering data at... more bound in the case when we use the 7 ω-ISTP in the odd waves However, the differences are very small: less than 15 keV for 3 H and about 40 keV Nucleon–Nucleon Interaction 261 Table 12 Non-zero matrix elements in ω units of the Version 2 ISTP matrix in the sd coupled waves ss Vnn matrix elements n 0 1 2 3 4 ss Vnn −0.466063146350 −0.2761680 294 73 −0.0 094 738036 59 0.1528737342 89 0.03754 792 9880 ss ss... details the transformed basis {|n⌫ } ≡ U {|n⌫ } Clearly the spectra of the Hamiltonians H and H are identical If the submatrix [U0 ] is small enough, the unitary transformation (67) leaves unchanged the last components N⌫ ⌫|λ of the eigenvectors n⌫|λ obtained by solving the algebraic problem (40) and hence it leaves unchanged the functions G⌫⌫ that completely determine the K -matrix, the S-matrix, the. .. Nijmegen group [3] We start the discussion from the ISTP in the coupled p f waves The non-zero potential energy matrix elements of the obtained 7 ω p f -ISTP are given in Table 9 (in ω = 40 MeV units) The description of the phase shifts δ p and δ f and of the mixing parameter ε is shown in Figs 18–20 The phenomenological data are seen to be well reproduced by the 7 ω ISTP up to the laboratory energy E . V fp n−1,n V pf nn = V fp nn 00.031138374332 1 −0.02731 096 5160 0.026548 899 815 2 −0.005320 397 951 −0.0070 399 0 097 8 30.0 099 068 396 70 Generally, for the coupled pf waves, we have four radial wave function. Eqs. (59e) and (59f), however the additional terms in Eqs. (59c) and (59d) include only the quantities calculated on the previous step. As a result, we obtain the following relations for the calculation. −2.224575 (9) 5.67(11) 1 .96 76(10) 0.8845(8) 0.0253(2) Compilation [44] −2.2245 89 — ⎧ ⎨ ⎩ 1 .96 35 1 .95 60 1 .95 0 0.8781 0.0272 We should improve the tensor component of the NN interaction to increase the
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