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Elastic and inelastic scattering of 180 MeV π± on 24Mg
Volume 113B, number 3
17 June 1982
PHYSICS LETTERS
ELASTIC AND INELASTIC SCATTERING OF 180 MeV n+ ON 24Mg M. GMITRO 1 J.KVASIL 2 Laboratory of Theoretical Physics, JINR Dubna, POB 79, SU I O1 000, Moscow, USSR and
R. MACH Institute of Nuclear Physics, CS/l ~ v, Rez, ~ ~ Czechoslovakia Received 15 February 1982
Equations of the coupled-channel method written in the momentum space are solved for the scattering of 180 MeV pions on 24Mg. The elastic and inelastic differential cross sections are calculated by Using the nuclear ground-state and transition densities, which describe correctly the (e, e') data for the same nuclear states. The results agree well with the recent 0r, n') data.
Recently, we have seen excellent experimental data on the elastic and inelastic scattering of 180 MeV n -+mesons on 24Mg [1]. In ref. [1] also a fit of the differential cross sections was performed within the DWBA and using a rough collective model of 24Mg. The values of the deformation parameters/3L dictated by the fit of the individual levels differ considerably from the corresponding values extracted from the scattering of 40 MeV protons [2]. Most unexpected is indeed the value 132 = 0.82 obtained [1] for the 2~(1.37 MeV) level. Among the possible explanations of this inconsistency, two seem to be most straightforward. One can expect that a coupled-channel (CC) calculation rather than DWBA should be appropriate for the mentioned fit. The other possible source of troubles is indeed a very crude collective model of 24Mg used in ref. [1]. To check those points we have performed a calculation within the CC formalism using a parametrization of the 24Mg nuclear densities as obtained from the (e, e') experiment [3] and processed in the spirit of the axial-symmetric rotor model of 24Mg.
Starting from the Watson multiple scattering theory we have obtained [4] the following system of integral equations
1 Also at: Institute of Nuclear Physics, CSAV, Rez, Czechoslovakia. 2 Also at: Department of Nuclear Physics, Charles University, Prague, CzechoSlovakia.
which depends on the pion initial momentum k and the final momentum k' in the 7r-N centre-of-mass system. The functions A i =-Ai(k' , k, co), i = 0, S, T, ST are
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
(Q'n IF(E)!0Q0) = ( a ' n Iv(E)10Q 0) Qt
1
tt
tt
Iv(E)ImQ )(Q m F(E)[OQ O)
(2rr) ~ m~=Of
E-
g
ft
m(Q ) + i e
X d3Q" c~(o,,),
(i)
n = 0, 1, 2 .... for the pion-nucleus amplitudes F ( E ) . Here, c/g (Q,,) is the pion-nucleus reduced mass [5]. To shorten the notation we use everywhere E = Eo(Qo). The potential ( a ' n l o ( E ) lmQ) = A (O'n [Wf(~o) lmQ) ,
(2)
may be simply expressed in terms of the 7r-N amplitude f(co) (k' If(co)lk) = A 0 + (t" x)A T + i ( , - [k X k ' ] / k k ' ) (A S + ( t . ¢ ) A s T ) ,
(3)
205
Volume 113B, number 3
PHYSICS LETTERS
constructed from the experimental ~rN amplitudes f{,t(co) (we consider l = 0, 1,2 waves and the phaseshift parametrization due to Salomon [6] ) and zrN form factors gl(k), which define the off-energy-sheU extrapolation of the 7rN amplitude. We have chosen a simple form
gl(k) = kl/(1 + o&2)2, which yields, using the value oe= 0.224 fm 2, almost the same off-shell extrapolation of rrN amplitude in the resonating P33 wave as the model of separable rrN potentials [7]. Unfortunately, the off-energy-shell extrapolation of the rrN amplitude is a rather arbitrary part of the present-day pion-nucleus calculations. When constructing the potential (2) we follow the prescription [8] for the relativistic transformation between the rr-N and 7r-nucleus centre-of-mass systems. In this way we end up (in the n-nucleus cms) with the quantifies Wf(co), O' and Q for the amplitude and the pion final and initial momentum, respectively. Further, the amplitudes fi, t(°°) were averaged over (nucleon) Fermi momenta and the remaining slowly varying parts of the potential matrix (2) were evaluated for effective momenta [9]. As a result, we retain a certain computational simplicity since the complete nuclear-structure input is still factorized in the form of the matrix elements
(Jn Tnn Illexp(iq " r l )g2 i IIIJmTmm) ,
(4)
where q = Q' Q and gZi = 1, r, o,or. In this way any microscopic or phenomenological model for the nuclear states IJn Tnn) can be simply Incorporated into our formalism. To solve the system(l) we perform the angular decomposition. In the Green functions [E0(Q0 ) -
E m ( a " ) + ie] - 1 = p lEo(Q0 ) Era(Q,,)] - 1
i7r6 (Eo(Qo) - E m ( a " ) ) ,
(5)
the principal-value integral is regularized [10] and represented by a gaussian quadrature formula. The system (1) is then solved in the model space m = 0, 1,2, ..., N using the matrix inversion method. No phenomenological corrections for the true pion absorption were included in (2). The Coulomb interaction effects were included [11 ] in the (0 Iv(E)10) term only. Now we proceed to describe the nuclear densities to be used in place of expressions (4). In 24Mg eight "va206
17 June 1982
lence" nucleons occupy the 2sld shell, the nucleus is therefore believed to be deformed. We postpone to our next communication the details of the collective axial-symmetric rotor model [12] we use for this nucleus. Here we consider for the g.s. band (K = 0, J~ = 0 +, 2 +, 4 + ,...) and "y-band" (K = 2, jTr = 2 +, 3 +, 4+,...) the rough formula (a small term is omitted for L ~ 4) for the reduced matrix elements of the tensor operator MLK of rank L
( J f g f l l g L IlYiKi) =[(2--6KiKf)(Zli+l)]l/2[Ji
X (Kf IMLK f_Ki [Ki) ,
L Jf ] K i K f - K i Kf (6)
where [: : :] stands for the Clebsch-Gordan coefficients. The intrinsic state [K) of 24Mg is the quasiparticle vacuum, [K = 0) = ]0),for the g.s. band, or a twoquasiparticle state, ]K = 2)= a-~a~ 10), for the 7band. Using eq. (6) and disregarding small "valence" terms in the expression for (K = 2 [MLo IK = 2) we were able to c6nstruct from the 'five measured [3] form factors the complete input for the CC meson-scat. tering program. This consist of 29 form factors if the 0+(g.s.), 2~(1.37), 4~(4.12), 2~(4.24), and 4~(6.0 MeV) levels of 24Mg are to be included explicitly in eq. (1). The results of this coupled-channel calculation are shown in fig. 1 for lr+ and 7r- elastic scattering on 24Mg. The comparison is made here with the experimental data and with the one-channel (optical model) calculations. It may be concluded that the calculated results reproduce qualitatively well the experimental data in the region of small scattering angles (0 ~< 90°). There are definite discrepancies in the region of the first minimum and the diffractive patterns of the calculated curves are shifted somewhat to smaller scattering angles in comparison with the experiment. The channel coupling tends to improve the agreement between the experiment and the calculated results, however, the role of virtual excitations into the low-lymg nuclear states turns out to be rather small. It is interesting to note that the difference between n + and rr differential cross sections due to the Coulomb force is appreciable in the large scattering-anNe region. The dif-
Volume 113B, number 3
PHYSICS LETTERS
\ 10
• !""i"
ELASTIC
SCATTERING
T
,t
'~ iP
°
%
,',;J
1
t
--'"
0.~
o.o ,
,
.
30
.
~
.
50
i
,
i
70
i
K
i ~l
90
,
i
,
i
110
(~crll.ldeg re es )
Fig. 1. Coupled-channel (full line) and optical model (dashed line) calculations of the ~r-+ elastic scattering are compared with the data of ref. [ 1 ].
terence is nicely reproduced by the CC calculation. The coupled channel results obtained for inelastic scattering are displayed in fig. 2 for final nuclear 2-~,
0
• • •
1
30
i
Ex =l"37MeV(2+l
•
'.j
50
70
90
O (degree.~)
Fig. 2. Coupled-channel (fullline) and DWlA (dashed line) calculations for the three inelastic transitions in the zr+-24Mg scattering. Data are from ref. [ 1 ].
17 June 1982
2~ and 4~ states and compared with the experiment and with the DWlA calculations. We do not show the results for 4-~(4,12 MeV) level, which is extremely weakly excited and actually has not been resolved from the strong 2~(4.24 MeV) level in the 0r, rr') experiment. The calculated curves agree well with the experiment in the small scattering-angle region, and it can be concluded that the nuclear form factors deduced from the electron scattering experiments provide a good over-all fit to the inelastic pion scattering without the necessity of adjusting parameters. If anything, the calculated cross sections overestimate slightly (e.g. in the first maximum of 2~ level) the data; the conclusion; just opposite to that of ref. [ 1]. The role of channel coupling is very small in the case of 2~ level, however, for the weakly excited 4~ level, the coupled-channel result agrees better with the experiment than the standard DWlA claculation, We have also calculated within the same assumptions the cross sections for the inelastic scattering of zt- mesons on 24Mg. The results are similar to those of fig. 2 and point to the same conclusions. Similarly to the elastic scattering there are discrepancies at large scattering angles, which can be caused by our ignorance of the detailed behaviour of nuclear form factors in a large momentum transfer region. Note that the electron scattering experiments were performed only for q <~ 2 f m - 1, which corresponds to the scattering angles 0 <~90 ° in pion scattering on 24Mg at 180 MeV. Also in the elastic scattering we observe a shift of diffractive patterns towards smaller scattering angles of a similar magnitude as in the elastic scattering. A possible origin of this discrepancy is the renormalization of the elementary amplitude in the nuclear medium (local field correction) which is not taken into account here, but the evaluation of such corrections is in progress. In conclusion, we have shown that the multiple scattering formalism is indeed capable to explain semiquantitatively the experimental data for the 180 MeV pion elastic and inelastic scattering if a realistic nuclear density P0 and transition densities Ptr are supplied as input. In testing runs we have seen that even a small inaccuracy in the fit of Ptr is blown up in the calculation of Or, ~r') cross sections and deteriorates the agreement with data considerably. Besides incorporating the local field corrections, as 207
Volume 113B, number 3
PHYSICS LETTERS
mentioned above, we also plan to improve upon the nuclear structure part that should allow us to avoid the fitting procedure for Ptr- A n interesting problem left is then a simultaneous calculation of (rr,zr') and (p,p') cross sections with the same microscopic set of the nuclear densities. We wish to thank A.B. Kurepin who has insisted on the importance of understanding the 24Mg(Tr, 7r') data, for his permanent interest in our investigation.
References
[3] [4] [5] [6] [7] [8]
[9] [10] [11] [12]
[1] C.A. Wiedner et al., Phys. Lett. 78B (1978) 26. [2] B. Zwieglinski, G.M. Crawley, H. Nann and J.A. Nolen Jr., Phys. Rev. C17 (1978) 872.
208
17 June 1982
H. Zarek et al., Phys. Lett. 80B (.1978) 26. M. Gmitro and R. Mach, Z. Phys. A290 (1979) 179. R. Mach, Nucl. Phys. A258 (1976) 513. G. Rowe, M. Salomon and R.H. Landau, Phys. Rev. C18 (1978) 584. J.T. Londergan, K.W. McVoy and E.J. Moniz, Ann. Phys. (NY) 86 (1974) 147. R. Heller. G.E. Bohannon and F. Tabakin, Phys. Rev. C13 (1976) 742; I.V. Falomkin et al.. Nuovo Cimento 57A (1980) 111. R.H. Landau and A.W. Thomas, Nucl. Phys. A302 (1978) 461. M.I. Haftel and F. Tabakin, Nucl. Phys. A158 (1970) 1. C.M. Vincent and S.C. Phatak, Phys. Rev. C10 (1974) 391. A. Bohr and B. Mottelson, Nuclear structure, Vol. II (Benjamin, New York, 1974).
17 June 1982
PHYSICS LETTERS
ELASTIC AND INELASTIC SCATTERING OF 180 MeV n+ ON 24Mg M. GMITRO 1 J.KVASIL 2 Laboratory of Theoretical Physics, JINR Dubna, POB 79, SU I O1 000, Moscow, USSR and
R. MACH Institute of Nuclear Physics, CS/l ~ v, Rez, ~ ~ Czechoslovakia Received 15 February 1982
Equations of the coupled-channel method written in the momentum space are solved for the scattering of 180 MeV pions on 24Mg. The elastic and inelastic differential cross sections are calculated by Using the nuclear ground-state and transition densities, which describe correctly the (e, e') data for the same nuclear states. The results agree well with the recent 0r, n') data.
Recently, we have seen excellent experimental data on the elastic and inelastic scattering of 180 MeV n -+mesons on 24Mg [1]. In ref. [1] also a fit of the differential cross sections was performed within the DWBA and using a rough collective model of 24Mg. The values of the deformation parameters/3L dictated by the fit of the individual levels differ considerably from the corresponding values extracted from the scattering of 40 MeV protons [2]. Most unexpected is indeed the value 132 = 0.82 obtained [1] for the 2~(1.37 MeV) level. Among the possible explanations of this inconsistency, two seem to be most straightforward. One can expect that a coupled-channel (CC) calculation rather than DWBA should be appropriate for the mentioned fit. The other possible source of troubles is indeed a very crude collective model of 24Mg used in ref. [1]. To check those points we have performed a calculation within the CC formalism using a parametrization of the 24Mg nuclear densities as obtained from the (e, e') experiment [3] and processed in the spirit of the axial-symmetric rotor model of 24Mg.
Starting from the Watson multiple scattering theory we have obtained [4] the following system of integral equations
1 Also at: Institute of Nuclear Physics, CSAV, Rez, Czechoslovakia. 2 Also at: Department of Nuclear Physics, Charles University, Prague, CzechoSlovakia.
which depends on the pion initial momentum k and the final momentum k' in the 7r-N centre-of-mass system. The functions A i =-Ai(k' , k, co), i = 0, S, T, ST are
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
(Q'n IF(E)!0Q0) = ( a ' n Iv(E)10Q 0) Qt
1
tt
tt
Iv(E)ImQ )(Q m F(E)[OQ O)
(2rr) ~ m~=Of
E-
g
ft
m(Q ) + i e
X d3Q" c~(o,,),
(i)
n = 0, 1, 2 .... for the pion-nucleus amplitudes F ( E ) . Here, c/g (Q,,) is the pion-nucleus reduced mass [5]. To shorten the notation we use everywhere E = Eo(Qo). The potential ( a ' n l o ( E ) lmQ) = A (O'n [Wf(~o) lmQ) ,
(2)
may be simply expressed in terms of the 7r-N amplitude f(co) (k' If(co)lk) = A 0 + (t" x)A T + i ( , - [k X k ' ] / k k ' ) (A S + ( t . ¢ ) A s T ) ,
(3)
205
Volume 113B, number 3
PHYSICS LETTERS
constructed from the experimental ~rN amplitudes f{,t(co) (we consider l = 0, 1,2 waves and the phaseshift parametrization due to Salomon [6] ) and zrN form factors gl(k), which define the off-energy-sheU extrapolation of the 7rN amplitude. We have chosen a simple form
gl(k) = kl/(1 + o&2)2, which yields, using the value oe= 0.224 fm 2, almost the same off-shell extrapolation of rrN amplitude in the resonating P33 wave as the model of separable rrN potentials [7]. Unfortunately, the off-energy-shell extrapolation of the rrN amplitude is a rather arbitrary part of the present-day pion-nucleus calculations. When constructing the potential (2) we follow the prescription [8] for the relativistic transformation between the rr-N and 7r-nucleus centre-of-mass systems. In this way we end up (in the n-nucleus cms) with the quantifies Wf(co), O' and Q for the amplitude and the pion final and initial momentum, respectively. Further, the amplitudes fi, t(°°) were averaged over (nucleon) Fermi momenta and the remaining slowly varying parts of the potential matrix (2) were evaluated for effective momenta [9]. As a result, we retain a certain computational simplicity since the complete nuclear-structure input is still factorized in the form of the matrix elements
(Jn Tnn Illexp(iq " r l )g2 i IIIJmTmm) ,
(4)
where q = Q' Q and gZi = 1, r, o,or. In this way any microscopic or phenomenological model for the nuclear states IJn Tnn) can be simply Incorporated into our formalism. To solve the system(l) we perform the angular decomposition. In the Green functions [E0(Q0 ) -
E m ( a " ) + ie] - 1 = p lEo(Q0 ) Era(Q,,)] - 1
i7r6 (Eo(Qo) - E m ( a " ) ) ,
(5)
the principal-value integral is regularized [10] and represented by a gaussian quadrature formula. The system (1) is then solved in the model space m = 0, 1,2, ..., N using the matrix inversion method. No phenomenological corrections for the true pion absorption were included in (2). The Coulomb interaction effects were included [11 ] in the (0 Iv(E)10) term only. Now we proceed to describe the nuclear densities to be used in place of expressions (4). In 24Mg eight "va206
17 June 1982
lence" nucleons occupy the 2sld shell, the nucleus is therefore believed to be deformed. We postpone to our next communication the details of the collective axial-symmetric rotor model [12] we use for this nucleus. Here we consider for the g.s. band (K = 0, J~ = 0 +, 2 +, 4 + ,...) and "y-band" (K = 2, jTr = 2 +, 3 +, 4+,...) the rough formula (a small term is omitted for L ~ 4) for the reduced matrix elements of the tensor operator MLK of rank L
( J f g f l l g L IlYiKi) =[(2--6KiKf)(Zli+l)]l/2[Ji
X (Kf IMLK f_Ki [Ki) ,
L Jf ] K i K f - K i Kf (6)
where [: : :] stands for the Clebsch-Gordan coefficients. The intrinsic state [K) of 24Mg is the quasiparticle vacuum, [K = 0) = ]0),for the g.s. band, or a twoquasiparticle state, ]K = 2)= a-~a~ 10), for the 7band. Using eq. (6) and disregarding small "valence" terms in the expression for (K = 2 [MLo IK = 2) we were able to c6nstruct from the 'five measured [3] form factors the complete input for the CC meson-scat. tering program. This consist of 29 form factors if the 0+(g.s.), 2~(1.37), 4~(4.12), 2~(4.24), and 4~(6.0 MeV) levels of 24Mg are to be included explicitly in eq. (1). The results of this coupled-channel calculation are shown in fig. 1 for lr+ and 7r- elastic scattering on 24Mg. The comparison is made here with the experimental data and with the one-channel (optical model) calculations. It may be concluded that the calculated results reproduce qualitatively well the experimental data in the region of small scattering angles (0 ~< 90°). There are definite discrepancies in the region of the first minimum and the diffractive patterns of the calculated curves are shifted somewhat to smaller scattering angles in comparison with the experiment. The channel coupling tends to improve the agreement between the experiment and the calculated results, however, the role of virtual excitations into the low-lymg nuclear states turns out to be rather small. It is interesting to note that the difference between n + and rr differential cross sections due to the Coulomb force is appreciable in the large scattering-anNe region. The dif-
Volume 113B, number 3
PHYSICS LETTERS
\ 10
• !""i"
ELASTIC
SCATTERING
T
,t
'~ iP
°
%
,',;J
1
t
--'"
0.~
o.o ,
,
.
30
.
~
.
50
i
,
i
70
i
K
i ~l
90
,
i
,
i
110
(~crll.ldeg re es )
Fig. 1. Coupled-channel (full line) and optical model (dashed line) calculations of the ~r-+ elastic scattering are compared with the data of ref. [ 1 ].
terence is nicely reproduced by the CC calculation. The coupled channel results obtained for inelastic scattering are displayed in fig. 2 for final nuclear 2-~,
0
• • •
1
30
i
Ex =l"37MeV(2+l
•
'.j
50
70
90
O (degree.~)
Fig. 2. Coupled-channel (fullline) and DWlA (dashed line) calculations for the three inelastic transitions in the zr+-24Mg scattering. Data are from ref. [ 1 ].
17 June 1982
2~ and 4~ states and compared with the experiment and with the DWlA calculations. We do not show the results for 4-~(4,12 MeV) level, which is extremely weakly excited and actually has not been resolved from the strong 2~(4.24 MeV) level in the 0r, rr') experiment. The calculated curves agree well with the experiment in the small scattering-angle region, and it can be concluded that the nuclear form factors deduced from the electron scattering experiments provide a good over-all fit to the inelastic pion scattering without the necessity of adjusting parameters. If anything, the calculated cross sections overestimate slightly (e.g. in the first maximum of 2~ level) the data; the conclusion; just opposite to that of ref. [ 1]. The role of channel coupling is very small in the case of 2~ level, however, for the weakly excited 4~ level, the coupled-channel result agrees better with the experiment than the standard DWlA claculation, We have also calculated within the same assumptions the cross sections for the inelastic scattering of zt- mesons on 24Mg. The results are similar to those of fig. 2 and point to the same conclusions. Similarly to the elastic scattering there are discrepancies at large scattering angles, which can be caused by our ignorance of the detailed behaviour of nuclear form factors in a large momentum transfer region. Note that the electron scattering experiments were performed only for q <~ 2 f m - 1, which corresponds to the scattering angles 0 <~90 ° in pion scattering on 24Mg at 180 MeV. Also in the elastic scattering we observe a shift of diffractive patterns towards smaller scattering angles of a similar magnitude as in the elastic scattering. A possible origin of this discrepancy is the renormalization of the elementary amplitude in the nuclear medium (local field correction) which is not taken into account here, but the evaluation of such corrections is in progress. In conclusion, we have shown that the multiple scattering formalism is indeed capable to explain semiquantitatively the experimental data for the 180 MeV pion elastic and inelastic scattering if a realistic nuclear density P0 and transition densities Ptr are supplied as input. In testing runs we have seen that even a small inaccuracy in the fit of Ptr is blown up in the calculation of Or, ~r') cross sections and deteriorates the agreement with data considerably. Besides incorporating the local field corrections, as 207
Volume 113B, number 3
PHYSICS LETTERS
mentioned above, we also plan to improve upon the nuclear structure part that should allow us to avoid the fitting procedure for Ptr- A n interesting problem left is then a simultaneous calculation of (rr,zr') and (p,p') cross sections with the same microscopic set of the nuclear densities. We wish to thank A.B. Kurepin who has insisted on the importance of understanding the 24Mg(Tr, 7r') data, for his permanent interest in our investigation.
References
[3] [4] [5] [6] [7] [8]
[9] [10] [11] [12]
[1] C.A. Wiedner et al., Phys. Lett. 78B (1978) 26. [2] B. Zwieglinski, G.M. Crawley, H. Nann and J.A. Nolen Jr., Phys. Rev. C17 (1978) 872.
208
17 June 1982
H. Zarek et al., Phys. Lett. 80B (.1978) 26. M. Gmitro and R. Mach, Z. Phys. A290 (1979) 179. R. Mach, Nucl. Phys. A258 (1976) 513. G. Rowe, M. Salomon and R.H. Landau, Phys. Rev. C18 (1978) 584. J.T. Londergan, K.W. McVoy and E.J. Moniz, Ann. Phys. (NY) 86 (1974) 147. R. Heller. G.E. Bohannon and F. Tabakin, Phys. Rev. C13 (1976) 742; I.V. Falomkin et al.. Nuovo Cimento 57A (1980) 111. R.H. Landau and A.W. Thomas, Nucl. Phys. A302 (1978) 461. M.I. Haftel and F. Tabakin, Nucl. Phys. A158 (1970) 1. C.M. Vincent and S.C. Phatak, Phys. Rev. C10 (1974) 391. A. Bohr and B. Mottelson, Nuclear structure, Vol. II (Benjamin, New York, 1974).