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The p-shell with one particle in the continuum
Nuclear Physics A 707 (2002) 65–80 www.elsevier.com/locate/npe
The p-shell with one particle in the continuum Dean Halderson Department of Physics, Western Michigan University, Kalamazoo, MI 49008-5252, USA Received 31 October 2001; received in revised form 22 February 2002; accepted 26 March 2002
Abstract A continuum shell model is now available for p-shell nuclei with exactly the same important properties as a bound-state p-shell model calculation. This means that one can obtain all solutions that correspond to coupling one nucleon to the possible p-shell states of a bound-state calculation. The resulting coupled-channels solutions are antisymmetric and contain no spurious components. This model can be used to describe a very broad range of nuclear phenomena, from capture reactions at astrophysical energies, to knockout reactions at low energies, to nucleon scattering and charge exchange at intermediate energies. Example calculations are shown for the compound systems 7 Be and 7 Li. Here it is demonstrated that agreement between theoretical and experimentally determined elastic and inelastic cross sections and analyzing power values can be improved by coupling to all open channels. 2002 Elsevier Science B.V. All rights reserved. PACS: 21.60.-n; 24.10.-I Keywords: Shell model; Nucleon scattering
1. Introduction Solutions for nucleon wave functions with one particle in the continuum are required in many nuclear investigations. Low energy charge exchange and capture reactions at astrophysical energies with stable or unstable targets, elastic, inelastic and charge exchange reactions at intermediate energies, and one-nucleon knockout reactions such as (e, e N) all require these wave functions. Typically, such reactions are addressed with optical model, distorted wave, or in rare cases with resonating group calculations. Now it is possible perform these calculations for p-shell nuclei in the framework of the recoil corrected continuum shell model (RCCSM) [1]. E-mail address: [email protected] (D. Halderson). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 2 ) 0 0 8 8 1 - 3
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The input to the RCCSM is the oscillator size parameter, the number of protons and neutrons in the compound system, and a realistic two-body effective interaction. The output is the coupled-channels, bound or unbound, solutions for a proton or neutron coupled to the possible core states. The effective interaction may be of any form that is consistent with translational invariance. It may contain central, spin–orbit, quadratic spin–orbit, and tensor terms. The coupled-channels solutions are antisymmetric and non-spurious since one works in the coordinate system connecting the core with the nucleon. The importance of such solutions was well illustrated in one application of the RCCSM in its 1p–1h incarnation. Wild deviations from the expected value of nine were measured for the ratio σ (π + , π + p)/σ (π − , π − p) on 4 He [2,3]. Such values were unexplained by likely suspects such as medium modification of the elementary interaction. However, the realistic coupled-channels continuum solutions of the RCCSM contained the answer [4]. One expects that RCCSM solutions in the p-shell will provide equally important dynamics.
2. Formalism The formalism for the RCCSM was developed by Philpott [1]. This formalism was incorporated in a 1h–1p code and applied to nucleon scattering and charge exchange [5, 6], capture [7], (e, e N) [8], (π + , π + ) [9], and many other reactions. In the p-shell, where the core states are not pure 1h states, one must first calculate the core state energies and wave functions with the chosen oscillator size parameter and effective interaction. Taking the 7 Be composite system as an example, one would have the core states of 6 Li and 6 Be, designated by |JA α. The oscillator size parameter, ν0 = 0.27 fm−2 , is chosen to fit the r.m.s. radius of 6 Li. In the above notation JA is the spin of a core state and α stands for the other quantum numbers necessary to specify the state. The Hamiltonian for the A − 1 system is given by HA =
A−1
pi2 /2m − Tc.m. +
A−1
νij − Hcore ,
(1)
i
i=1
where Hcore = 0|
4 i=1
pi2 /2m +
4
νij |0.
(2)
i
Therefore, all energies are measured with respect to the 4 He core energy. The expectation value JA α|Tc.m. |JA α is 34 h¯ ωδJA JA δαα , since the p-shell states are non-spurious. The Coulomb potential must be included in all calculations, and, therefore, it is necessary to carry out these calculations in a proton–neutron basis. However, throughout, m is taken as mp . The eigenvalues of HA , EjA α , are the threshold energies for the channels which correspond to an initial or residual state, |JA α. If one is using a standard shellmodel code to generate these wave functions and energies, then the single-particle energies are given by [J ] 5 ji jc τi τc (J )|ν ji jc τi τc (J ) + h¯ ω, εi = (3) [ji ] 4 J,{nlj τ }c
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where [K] = 2K + 1, {nlj τ }c = 0s1/2 ( p) or 0s1/2 (n), and the matrix elements are defined as: j1 j2 τ1 τ2 (J )|ν|j3 j4 τ3 τ4 (J ) = j1 j2 (J, T = 1)|ν|j3j4 including Coulomb for τ1 = τ2 = τ3 = τ4 = p; j1 j2 (J, T = 1)|ν|j3j4 excluding Coulomb for τ1 = τ2 = τ3 = τ4 = n; 1/2 1 (1 + δj1 j2 )(1 + δj3 j4 ) j1 j2 (J, T = 1)ν|j3 j4 + j1 j2 (J, T = 0)ν|j3 j4 2 for τ1 = τ3 = p and τ2 = τ4 = n, or τ1 = τ3 = n and τ2 = τ4 = p; and 1/2 1 (1 + δj1 j2 )(1 + δj3 j4 ) j1 j2 (J, T = 1)ν|j3 j4 − j1 j2 (J, T = 0)ν|j3 j4 2 for τ1 = τ4 = p and τ2 = τ3 = n, or τ1 = τ4 = n and τ2 = τ3 = p. The matrix elements, j1 j2 (J, T )|ν|j3 j4 , are the standard antisymmetrized, normalized, isospin matrix elements. Once the core states are determined, the states for A nucleons in the p-shell are calculated with the Hamiltonian HB =
A
pi2 /2m − Tc.m. +
i=1
A
νij − Hcore .
(4)
i
The eigenstates and eigenvalues of HB are designated by |JB β and EJB β . Again, the expectation value JB β|Tc.m. |JB β is 34 hωδ ¯ JB JB δββ . The next step is to calculate the + Jx J α , two-body densities J α [(a + a + )Jy Ty × a ] one-body densities JA α [akτ lτl A A k l k
+ JB (am an )Jx Tx ]J,T MT JA α , and overlaps JB β| [akτk |JA α] . Note that the first two are only once reduced matrix elements. These quantities will assist in calculating the matrix elements of the Hamiltonian and + J unit operator for the A-particle system in the shell-model basis, [anlj τ ⊗ |JA α] . In the example, these would be the basis states of the compound system 7 Be. The basis includes states for 2n + l up to ρmax (n starts at zero). In the calculations shown below, ρmax = 16 is the minimum required to accurately determine the appropriate cut-off radii in the method of Philpott and George [10]. The basis states are not normalized and most have spurious components. One can no longer replace Tc.m. by 34 h¯ ω in HB . Instead, one must use
A A A A pi · pj pi2 A − 1 pi2 − Tc.m. + + . νij − νij = 2m A 2m Am i
i=1
i=1
(5)
i
If the initial or final nlj (τ ) corresponds to 0s1/2 (p) or 0s1/2 (n), the matrix elements are zero. If both the initial and final nlj (τ ) are in the 0p-shell, then J
J JA α| ⊗ anlj τ HB an+ l j τ ⊗ JA α J + J + = EJβ Jβ| anlj τ ⊗ |JA α Jβ| an l j τ ⊗ JA α β
and
(6)
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D. Halderson / Nuclear Physics A 707 (2002) 65–80 J
J JA α| ⊗ anlj τ 1 an+ l j τ ⊗ JA α J + J + = Jβ| anlj . τ ⊗ |JA α Jβ| an l j τ ⊗ JA α
(7)
β
If the initial state and final state are outside the 0p-shell, then J J JA α| ⊗ anlj τ HB an+ l j τ ⊗ JA α
= EJA α + 34 h¯ ω/A δnlj τ,n l j τ δJA JA δαα [Jx ] A − 1 p2 jjc τ τc (Jx ) ν j jc τ τc (Jx ) + nlj | nlj + A 2m [j ] Jx ,{nlj τ }c
× δJA JA δαα δlj τ,l j τ
J + [J ]Jˆx JA α aj+1 τ1 aj2 τ2 x JA α j1 j τ1 τ (J )ν j2 j τ2 τ (J )
× W (j, J , Jx , j2 ; j1 , j )W JA , J, Jx , j ; j, JA ,
(8)
where νij = νij − pi · pj /Am, Jˆx = (2Jx + 1)1/2 , and the last sum is over j1 , j2 , τ1 , τ2 , J , and Jx . The matrix elements of the unit operator are just δJA JA δαα δnlj τ,n l j τ . For initial state (j ) in the 0p-shell and the final state (j ) outside the 0p-shell, J J JA α| ⊗ anlj τ HB an+ l j τ ⊗ JA α j3 j (Jx Tx MTx )ν |j1 j2 (1 + δj1 j2 )−1/2 =
J T J T × JA α aj+3 aj+ y y (aj1 aj2 )Jx Tx M JA α Jˆx Jˆy Jˆ T
× W j , J, J , JA ; JA , j W J , j, Jy , j3 ; j , Jx 12 12 mt3 mtj Ty MTy × 12 12 mt3 mtj Tx MTx Ty Tx MTy − MTx |T MT (−1)MTx −Tx +1+j +j +J
A − 1 p2 + nlj | |0lj Jˆx W j , JA , j, JA ; J, Jx A 2m Jx J × JA α aj+ τ a0lj τ x JA α (−1)1+j +JA −J J + [J ]Jˆx JA α aj+1 τ1 aj2 τ2 x JA α j1 j τ1 τ (J )ν j2 j τ2 τ (J )
× W j, J , Jx , j2 ; j1 , j W JA , J, Jx , j ; j, JA [Jx ] A − 1 p2 + nlj | nlj + jjc τ τc (Jx ) ν j jc τ τc (Jx ) A 2m [j ] Jx ,{nlj τ }c
× δJA JA δαα δlj τ,l j τ ,
(9)
where the first sum is over j1 j2 , j3 , τ3 , Jx , Jy , Tx , Ty , MTx , MTy , J , T , and MT , and the third sum is as in Eq. (8). The matrix elements of the unit operator are zero. The mixture of proton–neutron and isospin notation is a convenience for calculation and storage. The convention used above corresponds to the proton having mt = − 12 ; hence, MTx = −1 implies inclusion of the Coulomb potential.
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The matrix elements of the unit operator and Hamiltonian are then transformed as in [1] to the core–nucleon center-of-mass system, thereby removing the spurious components. + J Antisymmetry is preserved in the transformation. Since the [anlj τ ⊗ |JA α] basis is more than complete when nlj τ is in the 0p-shell, one must transform the Hamiltonian and unit matrix to the |JB β basis. One must also remove the zero matrix elements for the 0s-shell. The S-matrix elements are calculated in the R-matrix formalism of Ref. [11]. This requires matrix elements up to a cut-off radius, ac . The matrix elements one has go to infinity. Therefore, one must subtract the contributions that are long range. These subtractions are only made to orbits outside the 0p-shell. Those in the 0p-shell go to zero before the cut-off radius. For the unit matrix this just involves the integrals ∞ Rnl (r)Rn l (r)r 2 dr,
δJA JA δαα δlj τ,l j τ ac
where now the oscillator constant is ν = (A − 1)ν0/A. The Coulomb potential has a similar integral ∞ δJA JA δαα δlj τ,l j τ δτp
Rnl (r)
Zc e 2 Rn l (r)r 2 dr, r
ac
since the core is contained within the cut-off. One must also correct the kinetic energies with the integral ∞ δJA JA δαα δlj τ,l j τ
Rnl (r)
p2 Rn l (r)r 2 dr, 2µ
ac
where µ = m(A − 1)/A. Once the Hamiltonian and unit matrix are truncated at the cut-off, all cross sections are determined.
3. Examples The difficulty in applying this powerful tool is the lack of an effective interaction with a proper theoretical foundation. Most of the available interactions were designed for DWBA calculations, with no regard for bound-state properties. Those people deriving effective interactions for shell-model calculations were not concerned with the matrix elements of the high-lying particle states. One knows that effective interaction theory provides the correct answer to bound-state energies, provided one works in a non-spurious basis [12], but the ambiguities that arise when extracting high-lying particle state matrix elements for use in scattering calculations are disconcerting. One effective interaction was found which provided an excellent fit to both the 4 He bound-state energy and the p + 3 H, n + 3 He, p + 4 He cross sections [5,13]. This was the M3Y interaction of Ref. [14]. The agreement was very surprising, since the M3Y was designed for use in DWBA calculations with medium mass nuclei. However, when
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applied to p + 6 Li in DWBA calculations or when folded to produce potentials for coupledchannels calculations [15,16], the M3Y interaction required severe normalization. The normalization factors in Ref. [16] were denoted by the enhancement factors of the multipole terms 2L + 1 ν˜ L (r, r1 , ρ0 ) = 2
+1
ν R = |r − r 1 |, ρ0 PL (cos θ ) d(cos θ ).
(10)
−1
For L = 0 these factors are 1.2, 1.2, 1.4, 1.4, and 1.0 at Ep = 24.4, 49.75, 65, 80, and 136 MeV, and for L = 2 they are 4.6, 3.0, 2.5, 2.5, and 2.5 at the respective energies. In Ref. [15] the normalization factors are denoted by the reduction in the radial factors of the central interaction in the form g = g00 + g10 σ · σ + g01 τ · τ + g11 σ · σ τ · τ . Based on the 1+ –3+ excitation in 6 Li, g00 was reduced by 28% at 24.4 and 45 MeV. This reduction was made after multiplying the corresponding form factor by two to account for the factor of four between the calculated and experimental B(E2). Based on 7 Li + p data, g01 was reduced by 28% at 26 MeV and 21% at 45 MeV, and g11 was reduced by 57% at 26 MeV and 45% at 45 MeV. In Ref. [17] the entire central potential was replaced with the complex, density and energy dependent interaction of Ref. [18] with the imaginary part multiplied by 0.8. The spin-dependent components of M3Y were retained. The drastic normalization factors for the A = 7 composite system were somewhat surprising, given the excellent agreement obtained with M3Y for A = 4 and 5. Therefore, the elastic and inelastic calculations for the A = 7 composite system are repeated in this work with the RCCSM formalism. In Refs. [15–17] the Cohen and Kurath [19] wave functions are employed, whereas the wave functions used in RCCSM calculations are consistent with the interaction. However, results with M3Y and Cohen and Kurath wave functions differ little. In fact, in Ref. [15] it was shown that pure L–S coupled wave functions give similar results. Therefore, a fair comparison can be made between DWBA results and those of the RCCSM. Fig. 1 shows the 6 Li + p elastic RCCSM cross sections as solid lines. These calculations include only the 6 Li ground state. The data are from Refs. [20–22]. The dotted lines at 24.4, 35, and 49.75 MeV are the calculations of Ref. [17] and the dashes lines at 24.4, 49.75, 65, 80, and 136 MeV are the calculations of Ref. [16]. These dashed and dotted curves include the above normalization factors. At 65, 80, and 136 MeV the RCCSM-M3Y calculation does as well as the normalized folding calculation. At 24.4, 35, and 49.75 MeV the magnitude of the RCCSM-M3Y calculations seem correct, but one is acutely aware of the back-angle rise of the calculation which does not appear in the data. The general shape of these elastic cross sections, with the back-angle rise, is the same as was calculated for p + 3 H and p + 4 He. However, in these cases the data rose at back-angles so the agreement was good. Folding calculations will not produce as strong a back-angle rise, and in most cases, none at all. In fact, one can take the identical nuclear density and interaction used in an RCCSM calculation, perform a folding calculation with knockout-exchange, and the back-angle cross section will be much smaller than the corresponding RCCSM calculation. The difference between the two is the same difference that one sees between a resonating group calculation and the corresponding generator coordinate calculation. In one case the translationally invariant dynamical equations are solved, and in the other a potential is
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Fig. 1. Elastic scattering cross sections for 6 Li + p. Solid lines are one-core-state RCCSM calculation with M3Y interaction. Dotted and dashed lines are calculations of Refs. [16,17], respectively. Data are from Refs. [20,21].
generated as a function of distance between objects. This difference was most clearly demonstrated in the early days of constructing NN potentials from three quarks incident on three quarks. The potentials determined by calculating the potential energy as a function of distance between bags showed little repulsive core. However, when the resonating group kernel was expanded about its nonlocality, the repulsive core appeared. Therefore, the folding calculations are fortunate in that error incurred by not solving the dynamical equations goes in the same direction as whatever mechanism is missing from the RCCSM calculation for describing for 6 Li + p. This mechanism must be present in 6 Li + p and suppressed in p + 3 He and p + 4 He. The most likely candidate is one-nucleon knockout. The loosely bound valence nucleons in 6 Li have little chance of remaining in the nucleus when the momentum transfer is large as in back-angle scattering. Therefore the fault most likely lies with a missing mechanism and not with the M3Y interaction. Although one cannot include one-nucleon knockout in the RCCSM, one can include the inelastic channels corresponding to core excitation. The p-shell allows 15 A = 6 core states, not all of which are observed. If all 15 states are included, then the RCCSM produces the results shown in Fig. 2. The effect of the channel coupling is greatest at low energies and diminishes with increasing energy. The magnitudes of the cross sections at 24.4, 35, and 136 MeV are good and are a few percent low in the 50–80 MeV region. Considering
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Fig. 2. Elastic scattering cross sections for 6 Li + p. Solid lines are 15-core-state RCCSM calculation with M3Y interaction. Dotted and dashed lines are calculations of Refs. [16,17], respectively. Data are from Refs. [20,21].
the large energy range and considering that M3Y does not depend on energy, this result is impressive. It was noted in Ref. [16] that the calculated analyzing power in 6 Li(p, p)6 Li at Ep = 49.75 MeV in Ref. [17] was very much out of phase at back angles. This calculation is shown in Fig. 3 as a dotted line along with the M3Y calculation of Ref. [16] as a dashed line and the M3Y-RCCSM calculation with one core state as a solid line. The data are from Ref. [21]. All calculations use the same spin-dependent interaction but achieve different results. The M3Y-RCCSM calculation demonstrates a similar difficulty as in the calculation of Ref. [17] in that the analyzing power does not dip negative near 120◦ . Channel coupling was given as one possible explanation for the analyzing power problem in Ref. [17]. The present RCCSM code is the only available means of thoroughly checking this possibility, for it allows coupling of all 15 A = 6 core states. In Fig. 4 is shown the elastic analyzing power at 49.75, 65, and 80 MeV. The data are from Refs. [21,23]. The dashed lines are from calculations with the one-core-state M3Y-RCCSM, and the solid lines are from the 15-core-state M3Y-RCCSM. It is true that the channel coupling drives the 49.75 MeV analyzing power negative near 120◦ and also for 80 MeV. The 15channel calculation seems to provide somewhat better agreement with the data. However, the improved agreement is more dramatic if one employs the effective interaction derived in Ref. [13] from the Reid Soft core for 90 Zr (RSC-90 Zr). These results are shown in
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Fig. 3. Elastic analyzing power for 6 Li + p at Ep = 49.75 MeV. Solid lines are one-core-state RCCSM calculations with M3Y interaction. Dotted and dashed lines are calculations of Refs. [16,17], respectively. Data are from Ref. [21].
Fig. 4. Elastic analyzing powers for 6 Li + p. Dashed and solid lines are one-core-state and 15-core-state RCCSM calculations with M3Y interaction, respectively. Data are from Refs. [21,23].
Fig. 5 where one sees that the 15-core-state calculation does a good job of reproducing the data. One would have liked to have the back-angle data at 65 MeV to further test the correspondence. One needs all 15 core states to obtain this agreement. No one core state has the dominant effect. The 6 Li states with predominately [11] spatial symmetry have just as much effect as the predominately [2] spatial symmetry states, and the 6 Be states have much the same effect as the 6 Li states.
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Fig. 5. Elastic analyzing powers for 6 Li + p. Dashed and solid lines are one-core-state and 15-core-state RCCSM calculations with RSC-90 Zr interaction, respectively. Data are from Refs. [21,23].
In addition to examining the elastic analyzing power, the authors of Ref. [16] also compared calculations to then new analyzing power measurements for 6 Li(p, p )6 Li(3+ ) [23]. Distorted wave calculations with M3Y did very poorly. The authors demonstrated that the use of a momentum and density dependent interaction [24] improved the calculations considerably. A density dependent interaction cannot be used in the RCCSM because of the lack of translational invariance. So, once again this calculation is repeated in the RCCSM formalism with M3Y to see if equivalent agreement can be obtained. p )6 Li(3+ ) are shown in The inelastic analyzing powers and cross section for 6 Li(p, Figs. 6 and 7, respectively, along with the data of Ref. [23]. The final DWBA calculations of Ref. [16] with the density and momentum dependent potential are shown as dotted lines. The potential was given normalization factors for L = 0 of 1.00, 0.75, 0.90, 1.00, and 1.00 at Ep = 24.4, 49.75, 65, 80, and 136 MeV, and all scale factors were 1.00 for L = 2. The M3Y-RCCSM calculations are shown as dashed lines for two core states and solid lines for 15 core states. The 65 MeV asymmetry clearly shows that the coupling of all channels brings the M3Y-RCCSM calculation into phase with the data, although not quite large enough in magnitude. The DWBA calculation is a bit too large in magnitude and slightly out of phase. At 80 MeV the channel coupling is forcing the calculation toward being in phase, but the magnitude is much too small, while the DWBA calculation has the correct magnitude, but is out of phase. It is unfortunate that the 49.75 MeV asymmetry data is so poor and does not go to back angles, since it is there that the coupled-channels RCCSM calculation differs significantly from the DWBA calculation. Therefore, one has no strong reason to reject M3Y based on these data. Comparing the inelastic cross sections is more difficult. The M3Y, DWBA results of Ref. [16] do not seem to be consistent with those of Ref. [15] in the following way. When both use M3Y, Ref. [15] requires a 28% reduction in g00 , but in Ref. [16] requires a 3–4 factor enhancement in ν˜2 . Also, both have apparently taken the inelastic transition
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Fig. 6. Inelastic analyzing powers for 6 Li(p, p )6 Li(3+ ). Dotted lines are for DWBA calculations of Ref. [16]. Dashed and solid lines are with M3Y interaction for two and 15-channel RCCSM calculations, respectively. Data are from Ref. [23].
Fig. 7. Inelastic cross sections for 6 Li(p, p )6 Li(3+ ). Dashed and solid lines are with M3Y interaction for two and 15-core-state RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr. Data are from Refs. [15,20,21,23].
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Fig. 8. Cross sections for 7 Li composite system. Open circles are 6 Li(n, n )6 Li(3+ ) data of Ref. [24], and crosses are 6 Li(n, p)6 He(g.s.) data of Ref. [29]. Dashed and solid lines are two-core-state and 15-core-state M3Y-RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr.
density from electron scattering and assumed equal enhancement for protons and neutrons. If Ref. [16] had used the shell-model form factor, then the calculations would be consistent, but the agreement with the data found in that work with the density dependent interaction would disappear when the calculations are multiplied by four to be consistent with the measured B(E2). Therefore, it seems more appropriate to make a comparison with Ref. [17], where progress was made on solving the following, related dilemma which is illustrated in Fig. 7 along with the data of Refs. [15,20,21,23]. Since the M3Y-RCCSM calculation is equivalent to using the shell-model transition density, all inelastic cross sections should be approximately the same factor of four too small. This is not the case. The results obtained for two core states, shown as dashed lines, essentially agree with the data at 24.4 and 35 MeV. The coupling of all 15 core states, shown as solid lines, improves the two-corestate results, but solid lines are more like a factor of two too small at 24.4, 49.75, 65, and 80 MeV and a factor of one at 35 and 136 MeV. The RSC-90 Zr results, shown as dotted lines, are slightly better at 35 MeV, but still not as expected. The problem persists in 6 Li(n, n )6 Li(3+ ) at En = 24.4 MeV as shown by the upper curves in Fig. 8, when compared to the data of Ref. [25]. By using the density dependent central force of Ref. [18], the authors of Ref. [17] were able to obtain consistency between the elastic cross sections and the 3+ cross sections. These calculations, which include the expected factor of four enhancement, are shown as dot-dashed lines in Fig. 7 at 24.4, 35, and 49.75 MeV. At 24.4 and 49.75 MeV the agreement at forward angles is good with some discrepancy at 35 MeV. However, the
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Fig. 9. Inelastic cross sections for 6 Li(p, p )6 Li(0+ ). Data are from Refs. [15,20,21]. Dashed and solid lines are two-core-state and 15-core-state M3Y-RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr.
RCCSM calculation at 24.4 MeV shows a severe correction due to coupling many core states. Such a correction could negatively affect the results in Ref. [17], but nonetheless, the agreement is impressive, and provides one case where M3Y does not provide equivalent agreement. The 6 Li(p, p )6 Li(0+ ) cross sections are shown in Fig. 9 along with the data of Refs. [15,20,21]. The solid and dashed lines correspond to M3Y-RCCSM with 15 and 2 core states, respectively. The 24.4 MeV, 15-core-state results seem very diffractive. Therefore, the same calculation with RSC-90 Zr is also shown as a dotted line to demonstrate that the diffractive appearance is a result of the interaction employed. The M3Y calculations are above the data for Ep = 45.4, 49.75, and 136 MeV. This is unexpected since the calculated 1+ –0+ B(M1) is in good agreement with the experimental value. In Ref. [17] the calculations were a factor of two above the data at 45.4 and 49.75 MeV and a slightly larger factor at 24.4 MeV. It was suggested in Ref. [17] that perhaps inclusion of additional core states and proper treatment of the center of mass and antisymmetry would improve the agreement with experiment. Indeed, the coupling of additional core states does improve the comparison with data, but not enough to claim agreement.
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Fig. 10. Inelastic cross sections for 6 Li(p, n)6 Be(g.s.). Data are from Ref. [26]. Dashed and solid lines are two-core-state and 15-core-state M3Y-RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr.
The difficulty also appears in the charge-exchange reaction 6 Li(p, n)6 Be(g.s.) for which is the analog. In Fig. 10 are plotted that data of Ref. [26] with the same three curves as in Fig. 9. The M3Y-RCCSM calculation gives good forward angle agreement with the data at 30.2 MeV, but becomes progressively worse up to 120 MeV, where it is a factor of two too large. The RSC-90 Zr calculation exhibits the same difficulty that was noted in Ref. [17]. Their calculations at low energies were a factor of two too large, whereas calculations [27] using the t-matrix interaction of Ref. [28] produce agreement with forward angle data at 120 MeV. The RSC-90 Zr calculation exhibits the same behavior, but with one interaction. Unfortunately the 6 Li(p, p )6 Li(0+ ), 136 MeV data did not go to forward angles, and also, the measured values are somewhat lower that expected [27]. However, the same forward angle agreement is obtained with RSC-90 Zr for 6 Li(n, p)6 He(g.s.) at 120 MeV as shown in the lower curves of Fig. 8 and compared to the data of Ref. [29]. The disagreement between calculated and measured cross sections for this 1+ –0+ , spin– flip transition was considered the major puzzle for nucleon scattering from 6 Li in Ref. [17]. It is even more puzzling in light of the RCCSM calculation, where M3Y does better at low energies and worse at higher energies while RSC-90 Zr behaves in the opposite manner. This is more puzzling because the RCCSM calculations show that channel coupling becomes less important at higher energies, hence one would infer that channels missing from the calculation would becomes less important at higher energies. One might then understand the behavior of RSC-90 Zr and the calculations of Refs. [17,27] in terms of channels not included in the calculation. However, the M3Y calculation invalidates that argument, at least to the extent that one believes M3Y is a viable interaction. This may be one more reason to reject M3Y for p-shell nuclei. Then one must say that significant modifications 6 Li(p, p )6 Li(0+ )
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occur to the effective interaction when moving from A = 4 and 5 to A = 7 composite systems.
4. Conclusion The primary purpose of this paper was to demonstrate that the continuum shell model with correction for target recoil has been extended into the p-shell and to present the formalism required for implementing this very powerful tool. Example calculations were performed for nucleon scattering in the A = 7 composite systems. These calculations were chosen for two reasons. First they provided an opportunity to test the M3Y interaction. This interaction had produced very good results in the A = 4 and 5 composite systems, and A = 7 represents the next step up in mass. Second, the A = 6 core has 15 p-shell states so that one can test the effect of channel coupling. The M3Y interaction performed quite well for elastic scattering cross sections and reasonably well in describing asymmetries, both elastic and inelastic. However, as found in previous works [15,17], the magnitudes of the forward angle inelastic cross sections were too large at low and moderate energies for 1+ –3+ excitation and too large at moderate and intermediate energies for 1+ –0+ excitation. Another interaction with stronger Pauli corrections, RSC-90 Zr, tended to give more consistent results, but the over estimate of low and moderate energy inelastic cross sections remains a problem. One has several choices to explain this difficulty. It could be a missing mechanism, missing channels, naïve structure of the core states, or renormalization of the interaction for the A = 7 system. Since the RCCSM includes multiple scattering effects, and since the effects of channel coupling diminish around 80 MeV, the first two possibilities are less likely. However, given the rather wide variation of the results with the different effective interactions, what is more likely is that the effective interactions employed are not consistent with the model space. The first step toward improving these calculations would be a more careful construction of the interaction, perhaps by folded diagram techniques. However, even at this level, the coupled-channels wave functions derived from M3Y should be very good candidates for final states in knockout reactions.
Acknowledgement This work was supported by the National Science Foundation under grant PHY9732634.
References [1] [2] [3] [4] [5]
R.J. Philpott, Nucl. Phys. A 289 (1977) 109. M.K. Jones et al., Phys. Rev. C 46 (1992) 52. J. Langenbrunner, M.K. Jones, D. Dehnhard, C.L. Morris, W.R. Gibbs, Phys. Rev. Lett. 69 (1992) 1508. D. Halderson, V.A. Sadovnikova, Phys. Rev. C 56 (1997) 2688. D. Halderson, R.J. Philpott, Nucl. Phys. A 321 (1979) 295.
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[6] R.J. Philpott, D. Halderson, in: C.D. Goodman, S.M. Austin, S.D. Bloom, J. Rapaport, R.G. Satchler (Eds.), The (p, n) Reaction and the Nucleon–Nucleon Force, Plenum, New York, 1980, p. 491. [7] D. Halderson, R.J. Philpott, Nucl. Phys. A 359 (1981) 365. [8] D. Halderson, Phys. Rev. C 53 (1996) 2978. [9] D. Halderson, R.J. Philpott, J.A. Carr, F. Petrovich, Phys. Rev. C 24 (1981) 1094. [10] R.J. Philpott, J. George, Nucl. Phys. A 233 (1974) 164. [11] R.J. Philpott, Nucl. Phys. A 243 (1975) 260. [12] D. Halderson, P. Goldhammer, Phys. Rev. C 11 (1975) 237. [13] D. Halderson, J. Phys. G 20 (1994) 1461. [14] G. Bertsch, J. Borysowicz, H. McManus, W.G. Love, Nucl. Phys. A 284 (1977) 399, the interaction described on p. 412. [15] F. Petrovich, R.H. Howell, C.H. Poppe, S.M. Austin, G.M. Crawley, Nucl. Phys. A 383 (1982) 355. [16] R. Kanungo, C. Samanta, S. Roy, K.K. Samaddar, Nucl. Phys. A 581 (1995) 294. [17] F. Petrovich, S.K. Yoon, M.J. Threapleton, R.J. Philpott, J.A. Carr, F.S. Dietrich, L.F. Hansen, Nucl. Phys. A 563 (1993) 387. [18] J.P. Jeukenne, A. Lejeune, C. Mahaux, Phys. Rev. C 10 (1974) 1931; Phys. Rev. C 15 (1977) 10; Phys. Rev. C 16 (1977) 80. [19] S. Cohen, D. Kurath, Nucl. Phys. 73 (1965) 1. [20] K.H. Bray, M. Jain, K.S. Jayaraman, G. LoBianco, G.A. Moss, W.T.H. Van Oers, D.O. Wells, F. Petrovich, Nucl. Phys. A 189 (1972) 35. [21] G.S. Mani, D. Jacques, A.D.B. Dix, Nucl. Phys. A 165 (1971) 145; Nucl. Phys. A 172 (1971) 166. [22] R.S. Henderson, S.F. Coffins, B.M. Spicer, G.G. Shute, V.C. Officer, D.W. Devins, D.L. Friesel, W.P. Jones, Nucl. Phys. A 117 (1981) 372. [23] M. Tosaki, M. Fujiwara, K. Hosono, T. Noro, H. Ito, T. Yamazaki, H. Ikegami, M. Kamimura, RCNP Ann. Rep. (1985), as quoted in Ref. [15]. [24] D. Bandyopadhyay, C. Samanta, S.K. Samaddar, J.N. De, Nucl. Phys. A 511 (1990) 1. [25] L.F. Hansen, J. Rapaport, X. Wang, F.A. Barrios, F. Petrovich, A.W. Carpenter, M.J. Threapleton, Phys. Rev. C 38 (1988) 525. [26] C.J. Batty, B.E. Bonner, E. Friedman, C. Tschaler, L.E. Williams, Nucl. Phys. A 120 (1968) 297. [27] J. Rapaport et al., Phys. Rev. C 41 (1990) 1920. [28] W.G. Love, M. Franey, Phys. Rev. C 24 (1981) 1073. [29] D. Pocanic, K. Wang, C.J. Martoff, S.S. Hanna, R.C. Byrd, C.C. Foster, D.L. Friesel, J. Rapaport, Can. J. Phys. 65 (1087) 687.
The p-shell with one particle in the continuum Dean Halderson Department of Physics, Western Michigan University, Kalamazoo, MI 49008-5252, USA Received 31 October 2001; received in revised form 22 February 2002; accepted 26 March 2002
Abstract A continuum shell model is now available for p-shell nuclei with exactly the same important properties as a bound-state p-shell model calculation. This means that one can obtain all solutions that correspond to coupling one nucleon to the possible p-shell states of a bound-state calculation. The resulting coupled-channels solutions are antisymmetric and contain no spurious components. This model can be used to describe a very broad range of nuclear phenomena, from capture reactions at astrophysical energies, to knockout reactions at low energies, to nucleon scattering and charge exchange at intermediate energies. Example calculations are shown for the compound systems 7 Be and 7 Li. Here it is demonstrated that agreement between theoretical and experimentally determined elastic and inelastic cross sections and analyzing power values can be improved by coupling to all open channels. 2002 Elsevier Science B.V. All rights reserved. PACS: 21.60.-n; 24.10.-I Keywords: Shell model; Nucleon scattering
1. Introduction Solutions for nucleon wave functions with one particle in the continuum are required in many nuclear investigations. Low energy charge exchange and capture reactions at astrophysical energies with stable or unstable targets, elastic, inelastic and charge exchange reactions at intermediate energies, and one-nucleon knockout reactions such as (e, e N) all require these wave functions. Typically, such reactions are addressed with optical model, distorted wave, or in rare cases with resonating group calculations. Now it is possible perform these calculations for p-shell nuclei in the framework of the recoil corrected continuum shell model (RCCSM) [1]. E-mail address: [email protected] (D. Halderson). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 2 ) 0 0 8 8 1 - 3
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The input to the RCCSM is the oscillator size parameter, the number of protons and neutrons in the compound system, and a realistic two-body effective interaction. The output is the coupled-channels, bound or unbound, solutions for a proton or neutron coupled to the possible core states. The effective interaction may be of any form that is consistent with translational invariance. It may contain central, spin–orbit, quadratic spin–orbit, and tensor terms. The coupled-channels solutions are antisymmetric and non-spurious since one works in the coordinate system connecting the core with the nucleon. The importance of such solutions was well illustrated in one application of the RCCSM in its 1p–1h incarnation. Wild deviations from the expected value of nine were measured for the ratio σ (π + , π + p)/σ (π − , π − p) on 4 He [2,3]. Such values were unexplained by likely suspects such as medium modification of the elementary interaction. However, the realistic coupled-channels continuum solutions of the RCCSM contained the answer [4]. One expects that RCCSM solutions in the p-shell will provide equally important dynamics.
2. Formalism The formalism for the RCCSM was developed by Philpott [1]. This formalism was incorporated in a 1h–1p code and applied to nucleon scattering and charge exchange [5, 6], capture [7], (e, e N) [8], (π + , π + ) [9], and many other reactions. In the p-shell, where the core states are not pure 1h states, one must first calculate the core state energies and wave functions with the chosen oscillator size parameter and effective interaction. Taking the 7 Be composite system as an example, one would have the core states of 6 Li and 6 Be, designated by |JA α. The oscillator size parameter, ν0 = 0.27 fm−2 , is chosen to fit the r.m.s. radius of 6 Li. In the above notation JA is the spin of a core state and α stands for the other quantum numbers necessary to specify the state. The Hamiltonian for the A − 1 system is given by HA =
A−1
pi2 /2m − Tc.m. +
A−1
νij − Hcore ,
(1)
i
i=1
where Hcore = 0|
4 i=1
pi2 /2m +
4
νij |0.
(2)
i
Therefore, all energies are measured with respect to the 4 He core energy. The expectation value JA α|Tc.m. |JA α is 34 h¯ ωδJA JA δαα , since the p-shell states are non-spurious. The Coulomb potential must be included in all calculations, and, therefore, it is necessary to carry out these calculations in a proton–neutron basis. However, throughout, m is taken as mp . The eigenvalues of HA , EjA α , are the threshold energies for the channels which correspond to an initial or residual state, |JA α. If one is using a standard shellmodel code to generate these wave functions and energies, then the single-particle energies are given by [J ] 5 ji jc τi τc (J )|ν ji jc τi τc (J ) + h¯ ω, εi = (3) [ji ] 4 J,{nlj τ }c
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where [K] = 2K + 1, {nlj τ }c = 0s1/2 ( p) or 0s1/2 (n), and the matrix elements are defined as: j1 j2 τ1 τ2 (J )|ν|j3 j4 τ3 τ4 (J ) = j1 j2 (J, T = 1)|ν|j3j4 including Coulomb for τ1 = τ2 = τ3 = τ4 = p; j1 j2 (J, T = 1)|ν|j3j4 excluding Coulomb for τ1 = τ2 = τ3 = τ4 = n; 1/2 1 (1 + δj1 j2 )(1 + δj3 j4 ) j1 j2 (J, T = 1)ν|j3 j4 + j1 j2 (J, T = 0)ν|j3 j4 2 for τ1 = τ3 = p and τ2 = τ4 = n, or τ1 = τ3 = n and τ2 = τ4 = p; and 1/2 1 (1 + δj1 j2 )(1 + δj3 j4 ) j1 j2 (J, T = 1)ν|j3 j4 − j1 j2 (J, T = 0)ν|j3 j4 2 for τ1 = τ4 = p and τ2 = τ3 = n, or τ1 = τ4 = n and τ2 = τ3 = p. The matrix elements, j1 j2 (J, T )|ν|j3 j4 , are the standard antisymmetrized, normalized, isospin matrix elements. Once the core states are determined, the states for A nucleons in the p-shell are calculated with the Hamiltonian HB =
A
pi2 /2m − Tc.m. +
i=1
A
νij − Hcore .
(4)
i
The eigenstates and eigenvalues of HB are designated by |JB β and EJB β . Again, the expectation value JB β|Tc.m. |JB β is 34 hωδ ¯ JB JB δββ . The next step is to calculate the + Jx J α , two-body densities J α [(a + a + )Jy Ty × a ] one-body densities JA α [akτ lτl A A k l k
+ JB (am an )Jx Tx ]J,T MT JA α , and overlaps JB β| [akτk |JA α] . Note that the first two are only once reduced matrix elements. These quantities will assist in calculating the matrix elements of the Hamiltonian and + J unit operator for the A-particle system in the shell-model basis, [anlj τ ⊗ |JA α] . In the example, these would be the basis states of the compound system 7 Be. The basis includes states for 2n + l up to ρmax (n starts at zero). In the calculations shown below, ρmax = 16 is the minimum required to accurately determine the appropriate cut-off radii in the method of Philpott and George [10]. The basis states are not normalized and most have spurious components. One can no longer replace Tc.m. by 34 h¯ ω in HB . Instead, one must use
A A A A pi · pj pi2 A − 1 pi2 − Tc.m. + + . νij − νij = 2m A 2m Am i
i=1
i=1
(5)
i
If the initial or final nlj (τ ) corresponds to 0s1/2 (p) or 0s1/2 (n), the matrix elements are zero. If both the initial and final nlj (τ ) are in the 0p-shell, then J
J JA α| ⊗ anlj τ HB an+ l j τ ⊗ JA α J + J + = EJβ Jβ| anlj τ ⊗ |JA α Jβ| an l j τ ⊗ JA α β
and
(6)
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D. Halderson / Nuclear Physics A 707 (2002) 65–80 J
J JA α| ⊗ anlj τ 1 an+ l j τ ⊗ JA α J + J + = Jβ| anlj . τ ⊗ |JA α Jβ| an l j τ ⊗ JA α
(7)
β
If the initial state and final state are outside the 0p-shell, then J J JA α| ⊗ anlj τ HB an+ l j τ ⊗ JA α
= EJA α + 34 h¯ ω/A δnlj τ,n l j τ δJA JA δαα [Jx ] A − 1 p2 jjc τ τc (Jx ) ν j jc τ τc (Jx ) + nlj | nlj + A 2m [j ] Jx ,{nlj τ }c
× δJA JA δαα δlj τ,l j τ
J + [J ]Jˆx JA α aj+1 τ1 aj2 τ2 x JA α j1 j τ1 τ (J )ν j2 j τ2 τ (J )
× W (j, J , Jx , j2 ; j1 , j )W JA , J, Jx , j ; j, JA ,
(8)
where νij = νij − pi · pj /Am, Jˆx = (2Jx + 1)1/2 , and the last sum is over j1 , j2 , τ1 , τ2 , J , and Jx . The matrix elements of the unit operator are just δJA JA δαα δnlj τ,n l j τ . For initial state (j ) in the 0p-shell and the final state (j ) outside the 0p-shell, J J JA α| ⊗ anlj τ HB an+ l j τ ⊗ JA α j3 j (Jx Tx MTx )ν |j1 j2 (1 + δj1 j2 )−1/2 =
J T J T × JA α aj+3 aj+ y y (aj1 aj2 )Jx Tx M JA α Jˆx Jˆy Jˆ T
× W j , J, J , JA ; JA , j W J , j, Jy , j3 ; j , Jx 12 12 mt3 mtj Ty MTy × 12 12 mt3 mtj Tx MTx Ty Tx MTy − MTx |T MT (−1)MTx −Tx +1+j +j +J
A − 1 p2 + nlj | |0lj Jˆx W j , JA , j, JA ; J, Jx A 2m Jx J × JA α aj+ τ a0lj τ x JA α (−1)1+j +JA −J J + [J ]Jˆx JA α aj+1 τ1 aj2 τ2 x JA α j1 j τ1 τ (J )ν j2 j τ2 τ (J )
× W j, J , Jx , j2 ; j1 , j W JA , J, Jx , j ; j, JA [Jx ] A − 1 p2 + nlj | nlj + jjc τ τc (Jx ) ν j jc τ τc (Jx ) A 2m [j ] Jx ,{nlj τ }c
× δJA JA δαα δlj τ,l j τ ,
(9)
where the first sum is over j1 j2 , j3 , τ3 , Jx , Jy , Tx , Ty , MTx , MTy , J , T , and MT , and the third sum is as in Eq. (8). The matrix elements of the unit operator are zero. The mixture of proton–neutron and isospin notation is a convenience for calculation and storage. The convention used above corresponds to the proton having mt = − 12 ; hence, MTx = −1 implies inclusion of the Coulomb potential.
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The matrix elements of the unit operator and Hamiltonian are then transformed as in [1] to the core–nucleon center-of-mass system, thereby removing the spurious components. + J Antisymmetry is preserved in the transformation. Since the [anlj τ ⊗ |JA α] basis is more than complete when nlj τ is in the 0p-shell, one must transform the Hamiltonian and unit matrix to the |JB β basis. One must also remove the zero matrix elements for the 0s-shell. The S-matrix elements are calculated in the R-matrix formalism of Ref. [11]. This requires matrix elements up to a cut-off radius, ac . The matrix elements one has go to infinity. Therefore, one must subtract the contributions that are long range. These subtractions are only made to orbits outside the 0p-shell. Those in the 0p-shell go to zero before the cut-off radius. For the unit matrix this just involves the integrals ∞ Rnl (r)Rn l (r)r 2 dr,
δJA JA δαα δlj τ,l j τ ac
where now the oscillator constant is ν = (A − 1)ν0/A. The Coulomb potential has a similar integral ∞ δJA JA δαα δlj τ,l j τ δτp
Rnl (r)
Zc e 2 Rn l (r)r 2 dr, r
ac
since the core is contained within the cut-off. One must also correct the kinetic energies with the integral ∞ δJA JA δαα δlj τ,l j τ
Rnl (r)
p2 Rn l (r)r 2 dr, 2µ
ac
where µ = m(A − 1)/A. Once the Hamiltonian and unit matrix are truncated at the cut-off, all cross sections are determined.
3. Examples The difficulty in applying this powerful tool is the lack of an effective interaction with a proper theoretical foundation. Most of the available interactions were designed for DWBA calculations, with no regard for bound-state properties. Those people deriving effective interactions for shell-model calculations were not concerned with the matrix elements of the high-lying particle states. One knows that effective interaction theory provides the correct answer to bound-state energies, provided one works in a non-spurious basis [12], but the ambiguities that arise when extracting high-lying particle state matrix elements for use in scattering calculations are disconcerting. One effective interaction was found which provided an excellent fit to both the 4 He bound-state energy and the p + 3 H, n + 3 He, p + 4 He cross sections [5,13]. This was the M3Y interaction of Ref. [14]. The agreement was very surprising, since the M3Y was designed for use in DWBA calculations with medium mass nuclei. However, when
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applied to p + 6 Li in DWBA calculations or when folded to produce potentials for coupledchannels calculations [15,16], the M3Y interaction required severe normalization. The normalization factors in Ref. [16] were denoted by the enhancement factors of the multipole terms 2L + 1 ν˜ L (r, r1 , ρ0 ) = 2
+1
ν R = |r − r 1 |, ρ0 PL (cos θ ) d(cos θ ).
(10)
−1
For L = 0 these factors are 1.2, 1.2, 1.4, 1.4, and 1.0 at Ep = 24.4, 49.75, 65, 80, and 136 MeV, and for L = 2 they are 4.6, 3.0, 2.5, 2.5, and 2.5 at the respective energies. In Ref. [15] the normalization factors are denoted by the reduction in the radial factors of the central interaction in the form g = g00 + g10 σ · σ + g01 τ · τ + g11 σ · σ τ · τ . Based on the 1+ –3+ excitation in 6 Li, g00 was reduced by 28% at 24.4 and 45 MeV. This reduction was made after multiplying the corresponding form factor by two to account for the factor of four between the calculated and experimental B(E2). Based on 7 Li + p data, g01 was reduced by 28% at 26 MeV and 21% at 45 MeV, and g11 was reduced by 57% at 26 MeV and 45% at 45 MeV. In Ref. [17] the entire central potential was replaced with the complex, density and energy dependent interaction of Ref. [18] with the imaginary part multiplied by 0.8. The spin-dependent components of M3Y were retained. The drastic normalization factors for the A = 7 composite system were somewhat surprising, given the excellent agreement obtained with M3Y for A = 4 and 5. Therefore, the elastic and inelastic calculations for the A = 7 composite system are repeated in this work with the RCCSM formalism. In Refs. [15–17] the Cohen and Kurath [19] wave functions are employed, whereas the wave functions used in RCCSM calculations are consistent with the interaction. However, results with M3Y and Cohen and Kurath wave functions differ little. In fact, in Ref. [15] it was shown that pure L–S coupled wave functions give similar results. Therefore, a fair comparison can be made between DWBA results and those of the RCCSM. Fig. 1 shows the 6 Li + p elastic RCCSM cross sections as solid lines. These calculations include only the 6 Li ground state. The data are from Refs. [20–22]. The dotted lines at 24.4, 35, and 49.75 MeV are the calculations of Ref. [17] and the dashes lines at 24.4, 49.75, 65, 80, and 136 MeV are the calculations of Ref. [16]. These dashed and dotted curves include the above normalization factors. At 65, 80, and 136 MeV the RCCSM-M3Y calculation does as well as the normalized folding calculation. At 24.4, 35, and 49.75 MeV the magnitude of the RCCSM-M3Y calculations seem correct, but one is acutely aware of the back-angle rise of the calculation which does not appear in the data. The general shape of these elastic cross sections, with the back-angle rise, is the same as was calculated for p + 3 H and p + 4 He. However, in these cases the data rose at back-angles so the agreement was good. Folding calculations will not produce as strong a back-angle rise, and in most cases, none at all. In fact, one can take the identical nuclear density and interaction used in an RCCSM calculation, perform a folding calculation with knockout-exchange, and the back-angle cross section will be much smaller than the corresponding RCCSM calculation. The difference between the two is the same difference that one sees between a resonating group calculation and the corresponding generator coordinate calculation. In one case the translationally invariant dynamical equations are solved, and in the other a potential is
D. Halderson / Nuclear Physics A 707 (2002) 65–80
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Fig. 1. Elastic scattering cross sections for 6 Li + p. Solid lines are one-core-state RCCSM calculation with M3Y interaction. Dotted and dashed lines are calculations of Refs. [16,17], respectively. Data are from Refs. [20,21].
generated as a function of distance between objects. This difference was most clearly demonstrated in the early days of constructing NN potentials from three quarks incident on three quarks. The potentials determined by calculating the potential energy as a function of distance between bags showed little repulsive core. However, when the resonating group kernel was expanded about its nonlocality, the repulsive core appeared. Therefore, the folding calculations are fortunate in that error incurred by not solving the dynamical equations goes in the same direction as whatever mechanism is missing from the RCCSM calculation for describing for 6 Li + p. This mechanism must be present in 6 Li + p and suppressed in p + 3 He and p + 4 He. The most likely candidate is one-nucleon knockout. The loosely bound valence nucleons in 6 Li have little chance of remaining in the nucleus when the momentum transfer is large as in back-angle scattering. Therefore the fault most likely lies with a missing mechanism and not with the M3Y interaction. Although one cannot include one-nucleon knockout in the RCCSM, one can include the inelastic channels corresponding to core excitation. The p-shell allows 15 A = 6 core states, not all of which are observed. If all 15 states are included, then the RCCSM produces the results shown in Fig. 2. The effect of the channel coupling is greatest at low energies and diminishes with increasing energy. The magnitudes of the cross sections at 24.4, 35, and 136 MeV are good and are a few percent low in the 50–80 MeV region. Considering
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Fig. 2. Elastic scattering cross sections for 6 Li + p. Solid lines are 15-core-state RCCSM calculation with M3Y interaction. Dotted and dashed lines are calculations of Refs. [16,17], respectively. Data are from Refs. [20,21].
the large energy range and considering that M3Y does not depend on energy, this result is impressive. It was noted in Ref. [16] that the calculated analyzing power in 6 Li(p, p)6 Li at Ep = 49.75 MeV in Ref. [17] was very much out of phase at back angles. This calculation is shown in Fig. 3 as a dotted line along with the M3Y calculation of Ref. [16] as a dashed line and the M3Y-RCCSM calculation with one core state as a solid line. The data are from Ref. [21]. All calculations use the same spin-dependent interaction but achieve different results. The M3Y-RCCSM calculation demonstrates a similar difficulty as in the calculation of Ref. [17] in that the analyzing power does not dip negative near 120◦ . Channel coupling was given as one possible explanation for the analyzing power problem in Ref. [17]. The present RCCSM code is the only available means of thoroughly checking this possibility, for it allows coupling of all 15 A = 6 core states. In Fig. 4 is shown the elastic analyzing power at 49.75, 65, and 80 MeV. The data are from Refs. [21,23]. The dashed lines are from calculations with the one-core-state M3Y-RCCSM, and the solid lines are from the 15-core-state M3Y-RCCSM. It is true that the channel coupling drives the 49.75 MeV analyzing power negative near 120◦ and also for 80 MeV. The 15channel calculation seems to provide somewhat better agreement with the data. However, the improved agreement is more dramatic if one employs the effective interaction derived in Ref. [13] from the Reid Soft core for 90 Zr (RSC-90 Zr). These results are shown in
D. Halderson / Nuclear Physics A 707 (2002) 65–80
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Fig. 3. Elastic analyzing power for 6 Li + p at Ep = 49.75 MeV. Solid lines are one-core-state RCCSM calculations with M3Y interaction. Dotted and dashed lines are calculations of Refs. [16,17], respectively. Data are from Ref. [21].
Fig. 4. Elastic analyzing powers for 6 Li + p. Dashed and solid lines are one-core-state and 15-core-state RCCSM calculations with M3Y interaction, respectively. Data are from Refs. [21,23].
Fig. 5 where one sees that the 15-core-state calculation does a good job of reproducing the data. One would have liked to have the back-angle data at 65 MeV to further test the correspondence. One needs all 15 core states to obtain this agreement. No one core state has the dominant effect. The 6 Li states with predominately [11] spatial symmetry have just as much effect as the predominately [2] spatial symmetry states, and the 6 Be states have much the same effect as the 6 Li states.
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Fig. 5. Elastic analyzing powers for 6 Li + p. Dashed and solid lines are one-core-state and 15-core-state RCCSM calculations with RSC-90 Zr interaction, respectively. Data are from Refs. [21,23].
In addition to examining the elastic analyzing power, the authors of Ref. [16] also compared calculations to then new analyzing power measurements for 6 Li(p, p )6 Li(3+ ) [23]. Distorted wave calculations with M3Y did very poorly. The authors demonstrated that the use of a momentum and density dependent interaction [24] improved the calculations considerably. A density dependent interaction cannot be used in the RCCSM because of the lack of translational invariance. So, once again this calculation is repeated in the RCCSM formalism with M3Y to see if equivalent agreement can be obtained. p )6 Li(3+ ) are shown in The inelastic analyzing powers and cross section for 6 Li(p, Figs. 6 and 7, respectively, along with the data of Ref. [23]. The final DWBA calculations of Ref. [16] with the density and momentum dependent potential are shown as dotted lines. The potential was given normalization factors for L = 0 of 1.00, 0.75, 0.90, 1.00, and 1.00 at Ep = 24.4, 49.75, 65, 80, and 136 MeV, and all scale factors were 1.00 for L = 2. The M3Y-RCCSM calculations are shown as dashed lines for two core states and solid lines for 15 core states. The 65 MeV asymmetry clearly shows that the coupling of all channels brings the M3Y-RCCSM calculation into phase with the data, although not quite large enough in magnitude. The DWBA calculation is a bit too large in magnitude and slightly out of phase. At 80 MeV the channel coupling is forcing the calculation toward being in phase, but the magnitude is much too small, while the DWBA calculation has the correct magnitude, but is out of phase. It is unfortunate that the 49.75 MeV asymmetry data is so poor and does not go to back angles, since it is there that the coupled-channels RCCSM calculation differs significantly from the DWBA calculation. Therefore, one has no strong reason to reject M3Y based on these data. Comparing the inelastic cross sections is more difficult. The M3Y, DWBA results of Ref. [16] do not seem to be consistent with those of Ref. [15] in the following way. When both use M3Y, Ref. [15] requires a 28% reduction in g00 , but in Ref. [16] requires a 3–4 factor enhancement in ν˜2 . Also, both have apparently taken the inelastic transition
D. Halderson / Nuclear Physics A 707 (2002) 65–80
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Fig. 6. Inelastic analyzing powers for 6 Li(p, p )6 Li(3+ ). Dotted lines are for DWBA calculations of Ref. [16]. Dashed and solid lines are with M3Y interaction for two and 15-channel RCCSM calculations, respectively. Data are from Ref. [23].
Fig. 7. Inelastic cross sections for 6 Li(p, p )6 Li(3+ ). Dashed and solid lines are with M3Y interaction for two and 15-core-state RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr. Data are from Refs. [15,20,21,23].
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Fig. 8. Cross sections for 7 Li composite system. Open circles are 6 Li(n, n )6 Li(3+ ) data of Ref. [24], and crosses are 6 Li(n, p)6 He(g.s.) data of Ref. [29]. Dashed and solid lines are two-core-state and 15-core-state M3Y-RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr.
density from electron scattering and assumed equal enhancement for protons and neutrons. If Ref. [16] had used the shell-model form factor, then the calculations would be consistent, but the agreement with the data found in that work with the density dependent interaction would disappear when the calculations are multiplied by four to be consistent with the measured B(E2). Therefore, it seems more appropriate to make a comparison with Ref. [17], where progress was made on solving the following, related dilemma which is illustrated in Fig. 7 along with the data of Refs. [15,20,21,23]. Since the M3Y-RCCSM calculation is equivalent to using the shell-model transition density, all inelastic cross sections should be approximately the same factor of four too small. This is not the case. The results obtained for two core states, shown as dashed lines, essentially agree with the data at 24.4 and 35 MeV. The coupling of all 15 core states, shown as solid lines, improves the two-corestate results, but solid lines are more like a factor of two too small at 24.4, 49.75, 65, and 80 MeV and a factor of one at 35 and 136 MeV. The RSC-90 Zr results, shown as dotted lines, are slightly better at 35 MeV, but still not as expected. The problem persists in 6 Li(n, n )6 Li(3+ ) at En = 24.4 MeV as shown by the upper curves in Fig. 8, when compared to the data of Ref. [25]. By using the density dependent central force of Ref. [18], the authors of Ref. [17] were able to obtain consistency between the elastic cross sections and the 3+ cross sections. These calculations, which include the expected factor of four enhancement, are shown as dot-dashed lines in Fig. 7 at 24.4, 35, and 49.75 MeV. At 24.4 and 49.75 MeV the agreement at forward angles is good with some discrepancy at 35 MeV. However, the
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Fig. 9. Inelastic cross sections for 6 Li(p, p )6 Li(0+ ). Data are from Refs. [15,20,21]. Dashed and solid lines are two-core-state and 15-core-state M3Y-RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr.
RCCSM calculation at 24.4 MeV shows a severe correction due to coupling many core states. Such a correction could negatively affect the results in Ref. [17], but nonetheless, the agreement is impressive, and provides one case where M3Y does not provide equivalent agreement. The 6 Li(p, p )6 Li(0+ ) cross sections are shown in Fig. 9 along with the data of Refs. [15,20,21]. The solid and dashed lines correspond to M3Y-RCCSM with 15 and 2 core states, respectively. The 24.4 MeV, 15-core-state results seem very diffractive. Therefore, the same calculation with RSC-90 Zr is also shown as a dotted line to demonstrate that the diffractive appearance is a result of the interaction employed. The M3Y calculations are above the data for Ep = 45.4, 49.75, and 136 MeV. This is unexpected since the calculated 1+ –0+ B(M1) is in good agreement with the experimental value. In Ref. [17] the calculations were a factor of two above the data at 45.4 and 49.75 MeV and a slightly larger factor at 24.4 MeV. It was suggested in Ref. [17] that perhaps inclusion of additional core states and proper treatment of the center of mass and antisymmetry would improve the agreement with experiment. Indeed, the coupling of additional core states does improve the comparison with data, but not enough to claim agreement.
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Fig. 10. Inelastic cross sections for 6 Li(p, n)6 Be(g.s.). Data are from Ref. [26]. Dashed and solid lines are two-core-state and 15-core-state M3Y-RCCSM calculations, respectively. Dotted lines are 15-core-state results with RSC-90 Zr.
The difficulty also appears in the charge-exchange reaction 6 Li(p, n)6 Be(g.s.) for which is the analog. In Fig. 10 are plotted that data of Ref. [26] with the same three curves as in Fig. 9. The M3Y-RCCSM calculation gives good forward angle agreement with the data at 30.2 MeV, but becomes progressively worse up to 120 MeV, where it is a factor of two too large. The RSC-90 Zr calculation exhibits the same difficulty that was noted in Ref. [17]. Their calculations at low energies were a factor of two too large, whereas calculations [27] using the t-matrix interaction of Ref. [28] produce agreement with forward angle data at 120 MeV. The RSC-90 Zr calculation exhibits the same behavior, but with one interaction. Unfortunately the 6 Li(p, p )6 Li(0+ ), 136 MeV data did not go to forward angles, and also, the measured values are somewhat lower that expected [27]. However, the same forward angle agreement is obtained with RSC-90 Zr for 6 Li(n, p)6 He(g.s.) at 120 MeV as shown in the lower curves of Fig. 8 and compared to the data of Ref. [29]. The disagreement between calculated and measured cross sections for this 1+ –0+ , spin– flip transition was considered the major puzzle for nucleon scattering from 6 Li in Ref. [17]. It is even more puzzling in light of the RCCSM calculation, where M3Y does better at low energies and worse at higher energies while RSC-90 Zr behaves in the opposite manner. This is more puzzling because the RCCSM calculations show that channel coupling becomes less important at higher energies, hence one would infer that channels missing from the calculation would becomes less important at higher energies. One might then understand the behavior of RSC-90 Zr and the calculations of Refs. [17,27] in terms of channels not included in the calculation. However, the M3Y calculation invalidates that argument, at least to the extent that one believes M3Y is a viable interaction. This may be one more reason to reject M3Y for p-shell nuclei. Then one must say that significant modifications 6 Li(p, p )6 Li(0+ )
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occur to the effective interaction when moving from A = 4 and 5 to A = 7 composite systems.
4. Conclusion The primary purpose of this paper was to demonstrate that the continuum shell model with correction for target recoil has been extended into the p-shell and to present the formalism required for implementing this very powerful tool. Example calculations were performed for nucleon scattering in the A = 7 composite systems. These calculations were chosen for two reasons. First they provided an opportunity to test the M3Y interaction. This interaction had produced very good results in the A = 4 and 5 composite systems, and A = 7 represents the next step up in mass. Second, the A = 6 core has 15 p-shell states so that one can test the effect of channel coupling. The M3Y interaction performed quite well for elastic scattering cross sections and reasonably well in describing asymmetries, both elastic and inelastic. However, as found in previous works [15,17], the magnitudes of the forward angle inelastic cross sections were too large at low and moderate energies for 1+ –3+ excitation and too large at moderate and intermediate energies for 1+ –0+ excitation. Another interaction with stronger Pauli corrections, RSC-90 Zr, tended to give more consistent results, but the over estimate of low and moderate energy inelastic cross sections remains a problem. One has several choices to explain this difficulty. It could be a missing mechanism, missing channels, naïve structure of the core states, or renormalization of the interaction for the A = 7 system. Since the RCCSM includes multiple scattering effects, and since the effects of channel coupling diminish around 80 MeV, the first two possibilities are less likely. However, given the rather wide variation of the results with the different effective interactions, what is more likely is that the effective interactions employed are not consistent with the model space. The first step toward improving these calculations would be a more careful construction of the interaction, perhaps by folded diagram techniques. However, even at this level, the coupled-channels wave functions derived from M3Y should be very good candidates for final states in knockout reactions.
Acknowledgement This work was supported by the National Science Foundation under grant PHY9732634.
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