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Direct reaction analysis of continuum spectra and polarizations in the 48Ti(16O, 16O′) reaction
Volume 122B, number 5,6
PIIYSICS LETTERS
17 March 1983
DIRECT REACTION ANALYSIS OF CONTINUUM SPECTRA AND POLARIZATIONS IN THE 48Ti(160,160') REACTION '~ 11. L E N S K E . S. L A N D O W N E , H.H. W O L T E R , T. T A M U R A l Sektion Physik, Universitltt Mfinchen, D-8046 Garching. ICest Germant, and T. U D A G A W A Department of Physics, University o1 Texas, Austin, TX 78712, USA Received 17 November 1982 Revised manuscript received 22 December 1982
The multi-step direct reaction approach is used to calculate continuum cross sections and polarizations for the reaction 48Ti(160, 160') at Ela b = 100 MeV. The dynamical part of the c',dculations is done fully quantum mechanically, and includes one- and two-step inelastic excitations. The spectroscopic amplitudes are obtainc~ from correlated particle-hole response functions. A good description of the experimental data up to about 20 MeV energy loss is achieved this way.
Direct reaction theory is the generally accepted framework for analyzing nuclear reactions leading to low lying discrete states. It is also used extensively for analyzing giant resonances, which are in the contin u u m and require around 10 MeV o f excitation energy. Therefore it is tempting to extend the application o f direct reaction theory -- including multistep processes -to reactions exciting c o n t i n u u m states where a few tens o f MeV are lost. In this way one could try to explain, from a unified point o f view, reactions which are conventionally considered separately as belonging either to thc "quasi-elastic" or "deep-inelastic" parts o f the spectrum in heavy ion collisions. In a few recent papers we showed that a multi-step direct reaction (MSDR) m e t h o d can be used to describe c o n t i n u u m cross sections induced by both light [1,2] and heavy ions [3]. In such reactions, a large n u m b e r o f very complicated states are excited but
'~ Supported by the Bundesministerium l'fir Forschung und Technologie, Federal RepubLic of Germany, and by the US Department of Energy. I Alexander yon Ilumboldt Senior Scientist Awardec, on leave from the University of Texas, Austin, TX, USA.
0 031-9163/83/0000
0000/$ 03.00 © 1983 North-Holland
what is actually observed are s u m m e d (or averaged) cross sections, and not those for exciting the individual states. It is then sufficient to design a theory so as to calculate only the s u m m e d cross sections [ 1 -3 ]. In the detailed formulation presented in ref. [2] it is shown how the cross sections s u m m e d over complicated states can be replaced by those s u m m e d over simple states, the latter being those that are excited by direct-reaction processes. The key concept used in this tbrmulation is that the amplitudes o f the simple states are distributed statistically over the complicated states. Thus the final formulae on which our calculations are based refer explicitly only to the simple states. A immber o f related works, particularly for lightion c o n t i n u u m reactions, have been published by other groups [4]. The similarities and differences o f these works from ours have been discussed in ref. [2]. in contrast to the classical models o f collective excitation [5], friction [6], and statistical transport [7] which are often used to describe heavy-ion c o n t i n u u m reactions, in the MSDR approach the nuclear structure and the excitation mechanism are treated fully quantum mechanically. 333
The purpose of this article is to report on detailed MSDR calculations for heavy ion inelastic scattering to the continuum. For inelastic scattering the elementary process is considered to be the excitation o f a phonon (i.e. a correlated particle--hole pair). Tile first order double differential cross section for scattering to the continuum is expressed as [2]
= ~ Px, ( E l ) d°(xll) (E', E; 0)/dg2 ,
(l)
hi where E ( E ' ) is the incident (exit) energy of relative motion. 0 is the scattering angle,px~(El) is the spectroscopic p a r t i c l e - h o l e strength function at excitation energy E 1 = E - E' for multipolarity X1 , and is the usual first order DWBA cross section calculated with an average form factor for transfered angular momentum X1 . The second order cross section is given by a corresponding expression
dox([)/dgZ
E'
x~lx f dE"Px,(E1)Ph2(I'.'2) z 0
X
do(21)x2(E', E", E; 0 ) / d Q ,
(2)
where E" is the cm energy in the intermediate state and E 1 = E - E" and E 2 = E' - E" are the excitation energies for the first and second steps. Also, do(2t)x2 / dfZ is the second order DWBA cross section for transfeting angular momenta )t I and ~k2 in the first and second steps, respectively, and then summed over k = k I + k 2. It should be noted that the absence of interference between processes of the same and of different orders follows from the statistical assumptions about the continuum states [2]. In consistency with these assumptions it was also postulated that each step of th~ reaction leads to those components of the wave function which are written as a phonon created on the state reached in the previous step [2]. This resulted in eq. (2) which is expressed in terms of a simple product o f two one-phonon strength functions, convoluted wih a two-step direct reaction cross section. The above procedure can, in principle, be extended in a straightforward way to obtain the cross sections for more steps. For the inelastic excitation treated here, we take 334
the spectroscopic strength functions as tile distribution of the p a r t i c l e - h o l e strength of a given multipolarity in the continuum. To compute it we use the response function method given by Bohr and Mottelson [8] which accounts for correlations. Thus p x ( E ) = rr -1 Im x x ( E + i r / 2 ) ,
(3)
where Xx is the correlated response function and 1" is a spreading width. Following ref. [8],
d2o(l)(O)/d~2dE '
d2ot2)/d a d E ' =
17 March 1983
PIIYSICS LETTERS
Volume 122B, number 5,6
×ate) =
x(xO)(e)l[1-- ~:xX(x0)
- (e)l ,
(4)
where
x(xO)(e)= ~ 21S'phi(ph irhYx ]O)12/(Ep2h -e2), ph
(5)
is the uncorrelated p a r t i c l e - h o l e response function and Kx is a coupling constant which may be evaluated self-consistently [8]. In this work we calculate the reaction 48Ti(160, 160') at Ela b = 100 MeV and compare to corresponding measurements [9], which include determinations of the circular polarization ofT-rays emitted in coincidence with the ejectile. Not all o f these data have been published but they show similar features as tile 160 + 58Ni reactions reported in ref. [ 10]. We locus on the 160 + 48Ti system because more complete data were taken for this case but we expect that the results obtained should be typical for collisions of light heavy ions at energies well above the Coulomb barrier. The experiments [9] determine the charge o f the ejectile but do not distinguish its mass or state of excitation. By observing 7-rays for known transitions in 160, it was determined that projectile excitation accounts tor only about 25% of tile reaction cross section in the 160 + 58Ni case [10] and also for 160 + 48Ti [11]. The spectrum of Ti isotopes measured in coincidence with scattered O nuclei shows that neutron transfer reactions make a small contribution for energy losses less than 12 MeV [ 11 ]. Theretbre in these first calculations we treat the 160 projectile as being inert. The spectroscopic strength functions for the target have been calculated for 48Ca, which should be representative for nuclei in this region. They are shown in fig. 1. Transferred angular momenta up to X = 5 have been included. It has been checked that the calculations are well converged with respect to ?t in the range of energies considered. For the width I" we used a
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17 blarch 1983
to expect a potential which is diffe,ent from what is seen in the entrance chamrel. We find below that this is in tact the case. The average form factor is taken to be the derivative ot" the entrance channel optical potential, as in the collective model. This is a reasonable approximation for particle-hole excitations even when the particles are in the continuum because the hole orbitals are bound [2]. Thus the tbrm factor is not only state independent but also independent of the transfered angular momentum ~. and the excitation energy. The ~,-independence assumption considerably simplifies the two-step calculations [2]. The following correction was introduced to account for some of the X-dependence of the form factor: Using microscopic transition densities and the M3Y effective NN interaction, form facto,-s were calculated in the double folding approach [13]. Then onestep DWBA calculations were performed with both the derivative (DV) and the double folding (DF) form factors at different Q values. The ratios of the correspondin_g angle integrated cross sections % ( 0 ) = oraLlY(Q) are used as correction factors to muhiply the result with the derivative form ['actors. Test calculations indicate that this prescription tends to overestimate tim two-step cross sections by about a factor of 1.5. The calculated differential cross sections and polarizations are compared to the data of01a b = 25 c in fig. 2. The results in part (2a) are obtained when all potentials are taken to be the same as in the entrance channel. The total spectrum and polarization (solid curve) result from the sum of the one-step (dotted curve) and twostep (dashed curve) contributions. It may be noted that one-step cross section is confined by the reaction dynamics to excitation energies below 10 MeV. The two-step spectrum peaks at a higher energy and has a somewhat broader width than the one-step spectrum. It falls off rapidly for excitation energies above 20 MeV. The structure in the calculated cross sections can be traced to the relatively large X = 3 and X = 2 transition strengths (of. fig. 1). It should be noted that the calculations in fig. 2a fail to describe the polarization data in that they do not show the transition from negative to positive polarization at about Q = 20 MeV. This wrong tendency reveals itself already for Q-values where the one-step contribution is still dominating and it is found not to depend on the shape of the form factor. Therefore,
oDI"(Q)/
.....
5.0
10.0 15.0 2 0 . 0 25.0 E, [MeVl
I:ig. I. Spectroscopic strength functions for the different multipolarities used in the calculations. They are obtained from the correlated particle hole response function method 181.
value of r ~ 0.1 MeV for the low lying bound states, which increased linearly to I" ~ 6 MeV at the particle threshold and then stayed constant. The values of the coupling constants Ka in eq. (5) have been adjusted by about 5% relative to their self-consistent values [8 ], to get better agreement with the experimental spectrum. The resulting spectrum in fig. 1 is in good general agreement with respect to energy and transition rates for the low lying states in 48Ca. The position of the giant qt, adn, pole resonance is about 20% too low compared to systematics. For the DWBA calculations the optical potential in the entrance channel was determined from fits to the elastic scattering o f 160 + 48Ca over a range of incident energies [12]. With respect to the potential m the final (and intermediate) excited channels the usual procedure when considering inelastic excitations of low-lying states is to take it eqt, al to the entrance channel potential, t lowever, when considering highly excited states is the continuum it may be more natural
335
Volume 122B, number 5,6
PHYSICS LI.;TTERS
17 March 1983 18)
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Fig. 2. Double-differential cross sections (above) and polarizations (below) for the reaction 4~Ti(t60,160') at Ela b = 100 MeV for #lab = 25° as a function of outgoing energy. The data shown are from ref. [9], where only the charge of the outgoing particle was measured. The calculations include one- and two-step direct excitations. In part (A) on the left the optical potentials are the ,same as the elastic potential in all channels, while in part (B) the potentials and form factors are modified for excited states by increasing the radius of the real part with increasing excitation energy (see text). [he dotted and the dashed curves in the polarization plots are the calculated contributions of the one and two step processes to the total polarization (and not the polarizations of one- and two-step processes alone). The outgoing center of mass energy ECu~ = 75 MeV corresponds to the Q-value Q = 0 MeV.
as discussed above we consider the effect o f m o d i f y i n g the optical potentials for states in the c o n t i n u u m . In fig. 2b we show the results o f calculations in which the radius o f the real potential has been increased by 5%, 15% and 20% for Q = - 1 0 , --20 and - 3 0 MeV, respectively. It is seen that the description o f the polarization data can be improved considerably in this way. In particular the change o f sign at higher Q-values is now reproduced. This is mainly due to a corrseponding change o f sign in the two-step polarization. We also 336
note that the average form factor for tile second step is now taken as the derivative o f the optical potential in the intermediate channel. Its strength increases with increasing excitation energy. This raises the two-step cross sections at higher energies by about a factor o f two but is not i m p o r t a n t for the polarization. The effect o f increasing the nuclear attraction (by increasing the radius o f the real potential) is the same as that o f decreasing the C o u l o m b barrier, and the latter can be justified if most o f the states excited are
Volume 122B, number 5,6
PtlYSICS LI'71"TERS
well d e f o r m e d . Evidence of such a decreased C o u l o m b barrier has been known for some time in deep inelastic collisions [14]. In any case, the increased nuclear attraction results in negative angle scattering. This situation is similar to what is described by classical friction models [6]. We conclude that direct inelastic scattering processes up to second order are able to account for tire experimental data for the energy spectrum and the polarization up to about 20 MeV energy loss. The fact that two-step processes play an essential role for the larger energy losses is encouraging since it indicates that three- and higher order processes might be able to explain the spectrum at even higher Q-values. On the o t h e r hand the experimental data in the reaction studied also include projectile excitation, transfer, knock-out and break-up processes, whicb might contribute significantly at higher energies (see ref. [15 ]). In order to describe the trend o f the polarization data, it was found to be necessary to increase the range o f the attractive nuclear potential in the c o n t i n u u m states. It thus appears that analyses o f this kind are able to yield information about the nuclear interaction potential o f tire highly excited nucleus. To understand this quantitatively leads to an interesting subject o f the nuclear spectroscopy in the c o n t i n u u m . We are grateful to W. Dfinnweber and W. T r a u t m a n n for several discussions concerning the experimental data.
References
17 March 1983
T. Tamura and I . Udagawa, Phys. Lett. 7113 (1977) 273; 78B(1978) 189; T. Tamura, 11. Lenske and T. Udagawa, Phys. Rev. ('23 (1981) 2769. [2] T. Tamura, T. Udagawa and H. I.cnske, Phys. Rev. C26 (1982) 379. [3] T. Udagawa and T. Tamura, in: ('ontinuum spectra in heavy ion reactions, eds. by T. Tamura, J.B. Natowitz and D.It. Youngblood (Harwood Academic, New York, 1980) p. 155; T. Udagawa, I). Price and T. Tamura, Phys. Lett. 116B (1982) 311, and references therein. [4] G.F. Bertsch and S.I:. Tsai, Phys. Rep. C18 (1975) 125. G. Mantzouranis, II.A. Weidenmfiller and D. Agassi, Z. Phys. A276 (1976) 145; II. Feshbach, A.K. Kerman and S.E. Koonin, Ann. Phys. (NY) 125 (1980) 429; R. Bonetti, M. Camnasio, L. Colli-Milazzo and P.E. ltodgson, Phys. Rev. C24 (1981) 71. [5] R.A. Broglia, C.tt. I)asso and A. Winther, in: Proc. Intern. School of Physics E. Fermi Course LXXVII, eds. R.A. Broglia, C.It. l)asso and R.A. Ricci (North-llolland, Amsterdam, 1981 ). [6] D.It.I.:. Gross and tl. Kalinowski, Phys. Rep. 45 (1.978) 175. [ 7 ] II.A. Weidenmilller, in: Progress in nuclear and particle physics, ed. D.H. Wilkinson (Pergamon, London, 1979). [8] A. Bohr and B.R. Mottelson, Nuclear structure (Benjamin, London, 1975) Vol. II. [9l W. Trautmann et al., in: Proc. XIXth Intern. winter meeting on Nuclear physics, ed. 1. Iori (Bormio, 1981), R. Ritzka, Diploma thesis (Munich, 1981) unpublished. [101 C. Lauterbach et al., Phys. Rev. Lett. 41 (1981) 1774. [ 111 tl. Puchta, thesis, Munich 1980, unpublished, and private communication. [12] S.E. Vigdor et al., Phys. Rev. C20 (1979) 2147. [ 13] G.R.Satchler and W.G. Love, Phys. Rep. 55 (1979) 183. [14] J. Wilczynski, Phys. Left. 47B (1973) 484. [ 15] W. yon Oertzen, Nucl. Phys. A387 (1982) 930.
[ I ] T. Tamura, T. Udagawa, D.It. Feng and K.K. Kan, Phys. Lett. 66B (1977) 109;
337
PIIYSICS LETTERS
17 March 1983
DIRECT REACTION ANALYSIS OF CONTINUUM SPECTRA AND POLARIZATIONS IN THE 48Ti(160,160') REACTION '~ 11. L E N S K E . S. L A N D O W N E , H.H. W O L T E R , T. T A M U R A l Sektion Physik, Universitltt Mfinchen, D-8046 Garching. ICest Germant, and T. U D A G A W A Department of Physics, University o1 Texas, Austin, TX 78712, USA Received 17 November 1982 Revised manuscript received 22 December 1982
The multi-step direct reaction approach is used to calculate continuum cross sections and polarizations for the reaction 48Ti(160, 160') at Ela b = 100 MeV. The dynamical part of the c',dculations is done fully quantum mechanically, and includes one- and two-step inelastic excitations. The spectroscopic amplitudes are obtainc~ from correlated particle-hole response functions. A good description of the experimental data up to about 20 MeV energy loss is achieved this way.
Direct reaction theory is the generally accepted framework for analyzing nuclear reactions leading to low lying discrete states. It is also used extensively for analyzing giant resonances, which are in the contin u u m and require around 10 MeV o f excitation energy. Therefore it is tempting to extend the application o f direct reaction theory -- including multistep processes -to reactions exciting c o n t i n u u m states where a few tens o f MeV are lost. In this way one could try to explain, from a unified point o f view, reactions which are conventionally considered separately as belonging either to thc "quasi-elastic" or "deep-inelastic" parts o f the spectrum in heavy ion collisions. In a few recent papers we showed that a multi-step direct reaction (MSDR) m e t h o d can be used to describe c o n t i n u u m cross sections induced by both light [1,2] and heavy ions [3]. In such reactions, a large n u m b e r o f very complicated states are excited but
'~ Supported by the Bundesministerium l'fir Forschung und Technologie, Federal RepubLic of Germany, and by the US Department of Energy. I Alexander yon Ilumboldt Senior Scientist Awardec, on leave from the University of Texas, Austin, TX, USA.
0 031-9163/83/0000
0000/$ 03.00 © 1983 North-Holland
what is actually observed are s u m m e d (or averaged) cross sections, and not those for exciting the individual states. It is then sufficient to design a theory so as to calculate only the s u m m e d cross sections [ 1 -3 ]. In the detailed formulation presented in ref. [2] it is shown how the cross sections s u m m e d over complicated states can be replaced by those s u m m e d over simple states, the latter being those that are excited by direct-reaction processes. The key concept used in this tbrmulation is that the amplitudes o f the simple states are distributed statistically over the complicated states. Thus the final formulae on which our calculations are based refer explicitly only to the simple states. A immber o f related works, particularly for lightion c o n t i n u u m reactions, have been published by other groups [4]. The similarities and differences o f these works from ours have been discussed in ref. [2]. in contrast to the classical models o f collective excitation [5], friction [6], and statistical transport [7] which are often used to describe heavy-ion c o n t i n u u m reactions, in the MSDR approach the nuclear structure and the excitation mechanism are treated fully quantum mechanically. 333
The purpose of this article is to report on detailed MSDR calculations for heavy ion inelastic scattering to the continuum. For inelastic scattering the elementary process is considered to be the excitation o f a phonon (i.e. a correlated particle--hole pair). Tile first order double differential cross section for scattering to the continuum is expressed as [2]
= ~ Px, ( E l ) d°(xll) (E', E; 0)/dg2 ,
(l)
hi where E ( E ' ) is the incident (exit) energy of relative motion. 0 is the scattering angle,px~(El) is the spectroscopic p a r t i c l e - h o l e strength function at excitation energy E 1 = E - E' for multipolarity X1 , and is the usual first order DWBA cross section calculated with an average form factor for transfered angular momentum X1 . The second order cross section is given by a corresponding expression
dox([)/dgZ
E'
x~lx f dE"Px,(E1)Ph2(I'.'2) z 0
X
do(21)x2(E', E", E; 0 ) / d Q ,
(2)
where E" is the cm energy in the intermediate state and E 1 = E - E" and E 2 = E' - E" are the excitation energies for the first and second steps. Also, do(2t)x2 / dfZ is the second order DWBA cross section for transfeting angular momenta )t I and ~k2 in the first and second steps, respectively, and then summed over k = k I + k 2. It should be noted that the absence of interference between processes of the same and of different orders follows from the statistical assumptions about the continuum states [2]. In consistency with these assumptions it was also postulated that each step of th~ reaction leads to those components of the wave function which are written as a phonon created on the state reached in the previous step [2]. This resulted in eq. (2) which is expressed in terms of a simple product o f two one-phonon strength functions, convoluted wih a two-step direct reaction cross section. The above procedure can, in principle, be extended in a straightforward way to obtain the cross sections for more steps. For the inelastic excitation treated here, we take 334
the spectroscopic strength functions as tile distribution of the p a r t i c l e - h o l e strength of a given multipolarity in the continuum. To compute it we use the response function method given by Bohr and Mottelson [8] which accounts for correlations. Thus p x ( E ) = rr -1 Im x x ( E + i r / 2 ) ,
(3)
where Xx is the correlated response function and 1" is a spreading width. Following ref. [8],
d2o(l)(O)/d~2dE '
d2ot2)/d a d E ' =
17 March 1983
PIIYSICS LETTERS
Volume 122B, number 5,6
×ate) =
x(xO)(e)l[1-- ~:xX(x0)
- (e)l ,
(4)
where
x(xO)(e)= ~ 21S'phi(ph irhYx ]O)12/(Ep2h -e2), ph
(5)
is the uncorrelated p a r t i c l e - h o l e response function and Kx is a coupling constant which may be evaluated self-consistently [8]. In this work we calculate the reaction 48Ti(160, 160') at Ela b = 100 MeV and compare to corresponding measurements [9], which include determinations of the circular polarization ofT-rays emitted in coincidence with the ejectile. Not all o f these data have been published but they show similar features as tile 160 + 58Ni reactions reported in ref. [ 10]. We locus on the 160 + 48Ti system because more complete data were taken for this case but we expect that the results obtained should be typical for collisions of light heavy ions at energies well above the Coulomb barrier. The experiments [9] determine the charge o f the ejectile but do not distinguish its mass or state of excitation. By observing 7-rays for known transitions in 160, it was determined that projectile excitation accounts tor only about 25% of tile reaction cross section in the 160 + 58Ni case [10] and also for 160 + 48Ti [11]. The spectrum of Ti isotopes measured in coincidence with scattered O nuclei shows that neutron transfer reactions make a small contribution for energy losses less than 12 MeV [ 11 ]. Theretbre in these first calculations we treat the 160 projectile as being inert. The spectroscopic strength functions for the target have been calculated for 48Ca, which should be representative for nuclei in this region. They are shown in fig. 1. Transferred angular momenta up to X = 5 have been included. It has been checked that the calculations are well converged with respect to ?t in the range of energies considered. For the width I" we used a
Volume 122B, number 5,6 i
I
I
12.o~-
PItYSICS I.I¢'I"I'EI~.S 1 - - 1
~8Co
H.O[
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17 blarch 1983
to expect a potential which is diffe,ent from what is seen in the entrance chamrel. We find below that this is in tact the case. The average form factor is taken to be the derivative ot" the entrance channel optical potential, as in the collective model. This is a reasonable approximation for particle-hole excitations even when the particles are in the continuum because the hole orbitals are bound [2]. Thus the tbrm factor is not only state independent but also independent of the transfered angular momentum ~. and the excitation energy. The ~,-independence assumption considerably simplifies the two-step calculations [2]. The following correction was introduced to account for some of the X-dependence of the form factor: Using microscopic transition densities and the M3Y effective NN interaction, form facto,-s were calculated in the double folding approach [13]. Then onestep DWBA calculations were performed with both the derivative (DV) and the double folding (DF) form factors at different Q values. The ratios of the correspondin_g angle integrated cross sections % ( 0 ) = oraLlY(Q) are used as correction factors to muhiply the result with the derivative form ['actors. Test calculations indicate that this prescription tends to overestimate tim two-step cross sections by about a factor of 1.5. The calculated differential cross sections and polarizations are compared to the data of01a b = 25 c in fig. 2. The results in part (2a) are obtained when all potentials are taken to be the same as in the entrance channel. The total spectrum and polarization (solid curve) result from the sum of the one-step (dotted curve) and twostep (dashed curve) contributions. It may be noted that one-step cross section is confined by the reaction dynamics to excitation energies below 10 MeV. The two-step spectrum peaks at a higher energy and has a somewhat broader width than the one-step spectrum. It falls off rapidly for excitation energies above 20 MeV. The structure in the calculated cross sections can be traced to the relatively large X = 3 and X = 2 transition strengths (of. fig. 1). It should be noted that the calculations in fig. 2a fail to describe the polarization data in that they do not show the transition from negative to positive polarization at about Q = 20 MeV. This wrong tendency reveals itself already for Q-values where the one-step contribution is still dominating and it is found not to depend on the shape of the form factor. Therefore,
oDI"(Q)/
.....
5.0
10.0 15.0 2 0 . 0 25.0 E, [MeVl
I:ig. I. Spectroscopic strength functions for the different multipolarities used in the calculations. They are obtained from the correlated particle hole response function method 181.
value of r ~ 0.1 MeV for the low lying bound states, which increased linearly to I" ~ 6 MeV at the particle threshold and then stayed constant. The values of the coupling constants Ka in eq. (5) have been adjusted by about 5% relative to their self-consistent values [8 ], to get better agreement with the experimental spectrum. The resulting spectrum in fig. 1 is in good general agreement with respect to energy and transition rates for the low lying states in 48Ca. The position of the giant qt, adn, pole resonance is about 20% too low compared to systematics. For the DWBA calculations the optical potential in the entrance channel was determined from fits to the elastic scattering o f 160 + 48Ca over a range of incident energies [12]. With respect to the potential m the final (and intermediate) excited channels the usual procedure when considering inelastic excitations of low-lying states is to take it eqt, al to the entrance channel potential, t lowever, when considering highly excited states is the continuum it may be more natural
335
Volume 122B, number 5,6
PHYSICS LI.;TTERS
17 March 1983 18)
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_
70
Fig. 2. Double-differential cross sections (above) and polarizations (below) for the reaction 4~Ti(t60,160') at Ela b = 100 MeV for #lab = 25° as a function of outgoing energy. The data shown are from ref. [9], where only the charge of the outgoing particle was measured. The calculations include one- and two-step direct excitations. In part (A) on the left the optical potentials are the ,same as the elastic potential in all channels, while in part (B) the potentials and form factors are modified for excited states by increasing the radius of the real part with increasing excitation energy (see text). [he dotted and the dashed curves in the polarization plots are the calculated contributions of the one and two step processes to the total polarization (and not the polarizations of one- and two-step processes alone). The outgoing center of mass energy ECu~ = 75 MeV corresponds to the Q-value Q = 0 MeV.
as discussed above we consider the effect o f m o d i f y i n g the optical potentials for states in the c o n t i n u u m . In fig. 2b we show the results o f calculations in which the radius o f the real potential has been increased by 5%, 15% and 20% for Q = - 1 0 , --20 and - 3 0 MeV, respectively. It is seen that the description o f the polarization data can be improved considerably in this way. In particular the change o f sign at higher Q-values is now reproduced. This is mainly due to a corrseponding change o f sign in the two-step polarization. We also 336
note that the average form factor for tile second step is now taken as the derivative o f the optical potential in the intermediate channel. Its strength increases with increasing excitation energy. This raises the two-step cross sections at higher energies by about a factor o f two but is not i m p o r t a n t for the polarization. The effect o f increasing the nuclear attraction (by increasing the radius o f the real potential) is the same as that o f decreasing the C o u l o m b barrier, and the latter can be justified if most o f the states excited are
Volume 122B, number 5,6
PtlYSICS LI'71"TERS
well d e f o r m e d . Evidence of such a decreased C o u l o m b barrier has been known for some time in deep inelastic collisions [14]. In any case, the increased nuclear attraction results in negative angle scattering. This situation is similar to what is described by classical friction models [6]. We conclude that direct inelastic scattering processes up to second order are able to account for tire experimental data for the energy spectrum and the polarization up to about 20 MeV energy loss. The fact that two-step processes play an essential role for the larger energy losses is encouraging since it indicates that three- and higher order processes might be able to explain the spectrum at even higher Q-values. On the o t h e r hand the experimental data in the reaction studied also include projectile excitation, transfer, knock-out and break-up processes, whicb might contribute significantly at higher energies (see ref. [15 ]). In order to describe the trend o f the polarization data, it was found to be necessary to increase the range o f the attractive nuclear potential in the c o n t i n u u m states. It thus appears that analyses o f this kind are able to yield information about the nuclear interaction potential o f tire highly excited nucleus. To understand this quantitatively leads to an interesting subject o f the nuclear spectroscopy in the c o n t i n u u m . We are grateful to W. Dfinnweber and W. T r a u t m a n n for several discussions concerning the experimental data.
References
17 March 1983
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