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A microscopic multi-channel calculation for the reactions 7Li(p, p)7Li and7Li(p, n)7Be
Nuclear Physics A2g6 (1977)
89-100 ; © North-Holland Publlshlng Co ., Amsterdam
Not to be reproduced by photoprlnt or microSlm without written permission liom the publisher
A NIICROSCOPIC 1VIULTI-CHANNEL CALCULATION FOR THE REACTIONS 'Li(p, p)'LI AND 'Li(p, n)'Be H . STtSWE and W. ZAHN
Institut fia Theoretische Physik der Unimersit6t ~u Köln, Köln, Germant Received 20 October 1976 Abstract : Based on a refined resonating group method a miaoscopic m~hi-channel calculation for the reactions 'Li(p, p)'Li and 'Li(p, n)'Be has been performed . Diagonal S-matrix elements and differential tzoss sections are presented from the 'Li-p threshold up tô E = 2 .5 MeV c.m . and compared with experimental data .
1. Introdactlon The BBe scattering system enjoys wide popularity among light nuclei and has been treated both experimentally and theoretically. A comprehensive summary of eBe data is given by Ajzenberg-Selove and Lauritsen ' ). For excitation energies below 15 MeV only a-a channels are open and the scattering states are very simple . In the region above, however, the sequence of scattering states becomes overly involved. The reactions'Li(p, p)'Li (E,h~e,hold = 17.256 MeV) and'Li(p, n)'Be (Ethreahela = 18.900 MeV), supposed to provide direct access to the complicated multi-channel reaction mechanisms, were subjected to thorough experimental investigations. An excellent survey thereof up till 1973 is given in two publications of Brown et ul.) [cf. refs. 2, 3), and references therein] . They carried out polarization measurements for the'Li(p, p)'Li reaction in an energy region from 0.6 to 2.15 MeV c.m . above threshold. A subsequent phase shift analysis provided some clarifying insight into tht; material : For the 1 + resonantxs at 17.64 and 18.15 MeV equal amounts of channel spin mixing were required . However, for the 1 + resonance at 17.64 MeV with a width of T .. 10.7 keV no polarization data have been obtained. The assignment of quantum ntunbers J" = 2 - to the 18 .91 MeV state was confirmed and in ref.') this state was shown to be a threshold state with a width of about 50 keV. In ref. 2) the 19.06 MeV resonance was identified as "!~ = 3 + ; the deduced phase shifts gave evidence that the 3 + states at 19.06 and 19.22 MeV might be coupled via a chargedependent interaction and provided some indication for a 1 - state at 19.4 MeV. Rohrer and Brown a) have measured angular distributions of the proton analyzing power of the'Li(p, n)'Be reaction from the threshold to 1 MeV excitation energy . Their results as well as results of Burke et al.'), who measured'Li(p, n)'Be angular distributions from threshold up to 1 .7 MeV excitation energy, could not give any 89
9o
H . STÖWE AND W . ZAHN
clarification about the magnitude of the coupling of the 3 + states at 19.06 and 19.22 MeV. In addition earlier differential cross-section measurements for the reaction 'Li(p, p)'Li by Waiters et al . and Malmberg e) and corresponding measurements for the reaction'Li(p, n)'Be by Bevington et al.') could be mentioned in this context. To our knowledge rigorous theoretical investigations of the 'Li(p, p)'Li and 'Li(p, n)'Be reactions, which could file the experimental data above the corresponding thresholds into one coherent description, have not yet been performed. However, calculations in the shell-model approach covering mass numbers from A = 5 to 16. gave the first consolidated theoretical explanations . An intermediate coupling shell-model calculation was presented by Barker 8), utilizing simple effective interactions and mass-dependent parameters. In that work the assignment of Jx = 3+ to the 19.06 MTV state was assumed and the coupling to the 3 + state via a chargedependent interaction at 19.22 MeV was postulated. A comprehensive survey of 1 p shell nuclei was given e.g. by Cohen and Kurath 9) who performed effective interaction shell-model calculations . Kumar and Barker 1 °) presented a many-level Rmatrix analysis of'Li(p, a)4He data, the results being in good agréement with shellmodel calculations . Projected Hartree-Fork calculations by Bouten et al. ") were intended to demonstrate the limits of a pure shell model . A recent work by Kumar 1 s) emphasizes the limits of traditional shell-model interpretations. Deformed BBe nuclei are claimed to overcome the typical difficulties associated with the fitting of levels in the lighter 1 p shell nuclei . The reference to deformation of nuclei challenges investigations in the framework of the cluster model picture. Corresponding attempts to tackle high-lying resonances of the BBe system were made recently' 3 " ' 4). A microscopic two-channel calculation ' a) involving a-a and a-a* channels, predicted a whole rotational band in BBe, starting at the a* threshold (20.2 MeV). Given these experiences theoretical investigations above 'Li-p threshold in the framework of a microscopic nuclear theory became very much desirable and will be the subject of this paper. In sect. 2 we describe details of our calculations and in sect . 3 we present a compilation of elastic phase shifts and the corresponding inelastic parameters . Calculated differential cross sections are presented and compared with experimental data. Finally, in sect . 4 we shall give a~recapitulation of the most essential findings. 2. The alwlaflon
The eight-particle Schrôdinger equation is solved according to a generalized Kohn-Hulthén variational principle Here A denotes the reactance matrix ; the vector function W consists of linearly
MULTI-CHANNEL CALCULATION
9l
independent trial functions ~`i, . . . ~ corresponding to proper boundary conditions . The Hamiltonian contains kinetic energy, central, Coulomb, spin-orbit and tensor potentials and is the same as has been used as the nucleon-nucleon model potential in most of the calculations of the Hackenbroich group ; all potentials are approximated by sums of Gaussians and fitted exclusively to two-body data 1 s - l'). ps explained in ref.'s) this Hamiltonian should only be used in connection with Jastrow correlations which were chosen to be
with dl = 0.6 and dZ = 3.0 fm -Z. The approximate treatment offin the evaluation of matrix elements has been discussed in ref. ! s). Throughout this paper we shall pay attention to all systems consisting of two fragments which are stable under particle decay . That is to say we shall exclusively be concerned with the following six fragmentations : a + a, 'Li + p, 'Li* + p, 'Be +n, 'Be*+n and 6Li+d. Here and in what follows the asterisk denotes the first excited state of'Li(478 ke~~ or'Be(429 key. Our ansatz for the eight-nucleon wave functions reads `pt
= ~l~~k(Fk(RkNLk+al.kGk(Rkl+ [,,bl.k.nuYk.n~(Rkl)}~ k m
Here .sad denotes the antisymmetrization operator, k denotes the channel index and 1 the corresponding boundary. condition ; a and b are variational parameters . The functions Fare regular Coulomb functions, G irregular Coulomb functions multiplied by a regularization term ! s) and X square integrable functions of Gaussian type . The surface function ~ is built up in the form ~k
-
~~4rkk~14'k .2~ SkYLr(~Rk)~ 'Rkk '
(4)
The functions cp represent products of internal wave functions and appropriate spin isospin functions of the fragments and are given explicitly in the appendix . Further j is the spin of the fragment, S is the channel spin and L the angular momentum of the relative motion function, with S and L coupled to total angular momentum J. Because of limitations of computer capacity we restricted ourselves to angular momenta L = 0, 2 in the relative motion of a+a fragments, L = 0-2 for'Li+p, 'Li* + p, 'Be +n, 'Be* + n, and L = 0-3 in the relative motion of 6Li + d fragments. The numerical calculations have been performed at the CDC 7200/7600 of the computing center at the University of Cologne using Hackenbroich's "refined cluster model" program chain le). 3. Resalts We shall restrict our investigations to an energy region from the'Li-p threshold
H. STöWE AND W. ZAHN
92
up to E = 2.5 MeV c.m . where various experimental data exist. The quality of the calculated S-matrix is one of the more crucial problems emerging in connection with any microscopic multi-channel calculation. First, one has to allow for all relevant channels which contribute to the reaction mechanism under consideration, even if such a channel (in this case 6Li-d) is not yet open. Such a channel provides additional freedom for the variation in the interaction region and will be called a distortion channel in what follows. Second, in all channels the participating structures and the threshold energies must be described appropriately. With the wave functions and parameters listed in the appendix we obtain the threshold energies of table 1 . The calculated spectrum is given relative to the experimental 'Li-p threshold. Our calculated thresholds are lying somewhat closer and in consequence energy shifts in the S-matrix elements will result . 3.1 . CALCULATED PHASE SHIFTS AND INELASTIC PARAMETERS
We choose the convenient parametrization of the diagonal S-matrix in terms of phase shifts S~ and inelastic parameters rl',.s : S = nez~a. For future convenience we shall also introduce the notation zs+'L,, phase, using the symbols S, P, D for L = 0, 1, 2.
F
O
Fig. 1 . Phase shifts b and inelastic parameters n for J` = 0* .
i-p sP
Li
2 3 E I~1
Fig. 2 . Phase shifü b and inelastic parameters q for J` = 1 * .
P
~Ba - n 3
aa 'D~
0
1
~Li _ P 3 Pz
2 3 E (1~V I
Fg. 3 . Phase shifts d and inelastic parameters p for J` = 2* .
p
P~
94
H . STÖWE AND W. ZAHN
P
~O
.90 E (MeVI Fig . 4 . Phase shifts b and inelastic parameters q
E 1MeVj Fig. 5. Phase shifts b and inelastic parameters q
4 9 ~O
0
1
2 3 E [MeVI Fig. 6 . Phase shifts 8 and inelastic parameters q for J` = 1- . E 1MeVl
F
R 0 90
2 0
1
0 _1 _Z
Fig . 7 . Phase shifts b and inelastic parameters q for J` = 2 - .
3'
3 1 ~3 " 2""= 3`
3' 0.1 ;1 1;2'~ ?a
...~ 1 " . 2~ ."
____ ~U_p
exp.
cak .
_~_
Fig. 8. Calculated and experimental energy apdctra .
MULTI-CHANNEL CALCULATION
95
In figs. 1 to 7 we have compiled phase shifts and inelastic parameters according to the total J~ values 0 + up to 3+ and 0 - to 2- . In the lower parts of these figures phase shifts are plotted versus c.m . energy relative to the'Li-p threshold ;the relevant inelastic parameters are displayed in the upper parts. The figures are marked by the channel fragmentation and the respective quantum numbers . The D-waves are not displayed since they do not show relevant features. In fig. 8 we have proposed a spectrum which may be read off from the results given in figs . 1 to 7. This spectrum reveals some interesting aspects of our calculation. The calculated resonances are shifted in energy since our description of thresholds does not reproduce experimental findings correctly. The sequence of 2+, 1 + and 3 + resonance pairs appears in the correct way. This is only true if all relevant channels are properly taken into account. To examine this point closely we have performed a calculation omitting the 6Li-d distortion channel . This fragmentation turns out to have its most important influence for Jx = 3 + . The 'Li-p and 'Be-n channels are weakly coupled if only these two channels are allowed. After inclusion of the 6Li-d channel, which, above the 6Li-d threshold, is only weakly coupled to the'Li-p and 'Be-n structures, strongly coupled 'Li-p and 'Be-n channels result (cf. fig. 4). The sequence of the resonances depends crucially on spin-orbit and tensor forces which, like all other potentials, are taken from ref. "). The relative positions of the calculated resonances indicate a minor inadequacy of components of the utilized model potentials ; especially the overlap of the calculated 1 + and 2+ isospin pairs might be related to a too modest magnitude of the tensor potential. The negative parity states are connected by a common behavior in their S-waves. The'Li-p and 'Li*-p phase shifts show irregularities (resonances and Wigner cusps) at the'Be-n and 'Be*-n thresholds, respectively (cf: figs . 5-7). The 2- , 1 - states appear in the correct sequence . Finally, the occurrence of a 0- resonance and a second 1 - level is predicted (cf. figs. 5 and 6). These states have not yet been detected experimentally . 3.2 . DIFFERENTIAL CROSS SECTIONS
For the calculated differential cross sections wehave chosen a representation which takes care of the discrepancies between the calculated and experimental threshold energies displayed in table 1 . A transformation in the energy scale is performed such that the calculated thresholds coincide with experimental findings . We shall concentrate our main interest on results for the reactions'Li(p, p)'Li and'Li(p, n)'Be. In ßg. 9 calculated differential cross sections for the reaction 'Li(p, p)'Li are plotted versus c.m . excitation energy. Results are presented for c.m . angles 8 = 50°, 70°, 90°, 110°, 130° and 150` and compared with experimental data . The differential cross section (dQ/dla in b/sr) measurements are from Warters et al. for EP S 1 .4 MeV and from Malmberg for Ep > 1 .4 MeV [ref. 6 )]. In fig. 10 calculated differential cross sections are plotted for the reaction 'Li(p, n)'Be relative to the'Be-n threshold. The experimental data are taken from
96
H . STÜWE AND W. ZAHN TARLI : 1 Calculated and experimental threshold energies in MeV relative to the'Li-p threshold Experimental ') "Li +d 'ge'+n 'Be +n 'Li " +p 'Li +p a+a
Calculated
5 .026 2 .073 1 .644 0 .478 0 .0 -17 .348
3 .807 1 .694 1 .220 0 .483 0 .0 -16 .712
') Ref. ' ) .
E [MeVI
2
Fig. 9 . Calculated differential cross sections do/dß in b/sr for the reaction'Li(p, p)'Li for c.m . angles 8 = SO', 70', 90', 110', 130' and 150' relative to the'Li-p threshold. Data points are taken from ref. 6).
Burke et aI. s) and from Bevington et al.') who have performed cross-section measurements (dQ/dia in mb/sr) for several lab angles . Their results, shown in fig. 10, have been converted into the c.m . frame. The calculated differential cross sections show a similar structure as the underlying experimental data. Exact conformity cannot be expected since thecalculated spectrum does not coincide with experimental findings in detail ; adjustments concerning the calculated spectrum have not been performed. For scattering angles in the forward
MULTI-CHANNEL CALCULATION
97
50
Fig. 10 . Calculated differential cross sections da/dß in mb/sr for the reaction 'Li(p, n)'Be for c.m . angles 8 = 50', 70', 90°, 110°, 130' and 150' relative to the'He-n threshold. Data points are taken from refs.'') and wnverted into the c.m . frame.
direction experiment and theory show a rise of differential cross sections for both reactions under consideration. For the'Li(p, p)'Li reaction calculated and measured results show the same tendency, the rise of cross sections for smaller angles being due to Coulomb effects. For the reaction 'Li(p, n)'Be we find satisfactory agreement of calculated and measured data for scattering angles in the backward direction. In the forward direction, however, our results predict a considerably stronger rise of cross sections compared to experimental findings . A calculation where the 6Li-d structure has been omitted displays a less pronounced forward-peaked behavior. This is an additional indication of a crucial influence of the 6Li-d distortion channel. A detailed exhibition of the'Li(p, p')'Li* and'Li(p, n')'Be* results is renounced since they would not supply us with new information. As a marginal note it is perhaps worth mentioning that our results reproduce the experimental data of Bevington et al. ') for the 'Li(p, n')'Be* differential cross sections quite nicely . Calculated polarization analyzing powers are not reported . Because of inadequacies ofthe model potential and the calculated energy spectrtun conclusive results can not be expected. 4. Conclosion
In the energy region above 'Li-p threshold to which we have paid attention in this paper, the states of the eight-nucleon compound system are generated by a
98
H. STÖWE AND W . ZAHN
complicated multi-channel reaction mechanism. All systems consisting of two fragments which are stable under particle decay such as a-a,'Li-p,'Li*-p,'Be-n,'Be*-n and 6Li-d, yield a significant contribution. The a-a fragmentation plays a predominant role whenever spin and parity allow its formation. The'Li-p and'Li*-p systems show resonant behavior at the corresponding'Be-n and'Be*-n thresholds, respectively. The 6Li~ fragmentation included as a distortion channel turns out to be of considerable influence on the coupling of the 'Li-p and 'Be-n channels and the differential cross sections for forward angles . The experimental situation is comprehensively described in the literature [cf., e.g. refs. z' a)]. Cross-section measurements and polarization analyzing power measurements have been presented by various authors and are in good agreement. The identification of the experimentally detected levels does not raisç questions about its reliability. Our calculation reproduces substantially the experimental findings and moreover predicts the occurrence of two additional levels in the energy spectrum . As a first microscopic multi-channel scattering calculation for more than six nucleons it demonstrates the high efficiency of the utilized computer program chain 1 e) and gives rise to the hope that similarly complicated systems can be investigated by our method in the near future . The authors want to thank W. Weyer for helpful discussions. Financial support of the "Bundesministerium für Forschung und Technologie" is gratefully acknowledged. Appendix
Here we shall give explicit expressions for the wave functions of the fragments and thecorresponding sets of parameters which have been used for the present calculation. The deuteron function z with ßi = 0.3204, c, = 0.5320,
ßz = 0.0403, cz = 0.1492,
yields a binding energy EB = -1 .352 MeV . Eiere and in what follows ~ denotes the appropriate spin-isospin functions. The a-function cp(r l , . . ., r4) _ ~ c~, exp { - âß~ ~ (ri-rl)z}~ ~=i
t
(A .2)
99
MULTI-CHANNEL CALCULATION
with
ßi = 0.3240,
ßz = 0.0840,
cl = 5.2018,
c z = 0.0658,
yields EH = -21 .476 MeV. The 6Li function 4 Z ~P(ri, . . ., r 6) = exp { T~ ~ (ri - r;) } exP { - ~rs - rb)z } ~P { - SRz}~, +<; with S = 0.0420, y = 0.2369, ß = 0.2768,
(A.3)
iz ri+rz+r3+r4) - V s~rs+re),
R =
yields EH = -21 .086 MeV. The'Li(j = ~) and~'Li*(j = ~) functions
x exp { - 3Y~ Wlth
ß l = 0.3169, S 1 = 0.0434, R
_
~
s.st< ;s~
(ri-r;)z} exp { -S~,Rz}R[Yi(~~~~,
ßz = 0.2207,
yl = 0.2895,
(A.4)
yz = 0.1201,
cz = 0.4519, cl = 8.2287, 2 3 (rs+r6+r ~~ (rl+rz+r3+r4)21 21 S z = 0.0298,
yield EB = -26.240 MeV for'Li and EB = -25 .757 MeV for'Li* . The 'Be(j = ~) and 'Be*(j = }) functions
x exp { - 3?'~ With
ß l = 0.2439, Yi = 0.1070, Sl = 0.0259, cl = 0.3898, R
321
~
sgt< ;s~
;)z} exp { -S~,Rz}R[Yl(A,e~^]~, (ri-r
ßz = 0.3076,
ßa = 0.2910,
S z = 0.0142,
S 3 = 0.0708,
yz = 0.3234, cz = 1.2116,
(rl +rz +r3 +r4)-
ya = 0.2473,
J 21
c3 = 6.9461, (rs+r6+r7~
yield Es = - 25 .020 MeV for 'Be and EH = - 24.546 MeV for 'Be* .
(A.5)
100
H. STÖWE AND W. ZAHN
References 1) F. Ajzenberg-Selove and T. Lauritsen, Nucl . Phys . A227 (1974) 1 2) . L. Brown, E. Steiner, L. G. Arnold and R. G. Seyler, Nucl. Phys. A206 (1973) 353 3) L. G. Arnold, R. G. Seyler, L. Hrown, T. I. Honner and E. Steiner, Phys. Rev. Lett . 32 (1974) 895 4) U. Rohrer and L. Brown, Nucl . Phys. A217 (1973) 525 5) C. A. Burke, M. T. Lunnon and H. W. Lefevre, Phys . Rev. C10 (1974) 1299 6) W. D. Waiters, W. A. Fowler and C. C. Lauritsen, Phys . Rev. 91 (1953) 917 ; P. R. Malmberg, Phys . Rev. 101 (1956) 114 7) P. R. Hevington, W. W. Rolland and H. W. Lewis, Phys. Rev. 121 (1961) 871 8) F. C. Barker, Nucl . Phys. 83 (1966) 418 9) S. Cohen and D. Kurath, Nucl . Phys. 73 (1965) 1 ; 89 (1966) 707; A101(1967) l ; A141(1970) 145 10) N. Kumar and F. C. Barker, Nucl . Phys. A167 (1971) 434 11) M. Bouter, P. van Leuven, H. Depuydt and L. Schotsmans, Nucl. Phys . A100 (1967) 90 12) N. Kumar, Nucl . Phys. A225 (1974) 221 13) D. Fick, H. H. Hackenbroich, T. H. Seligman and W. Zahn, Phys . Lett . 62B (1976) 121 14) H. StBwe and W. Zahn, to be published 15) H. H. Hackenbroich, Symp. on present status and novel developments in the nuclear many-body problem, Rome, 1973 (Editrice Compositors, Bologna) 16) H. Eikemeier and H. H. Hackcnbroich, Z. Phys . 197 (1966) 412 17) H. Eikemeier and H. H. Hackenbroich, Nucl . Phys . A169 (1971) 407 18) H. H. Hackenbroich, Z. Phys . 231 (1970) 216
89-100 ; © North-Holland Publlshlng Co ., Amsterdam
Not to be reproduced by photoprlnt or microSlm without written permission liom the publisher
A NIICROSCOPIC 1VIULTI-CHANNEL CALCULATION FOR THE REACTIONS 'Li(p, p)'LI AND 'Li(p, n)'Be H . STtSWE and W. ZAHN
Institut fia Theoretische Physik der Unimersit6t ~u Köln, Köln, Germant Received 20 October 1976 Abstract : Based on a refined resonating group method a miaoscopic m~hi-channel calculation for the reactions 'Li(p, p)'Li and 'Li(p, n)'Be has been performed . Diagonal S-matrix elements and differential tzoss sections are presented from the 'Li-p threshold up tô E = 2 .5 MeV c.m . and compared with experimental data .
1. Introdactlon The BBe scattering system enjoys wide popularity among light nuclei and has been treated both experimentally and theoretically. A comprehensive summary of eBe data is given by Ajzenberg-Selove and Lauritsen ' ). For excitation energies below 15 MeV only a-a channels are open and the scattering states are very simple . In the region above, however, the sequence of scattering states becomes overly involved. The reactions'Li(p, p)'Li (E,h~e,hold = 17.256 MeV) and'Li(p, n)'Be (Ethreahela = 18.900 MeV), supposed to provide direct access to the complicated multi-channel reaction mechanisms, were subjected to thorough experimental investigations. An excellent survey thereof up till 1973 is given in two publications of Brown et ul.) [cf. refs. 2, 3), and references therein] . They carried out polarization measurements for the'Li(p, p)'Li reaction in an energy region from 0.6 to 2.15 MeV c.m . above threshold. A subsequent phase shift analysis provided some clarifying insight into tht; material : For the 1 + resonantxs at 17.64 and 18.15 MeV equal amounts of channel spin mixing were required . However, for the 1 + resonance at 17.64 MeV with a width of T .. 10.7 keV no polarization data have been obtained. The assignment of quantum ntunbers J" = 2 - to the 18 .91 MeV state was confirmed and in ref.') this state was shown to be a threshold state with a width of about 50 keV. In ref. 2) the 19.06 MeV resonance was identified as "!~ = 3 + ; the deduced phase shifts gave evidence that the 3 + states at 19.06 and 19.22 MeV might be coupled via a chargedependent interaction and provided some indication for a 1 - state at 19.4 MeV. Rohrer and Brown a) have measured angular distributions of the proton analyzing power of the'Li(p, n)'Be reaction from the threshold to 1 MeV excitation energy . Their results as well as results of Burke et al.'), who measured'Li(p, n)'Be angular distributions from threshold up to 1 .7 MeV excitation energy, could not give any 89
9o
H . STÖWE AND W . ZAHN
clarification about the magnitude of the coupling of the 3 + states at 19.06 and 19.22 MeV. In addition earlier differential cross-section measurements for the reaction 'Li(p, p)'Li by Waiters et al . and Malmberg e) and corresponding measurements for the reaction'Li(p, n)'Be by Bevington et al.') could be mentioned in this context. To our knowledge rigorous theoretical investigations of the 'Li(p, p)'Li and 'Li(p, n)'Be reactions, which could file the experimental data above the corresponding thresholds into one coherent description, have not yet been performed. However, calculations in the shell-model approach covering mass numbers from A = 5 to 16. gave the first consolidated theoretical explanations . An intermediate coupling shell-model calculation was presented by Barker 8), utilizing simple effective interactions and mass-dependent parameters. In that work the assignment of Jx = 3+ to the 19.06 MTV state was assumed and the coupling to the 3 + state via a chargedependent interaction at 19.22 MeV was postulated. A comprehensive survey of 1 p shell nuclei was given e.g. by Cohen and Kurath 9) who performed effective interaction shell-model calculations . Kumar and Barker 1 °) presented a many-level Rmatrix analysis of'Li(p, a)4He data, the results being in good agréement with shellmodel calculations . Projected Hartree-Fork calculations by Bouten et al. ") were intended to demonstrate the limits of a pure shell model . A recent work by Kumar 1 s) emphasizes the limits of traditional shell-model interpretations. Deformed BBe nuclei are claimed to overcome the typical difficulties associated with the fitting of levels in the lighter 1 p shell nuclei . The reference to deformation of nuclei challenges investigations in the framework of the cluster model picture. Corresponding attempts to tackle high-lying resonances of the BBe system were made recently' 3 " ' 4). A microscopic two-channel calculation ' a) involving a-a and a-a* channels, predicted a whole rotational band in BBe, starting at the a* threshold (20.2 MeV). Given these experiences theoretical investigations above 'Li-p threshold in the framework of a microscopic nuclear theory became very much desirable and will be the subject of this paper. In sect. 2 we describe details of our calculations and in sect . 3 we present a compilation of elastic phase shifts and the corresponding inelastic parameters . Calculated differential cross sections are presented and compared with experimental data. Finally, in sect . 4 we shall give a~recapitulation of the most essential findings. 2. The alwlaflon
The eight-particle Schrôdinger equation is solved according to a generalized Kohn-Hulthén variational principle Here A denotes the reactance matrix ; the vector function W consists of linearly
MULTI-CHANNEL CALCULATION
9l
independent trial functions ~`i, . . . ~ corresponding to proper boundary conditions . The Hamiltonian contains kinetic energy, central, Coulomb, spin-orbit and tensor potentials and is the same as has been used as the nucleon-nucleon model potential in most of the calculations of the Hackenbroich group ; all potentials are approximated by sums of Gaussians and fitted exclusively to two-body data 1 s - l'). ps explained in ref.'s) this Hamiltonian should only be used in connection with Jastrow correlations which were chosen to be
with dl = 0.6 and dZ = 3.0 fm -Z. The approximate treatment offin the evaluation of matrix elements has been discussed in ref. ! s). Throughout this paper we shall pay attention to all systems consisting of two fragments which are stable under particle decay . That is to say we shall exclusively be concerned with the following six fragmentations : a + a, 'Li + p, 'Li* + p, 'Be +n, 'Be*+n and 6Li+d. Here and in what follows the asterisk denotes the first excited state of'Li(478 ke~~ or'Be(429 key. Our ansatz for the eight-nucleon wave functions reads `pt
= ~l~~k(Fk(RkNLk+al.kGk(Rkl+ [,,bl.k.nuYk.n~(Rkl)}~ k m
Here .sad denotes the antisymmetrization operator, k denotes the channel index and 1 the corresponding boundary. condition ; a and b are variational parameters . The functions Fare regular Coulomb functions, G irregular Coulomb functions multiplied by a regularization term ! s) and X square integrable functions of Gaussian type . The surface function ~ is built up in the form ~k
-
~~4rkk~14'k .2~ SkYLr(~Rk)~ 'Rkk '
(4)
The functions cp represent products of internal wave functions and appropriate spin isospin functions of the fragments and are given explicitly in the appendix . Further j is the spin of the fragment, S is the channel spin and L the angular momentum of the relative motion function, with S and L coupled to total angular momentum J. Because of limitations of computer capacity we restricted ourselves to angular momenta L = 0, 2 in the relative motion of a+a fragments, L = 0-2 for'Li+p, 'Li* + p, 'Be +n, 'Be* + n, and L = 0-3 in the relative motion of 6Li + d fragments. The numerical calculations have been performed at the CDC 7200/7600 of the computing center at the University of Cologne using Hackenbroich's "refined cluster model" program chain le). 3. Resalts We shall restrict our investigations to an energy region from the'Li-p threshold
H. STöWE AND W. ZAHN
92
up to E = 2.5 MeV c.m . where various experimental data exist. The quality of the calculated S-matrix is one of the more crucial problems emerging in connection with any microscopic multi-channel calculation. First, one has to allow for all relevant channels which contribute to the reaction mechanism under consideration, even if such a channel (in this case 6Li-d) is not yet open. Such a channel provides additional freedom for the variation in the interaction region and will be called a distortion channel in what follows. Second, in all channels the participating structures and the threshold energies must be described appropriately. With the wave functions and parameters listed in the appendix we obtain the threshold energies of table 1 . The calculated spectrum is given relative to the experimental 'Li-p threshold. Our calculated thresholds are lying somewhat closer and in consequence energy shifts in the S-matrix elements will result . 3.1 . CALCULATED PHASE SHIFTS AND INELASTIC PARAMETERS
We choose the convenient parametrization of the diagonal S-matrix in terms of phase shifts S~ and inelastic parameters rl',.s : S = nez~a. For future convenience we shall also introduce the notation zs+'L,, phase, using the symbols S, P, D for L = 0, 1, 2.
F
O
Fig. 1 . Phase shifts b and inelastic parameters n for J` = 0* .
i-p sP
Li
2 3 E I~1
Fig. 2 . Phase shifü b and inelastic parameters q for J` = 1 * .
P
~Ba - n 3
aa 'D~
0
1
~Li _ P 3 Pz
2 3 E (1~V I
Fg. 3 . Phase shifts d and inelastic parameters p for J` = 2* .
p
P~
94
H . STÖWE AND W. ZAHN
P
~O
.90 E (MeVI Fig . 4 . Phase shifts b and inelastic parameters q
E 1MeVj Fig. 5. Phase shifts b and inelastic parameters q
4 9 ~O
0
1
2 3 E [MeVI Fig. 6 . Phase shifts 8 and inelastic parameters q for J` = 1- . E 1MeVl
F
R 0 90
2 0
1
0 _1 _Z
Fig . 7 . Phase shifts b and inelastic parameters q for J` = 2 - .
3'
3 1 ~3 " 2""= 3`
3' 0.1 ;1 1;2'~ ?a
...~ 1 " . 2~ ."
____ ~U_p
exp.
cak .
_~_
Fig. 8. Calculated and experimental energy apdctra .
MULTI-CHANNEL CALCULATION
95
In figs. 1 to 7 we have compiled phase shifts and inelastic parameters according to the total J~ values 0 + up to 3+ and 0 - to 2- . In the lower parts of these figures phase shifts are plotted versus c.m . energy relative to the'Li-p threshold ;the relevant inelastic parameters are displayed in the upper parts. The figures are marked by the channel fragmentation and the respective quantum numbers . The D-waves are not displayed since they do not show relevant features. In fig. 8 we have proposed a spectrum which may be read off from the results given in figs . 1 to 7. This spectrum reveals some interesting aspects of our calculation. The calculated resonances are shifted in energy since our description of thresholds does not reproduce experimental findings correctly. The sequence of 2+, 1 + and 3 + resonance pairs appears in the correct way. This is only true if all relevant channels are properly taken into account. To examine this point closely we have performed a calculation omitting the 6Li-d distortion channel . This fragmentation turns out to have its most important influence for Jx = 3 + . The 'Li-p and 'Be-n channels are weakly coupled if only these two channels are allowed. After inclusion of the 6Li-d channel, which, above the 6Li-d threshold, is only weakly coupled to the'Li-p and 'Be-n structures, strongly coupled 'Li-p and 'Be-n channels result (cf. fig. 4). The sequence of the resonances depends crucially on spin-orbit and tensor forces which, like all other potentials, are taken from ref. "). The relative positions of the calculated resonances indicate a minor inadequacy of components of the utilized model potentials ; especially the overlap of the calculated 1 + and 2+ isospin pairs might be related to a too modest magnitude of the tensor potential. The negative parity states are connected by a common behavior in their S-waves. The'Li-p and 'Li*-p phase shifts show irregularities (resonances and Wigner cusps) at the'Be-n and 'Be*-n thresholds, respectively (cf: figs . 5-7). The 2- , 1 - states appear in the correct sequence . Finally, the occurrence of a 0- resonance and a second 1 - level is predicted (cf. figs. 5 and 6). These states have not yet been detected experimentally . 3.2 . DIFFERENTIAL CROSS SECTIONS
For the calculated differential cross sections wehave chosen a representation which takes care of the discrepancies between the calculated and experimental threshold energies displayed in table 1 . A transformation in the energy scale is performed such that the calculated thresholds coincide with experimental findings . We shall concentrate our main interest on results for the reactions'Li(p, p)'Li and'Li(p, n)'Be. In ßg. 9 calculated differential cross sections for the reaction 'Li(p, p)'Li are plotted versus c.m . excitation energy. Results are presented for c.m . angles 8 = 50°, 70°, 90°, 110°, 130° and 150` and compared with experimental data . The differential cross section (dQ/dla in b/sr) measurements are from Warters et al. for EP S 1 .4 MeV and from Malmberg for Ep > 1 .4 MeV [ref. 6 )]. In fig. 10 calculated differential cross sections are plotted for the reaction 'Li(p, n)'Be relative to the'Be-n threshold. The experimental data are taken from
96
H . STÜWE AND W. ZAHN TARLI : 1 Calculated and experimental threshold energies in MeV relative to the'Li-p threshold Experimental ') "Li +d 'ge'+n 'Be +n 'Li " +p 'Li +p a+a
Calculated
5 .026 2 .073 1 .644 0 .478 0 .0 -17 .348
3 .807 1 .694 1 .220 0 .483 0 .0 -16 .712
') Ref. ' ) .
E [MeVI
2
Fig. 9 . Calculated differential cross sections do/dß in b/sr for the reaction'Li(p, p)'Li for c.m . angles 8 = SO', 70', 90', 110', 130' and 150' relative to the'Li-p threshold. Data points are taken from ref. 6).
Burke et aI. s) and from Bevington et al.') who have performed cross-section measurements (dQ/dia in mb/sr) for several lab angles . Their results, shown in fig. 10, have been converted into the c.m . frame. The calculated differential cross sections show a similar structure as the underlying experimental data. Exact conformity cannot be expected since thecalculated spectrum does not coincide with experimental findings in detail ; adjustments concerning the calculated spectrum have not been performed. For scattering angles in the forward
MULTI-CHANNEL CALCULATION
97
50
Fig. 10 . Calculated differential cross sections da/dß in mb/sr for the reaction 'Li(p, n)'Be for c.m . angles 8 = 50', 70', 90°, 110°, 130' and 150' relative to the'He-n threshold. Data points are taken from refs.'') and wnverted into the c.m . frame.
direction experiment and theory show a rise of differential cross sections for both reactions under consideration. For the'Li(p, p)'Li reaction calculated and measured results show the same tendency, the rise of cross sections for smaller angles being due to Coulomb effects. For the reaction 'Li(p, n)'Be we find satisfactory agreement of calculated and measured data for scattering angles in the backward direction. In the forward direction, however, our results predict a considerably stronger rise of cross sections compared to experimental findings . A calculation where the 6Li-d structure has been omitted displays a less pronounced forward-peaked behavior. This is an additional indication of a crucial influence of the 6Li-d distortion channel. A detailed exhibition of the'Li(p, p')'Li* and'Li(p, n')'Be* results is renounced since they would not supply us with new information. As a marginal note it is perhaps worth mentioning that our results reproduce the experimental data of Bevington et al. ') for the 'Li(p, n')'Be* differential cross sections quite nicely . Calculated polarization analyzing powers are not reported . Because of inadequacies ofthe model potential and the calculated energy spectrtun conclusive results can not be expected. 4. Conclosion
In the energy region above 'Li-p threshold to which we have paid attention in this paper, the states of the eight-nucleon compound system are generated by a
98
H. STÖWE AND W . ZAHN
complicated multi-channel reaction mechanism. All systems consisting of two fragments which are stable under particle decay such as a-a,'Li-p,'Li*-p,'Be-n,'Be*-n and 6Li-d, yield a significant contribution. The a-a fragmentation plays a predominant role whenever spin and parity allow its formation. The'Li-p and'Li*-p systems show resonant behavior at the corresponding'Be-n and'Be*-n thresholds, respectively. The 6Li~ fragmentation included as a distortion channel turns out to be of considerable influence on the coupling of the 'Li-p and 'Be-n channels and the differential cross sections for forward angles . The experimental situation is comprehensively described in the literature [cf., e.g. refs. z' a)]. Cross-section measurements and polarization analyzing power measurements have been presented by various authors and are in good agreement. The identification of the experimentally detected levels does not raisç questions about its reliability. Our calculation reproduces substantially the experimental findings and moreover predicts the occurrence of two additional levels in the energy spectrum . As a first microscopic multi-channel scattering calculation for more than six nucleons it demonstrates the high efficiency of the utilized computer program chain 1 e) and gives rise to the hope that similarly complicated systems can be investigated by our method in the near future . The authors want to thank W. Weyer for helpful discussions. Financial support of the "Bundesministerium für Forschung und Technologie" is gratefully acknowledged. Appendix
Here we shall give explicit expressions for the wave functions of the fragments and thecorresponding sets of parameters which have been used for the present calculation. The deuteron function z with ßi = 0.3204, c, = 0.5320,
ßz = 0.0403, cz = 0.1492,
yields a binding energy EB = -1 .352 MeV . Eiere and in what follows ~ denotes the appropriate spin-isospin functions. The a-function cp(r l , . . ., r4) _ ~ c~, exp { - âß~ ~ (ri-rl)z}~ ~=i
t
(A .2)
99
MULTI-CHANNEL CALCULATION
with
ßi = 0.3240,
ßz = 0.0840,
cl = 5.2018,
c z = 0.0658,
yields EH = -21 .476 MeV. The 6Li function 4 Z ~P(ri, . . ., r 6) = exp { T~ ~ (ri - r;) } exP { - ~rs - rb)z } ~P { - SRz}~, +<; with S = 0.0420, y = 0.2369, ß = 0.2768,
(A.3)
iz ri+rz+r3+r4) - V s~rs+re),
R =
yields EH = -21 .086 MeV. The'Li(j = ~) and~'Li*(j = ~) functions
x exp { - 3Y~ Wlth
ß l = 0.3169, S 1 = 0.0434, R
_
~
s.st< ;s~
(ri-r;)z} exp { -S~,Rz}R[Yi(~~~~,
ßz = 0.2207,
yl = 0.2895,
(A.4)
yz = 0.1201,
cz = 0.4519, cl = 8.2287, 2 3 (rs+r6+r ~~ (rl+rz+r3+r4)21 21 S z = 0.0298,
yield EB = -26.240 MeV for'Li and EB = -25 .757 MeV for'Li* . The 'Be(j = ~) and 'Be*(j = }) functions
x exp { - 3?'~ With
ß l = 0.2439, Yi = 0.1070, Sl = 0.0259, cl = 0.3898, R
321
~
sgt< ;s~
;)z} exp { -S~,Rz}R[Yl(A,e~^]~, (ri-r
ßz = 0.3076,
ßa = 0.2910,
S z = 0.0142,
S 3 = 0.0708,
yz = 0.3234, cz = 1.2116,
(rl +rz +r3 +r4)-
ya = 0.2473,
J 21
c3 = 6.9461, (rs+r6+r7~
yield Es = - 25 .020 MeV for 'Be and EH = - 24.546 MeV for 'Be* .
(A.5)
100
H. STÖWE AND W. ZAHN
References 1) F. Ajzenberg-Selove and T. Lauritsen, Nucl . Phys . A227 (1974) 1 2) . L. Brown, E. Steiner, L. G. Arnold and R. G. Seyler, Nucl. Phys. A206 (1973) 353 3) L. G. Arnold, R. G. Seyler, L. Hrown, T. I. Honner and E. Steiner, Phys. Rev. Lett . 32 (1974) 895 4) U. Rohrer and L. Brown, Nucl . Phys. A217 (1973) 525 5) C. A. Burke, M. T. Lunnon and H. W. Lefevre, Phys . Rev. C10 (1974) 1299 6) W. D. Waiters, W. A. Fowler and C. C. Lauritsen, Phys . Rev. 91 (1953) 917 ; P. R. Malmberg, Phys . Rev. 101 (1956) 114 7) P. R. Hevington, W. W. Rolland and H. W. Lewis, Phys. Rev. 121 (1961) 871 8) F. C. Barker, Nucl . Phys. 83 (1966) 418 9) S. Cohen and D. Kurath, Nucl . Phys. 73 (1965) 1 ; 89 (1966) 707; A101(1967) l ; A141(1970) 145 10) N. Kumar and F. C. Barker, Nucl . Phys. A167 (1971) 434 11) M. Bouter, P. van Leuven, H. Depuydt and L. Schotsmans, Nucl. Phys . A100 (1967) 90 12) N. Kumar, Nucl . Phys. A225 (1974) 221 13) D. Fick, H. H. Hackenbroich, T. H. Seligman and W. Zahn, Phys . Lett . 62B (1976) 121 14) H. StBwe and W. Zahn, to be published 15) H. H. Hackenbroich, Symp. on present status and novel developments in the nuclear many-body problem, Rome, 1973 (Editrice Compositors, Bologna) 16) H. Eikemeier and H. H. Hackcnbroich, Z. Phys . 197 (1966) 412 17) H. Eikemeier and H. H. Hackenbroich, Nucl . Phys . A169 (1971) 407 18) H. H. Hackenbroich, Z. Phys . 231 (1970) 216