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Neutron inelastic scattering from 40Ca at 21.7 and 25.5 MeV
Nuclear Physics A462 (1987) 445-454 North-Holland, Amsterdam
NEUTRON
INELASTIC SCAmERING 21.7 AND 25.5 MeV* R. ALARCON’ Ohio University, Received (Revised
and Athens,
FROM 40Ca AT
J. RAPAPORT OH 45701 USA
18 April 1986 22 July 1986)
Abstract: Differential cross sections for the scattering of neutrons from 40Ca for states up to 5 MeV excitation energy have been measured at 21.7 and 25.5 MeV. The analysis is done using a coupled channel formalism in terms of a vibration collective model. The results using a usual standard Woods-Saxon form factor for the transitions are compared with those obtained using form factors derived from a model-independent analysis to the elastic cross sections.
E
NUCLEAR
REACTIONS Ca(n, n’), E = 21.7, 25.5 MeV, ?Za levels deduced parameters, octupole moments, B(h). CC analysis.
deformation
1. Introduction In an accompanying paper ’ we present an analysis of nucleon elastic scattering from 40Ca between 11 and 48 MeV. One of the motivations was to study the effect of the traditional Woods-Saxon (WS) radial dependence of the optical model potentials by adopting a model-independent analysis (MIA). In this model, the potential shape is modified with the addition of spherical Fourier-Bessel (FB) functions.
The expansion
coefficients
are then adjusted
to provide
the lowest chi-
square. The inclusion of FB terms as proposed by Friedman and Batty ‘) provides a flexibility in the potential shape that leads to a substantial improvement in the scattering data and is being extended to probes other than nucleons ‘). The analysis usually assumes that a conventional OM analysis has been performed and then the FB terms are added and the expansion coefficients searched for. This procedure was followed in ref. ‘) under the assumption of a spherical OMP. In this paper we study the changes in the OMP parameters when coupling to the excited states in 4oCa are introduced (coupled channel calculations, CC), with the assumption of WS radial form factors. We also study the effects of using the MIA form factors to the shape of calculated distorted waves (DW) inelastic transitions. In a recent paper Honore et al. “) study the CC analysis of nucleon scattering from 40Ca up to 80 MeV. They analyzed an extended set of data including their own differential and analyzing powers for neutron scattering in the energy range ’ Supported in part by the National ’ Now at University 03759474/87/%03.50 0 (North-Holland Physics
of Illinois,
Science Foundation. Nuclear Physics Laboratory,
Elsevier Science Publishers Publishing Division)
B.V.
Champaign,
IL 61820.
446
R Alarcon, J. Rapaport / Neutron inelastic scattering
from 11 to 17 MeV. However, the experimental resolution was not adequate to separate the 3- and !!I-transitions. The work of Honore et al ‘) provides an excellent bibliography to previous studies of nucleon scattering from 4oCa. 2. Experimental
results
The experimental conditions under which the measurements were performed are described in detail in ref. ‘). In fig. 1 of ref. ‘), energy-converted backgroundsubtracted time-of-flight spectra are presented at En = 25.5 and 2 1.7 MeV. The spectra show the obtained resolution, about 450 keV (FWHM), enough to separate the 3and 5 transitions. The 2: state in 40Ca at E, = 3.904 MeV is 0.168 MeV above the 3; state. These two transitions were not resolved in the present experiment. The transition to the first excited state in 40Ca (E, = 3.35 MeV, J” = 0”) was not detected in these measurements.
3. Analysis of the inelastic scattering data The low-lying excited states of 40Ca up to 5 MeV excitation
energy are studied in this section under the framework of the vibrational model for nuclei of even mass 5)_The vibrational model considers shape changes produced by allowing the nuclear surface to be deformed. The nuclear radius R is expressed:
A Taylor expansion of the optical potential U(r, R) then gives to lowest order a non-spherical interaction term which is peaked at the nuclear in the operators (Ye@ surface (2) The operator ohP can be expressed in terms of the deformation parameters Ph and the phonon creation and annihilation operators (see ref. 6), eq. 32). From these phonon operators the wavefunctions of the vibrational states are constructed and used to calculate the matrix elements of eq. (2). The ground state is the no-phonon state from which phonon states of different multipola~ties are constructed by applying creation operators. When a WS shape is used for the optical potential U(r, R), then the radial part of eq. (2) is given by F,(r)=p,(dldr)[R,V(r)+
&W(r)1 t
(3)
where the potentials and nuclear radii have the usual definition ‘) and no deformation of the spin-orbit potential has been considered in eq. (3). For potentials with a non-WS shape, like the MIA potentials ‘), one can generalize eq. (3) by simply considering a deformation of the central potential of the same
R. Alarcon, J. Rapaport / Neutron inelastic scattering
form as the deformations
that distort
the radius
parameter
447
R (eq. (1)). Thus one
may write that K(r) where 6* are deformation
iAIWMdr)l,
= &(dldr)[AR&r,(r)+ lengths
to be determined
(4)
by fitting the experimental
data,
i.e., in the same way as the deformation parameters /3* of eq. (3) are found. The normalizing parameters AR and A1 represent the adjustment of the MIA potentials determined in elastic scattering and to be used in inelastic scattering. In the case of WS potentials this is accomplished by adjusting the potential depths in the inelastic scattering calculations. The aim is to perform CC calculations using the computer code ECIS79 [ref. “)I for the coupling schemes Ccl, CC2 and CC3 shown in fig. 1. The coupling scheme CC1 is studied using transition potentials given by eqs. (3) and (4) as external inputs into the code. For the WS potentials, searches on the real and imaginary potential depths are performed using as a starting set the best fits of table 1 [ref. ‘)I (this adjustment is mandatory because of the coupling of the g.s. to the inelastic states). Then a final search on the deformation parameters Ph is performed. In the case of the MIA potentials the search process is analogous. Thus it is possible to study the predictions for inelastic scattering using non-WS potential shapes and to study the possible consequences of ambiguities in their shapes. In the coupling schemes CC2 and CC3 the transitions correspond to the measured y-ray transition strengths 7P8). The purpose of studying these coupling schemes is to evaluate the effect (if any) of the remaining coupling in the measured cross sections for the 3; and 5; states. The measured neutron cross sections for the 3; state have been corrected for contributions due to the excitation of the 2: state at 3.90 MeV [ref. 9)]. The final optical model parameters obtained in the CC calculations are listed in table 1 (WS potentials), obtained
and in table 2 (MIA potentials);
with these two methods
is presented
“Ca
cc1 Fig. 1. Coupling
Couplmg
a comparison
of chi-squares
in table 3. The deformation
Schemes
cc3
cc2 schemes for the %a(n,
n’)?Za
reaction.
lengths
R. Alarcon, J. Rapaport / Neutron inelastic scattering
448
TABLE 1
WS optical model parameters for the 40Ca(n, n’) reaction Set
(h%,
va
V”
WD
83
Ps
rz,
(2)
elastic
21.7 25.5
47.0 45.3
0.0 0.5
7.30 7.40
cc1
21.7 25.5
45.3 43.63
0.0 0.5
6.09 6.07
0.33 0.32
0.23 0.23
1.34 1.32
0.93 0.93
cc2
21.7
45.62
0.0
5.88
0.33
0.23
1.34
0.93
cc3
21.7
44.98
0.0
5.04
0.33
0.23
1.34
0.93
Potential depths are in MeV, the geometrical parameters are those found in elastic scattering and are tabulated in table 1 (ref. I)), The spin orbit parameters were kept fixed at the values V, o = 5.90 MeV, r,, =l.l2fm, as0 =0.5Ofm.
TABLE 2
MIA optical mode1 parameters for the @Ca(n, n’) reaction
Set
S,, (MeV)
elastic
21.7 25.5
1.0 1.0
1.0 1.0
cc1
21.7 25.5
0.98 0.97
0.84 0.86
AR
AI
8, (fm)
& (fm)
1.17 1.00
0.82 0.74
TABLE 3
Values of chi-square per point x2/N, for the different coupled-channel calculations Set
E, (MeV)
gs.
3;
5;
ccl-ws Ccl-MIA ccl-ws Ccl-MIA ccz-ws cc3-ws
21.7 21.7 25.5 25.5 2.17 21.7
8.5 3.1 10.2 5.3 14.2 28.5
2.1 2.5 3.2 2.4 5.3 11.4
3.4 2.8 5.0 3.3 6.0 7.9
6, in table 1 are defined in the usual way, i.e., 8, = j3.,RR where RR is the radius of the reaI WS potential. Those of table 2 are the ones defined in eq. (4). The calculated cross sections (CC1 coupling scheme) using the two methods are compared to the experimental data at 21.7 MeV and 25.5 MeV in fig. 2. It can be observed that the MIA gives a slightly better fit to the elastic angular distribution at backward angles,
449
R. Alarcon. J. Rapaport / Neutron inelastic scattering
4oCo (n , n’)
Fig. 2. Neutron
2 I .7MeV
j
elastic and inelastic angular distributions. Results for the coupling MIA (solid) and WS (dashed) transition potentials.
scheme
CC1 using
and a slightly better fit to the 5; inelastic angular distribution in the same region. The 3; transition is better fit at larger angles using the WS potentials but the forward angles are better fit with the MIA potentials. The reduction in potential depths resulting when the CC1 approach is used instead of the spherical OMP is very similar for both the WS and MIA methods (tables 1 and 2). However, the deformation lengths are different. The reason for the discrepancy lies in the different potentials used in eqs. (3) and (4) to calculate the transition amplitudes Fh (r). In the case of a non-WS potential, it is inappropriate to compare quantities that depend strongly on the geometry of the optical potential. Instead it is more appropriate to compare the transition amplitudes FA (r) used in the WS and MIA methods. Fig. 3 shows the real (imaginary) F3(r) transition amplitudes as given by eqs. (3) and (4). It may be observed that these two methods give F3(r) values different only at very small r; however, the region sensitive to the neutron inelastic scattering is between 2 and 7 fm. This may be seen in fig. 4, which shows the ratio of the calculated total inelastic cross section between R and (R +0.5) [fm] to the total inelastic cross section calculated between R = 0 and R = 10 fm. This example is for the 3; state at 21.7 MeV neutron bombarding energy assuming the WS transition potentials. The calculation shows clearly that at the incident neutron energies here studied the interaction takes place mainly near the nuclear surface.
R. Alarcon, 1. Rapport / Neutron inelastic scattering
450 1
?a
(n,d)
‘OCa
3;state E,: 21.7 MeV
( n. d
3;state
-
En=25.5MeV MIA --_ws
0
2
4
6
_ ,; /
8
10
r (fm)
-5 4 0
2
4 r (fm)
I 10
8
6
Fig. 3. Real and imaginary MIA (solid) and WS (dashed) S(r) transition amplitudes. To compare method field:
deformation
of Hamilton
parameters
and Mackintosh
integral
in both methods,
lo) by using the moment
qhO = PA
and the volume
obtained
given by 5=&r
J J
The ratio of eqs. (5) and (6) defines
we follow
of the real vibrational
(5)
rA+’ Re F3(r) dr
r’V(r)
dr.
a “normalized
Qho= aolJ.
the
(6) moment”
Q, i.e., (7)
This normalized multipole moment was defined in ref. lo) for a rotational model, and by using Satchler’s theorem 11) it becomes equal to the multipole moment of the nuclear density.
451
R Alarcon, J. Rapaport / Neutron inelastic scatteting
'"Ca (n,d) 30
En= 21.7 MeV
3; state % of
Total
E,= 3.73 MeV Inelastic
0.5 fm
Cross Section
0;
bins
r (fm) Fig. 4. Ratio
of the 3; total
cross
section in radial intervals of 0.5 fm to the total calculated between 0 and 10 fm.
3;
cross section
The results obtained for the 3; state are presented in table 4 for two different integration limits: between 2 and 7 fm, and from 0.1 to 10 fm. The tabulated values show that at 21.7 MeV there is good agreement between the QSOvalues when they are calculated between 2-7 fm, but a discrepancy is found for the 0.1 to 10 fm range. This is due to the negative values of the MIA F,(r) at large radii (see fig. 3). At 25.5 MeV the agreement in the 2-7 fm range is not as good as in the 21.7 MeV case; also at 25.5 MeV the MIA &(r) gives an unusually large value of Q30 for the 0.1 to 10 fm range, likely due to the oscillation of &(r) at large r (see fig. 3). Also tabulated in table 4 are the B(E3) values obtained from B(E3) = (ZQJ0)‘e2 TABLE
Normalized
octupole
moments,
m
4
values for the g.s. + 3; transition
JBO E, WV)
Method
hi@j (e . fm3) 0.1-10 fm
(e. fm3) 2-7 fm
21.7
MIA ws
8.0* 0.7 8.2 f 0.7
160* 14 164+ 14
5.9*0.5 8.8*0.8
119* 10 176* 15
25.5
MIA ws
6.4+0.5 8.1 ho.7
127+11 162* 14
11.0* 1.0 8.6kO.7
220* 19 171*15
The EM value is that of ref. ‘). The quoted errors reflect lengths found in the CC1 calculation (tables 1 and 2).
only the uncertainty
in 40Ca
v@G (e . fm’) EM
132*4
of the deformation
wosytrdtuo:, ly%.queatu avow e alq!ssod aqeur pInoM uo~~a.~s!q$ III suo!~as sso.13 ~gse~au~30 sw3uIasnsaaur za)$aq ‘zafiaMoH *satyea ~33 aq%uI0.g iuaJag!p a3Bsa@uc pzwtqD??q w SUO~XXSSOJD~33 pue 233 aql30 sadeqs aqt osfv *arnaqDs8ugdno:, 133 aql II! palctIpa.td sanp?A aq$ 0%padsaJ ~J!M suo!yas SSOJ~a@~utlplaM.103 Js puz 1 E aql30 sanIeA aql30 uo!pnpoJdaJ ay~~o3 uaas aq ue3 q%noql swauIaaoJdwr I~XIIS -saws _Ig pue Jc aql30 uoy~pxa aql ‘03 alq!suodw ~C~U~XI.I aJE sassaDoJd dais-auo aql leql i?uge~pu! (z ‘%yaaS) 133 auIaqX aql ~I!M sf! ~00% se IOUaJe slpwal ayl wyl map s! 11 ‘(1 +i?y oqt! aas) (8 ‘3al u! pals!1 suo!~!sutw 11”30 asn Buppzur snql put? 233 u! alQ?)s;O aql Sa)vlOd.IOC?U! q%q~ ~33 sauIaq~s %ugdnoa aql ql!M s$ptsaJ aql a.n? (SaUg paqszp sa) u~oqs osjv ‘(saug pyos) 5 ‘8y u! ewp pwanrpadxa aql 01 paladuxo3 a.~?233 auraq% kIdno:, aq$ q$!M suog%~ SSOJSpav+wpz3 aqJ, ‘(1 aIqe1 u! a.nz s.xalaurr.wd ~XIIJaq)) Ougdno:, aJotu 30 uo!snpu! aq) IDagal 01 palsnfpv aJaM sqldap lagualod SM aql &IO *Zd 30 arqaA aq) u! auop stz~ yaleas ou snqi ‘awls Jz aql 103 aIqBIp5w ale wep ou wnu -nadxa )uasaJd aq$ UI *(EIguuaww uo$o.rd AaW OS-0~ 1.uo.13 pule %ugalws uo.wIau AaH 11 uioi3 SatyeA palnst?aw q$;M ~uauraa.& ~00% bran u! 01’0 = Zd .ta$auwzd uogzuuo3ap B SW@ q9tqM pasn uaaq seq ~3 OE**= fzg)“g anpz” e satye” (za)g aq$ ~013 s)uau~a~a x~rwu~ ~3 aql awzlnyw OJ. *auraqDs %ugdno:, 133 aq$ uy JOU suog~sue.~l ~3 0~1 saq aI.uaqDs %ugdno:, 233 aqJ, w3 fjz’p =:(sa)“x s! uo~l~Sue3~ Js +T% aq) .x03 pue tu3 LE-P= (~a)“‘)i 3;[ uogye~~ JE t-~% aql “03 tInsa au $,I luapuadap uogyw aq 01 uiklouq sj qXqM sn!peJ uop~suw aql awItya3 uci au0 ‘6“ sueazu $4~3 6q parnseam asoql q$!M sanp2A (?~)“a asaql Buyznba Ag *sanII?A (Q)ug ~0 sql8uam uo!l~suw~ xJst?lau! uoJmau aql al2fmIm 01 pasn ale (1 a1qel aaS) I\agbJL’TZ1~ pau??lqo ‘ff put! Ed s.xalauwxd u09?uL103ap aq&. ‘1 ‘%gu! umoqs s! 11puz 233 palIe:, s! aruaqDs ifuydno:, flugnsa3 aqi ‘(Aagq 06~ = ‘2) aws Jz aql 30 Daga aql ampowi! 03 SEM133 aUIaq% %ugdno~ aql 01 uogt?~yporu )s.~y aqL *&jawoa% SM aql pm s~alammd uoymto3ap aql lallaq auyap 01 pasn aq LIED0’0 aql axqs sIspua$od SM amd 1x10~3synm saledwo:, auo uaqkt In3asn os~a sr s!q~. ‘as83 luasald ay$ u! se sadeqs Iagua$od aq, u; sagn%~qure ilu~Apn$s~03 100%ahyisuas e aq OXswaas 0’0 aq%30 asn aq.& *I%yuaJod VIW aq$30 aage+ap aqz 8ugw dq pawpIcqe~ STJ! astwaq c1ow3 two3 DgszIaut aql~o3 pastwqua az A aZnzj 1%suo~yz~~pso au wep IwuaurPadxa aqt 0%$LJaq, ‘alojaraql ‘pue s[egualod VII/V aql30 adeqs aq$Ou!va$$% 61a8.y lnoql!M J a8.y le SUO~~Z~I~~SO IIEUISaql sampas lnq axxapuadap lapour ~pxus e sampow! u.136 30 smpw go-in:, B hapas lassaa-Japnod aql ~03 w3 01 30 srupm go-ltw ‘I?%u!sn (~ ‘3a.1r.q %u!Jallms ct!lwja .103pun03 alaM spxlualod asaw -saxws~p 1112 le pauyap 11aMspxlualod VIM %u!aoq 30 paau aql Moqs p aIqel30 s1Insa.Taq_L G‘ sanp?A ~r;ra pavodar aqj wq$ ~a%lal am sanp% (ES& palt?pmIm wasaId au -p a1qw u! papaga3 s! sg~_ *Q!suap ~0~~~s~~~~ aq) 30 sluaurom aIodf$Inur aql ueql xa%leI axe legualod paurrojap aql30 sluaurosu aIod -ynur pazgemou aql ‘paprqm! s! amapuadap-Qsuap 31 *(aiLmy:, snapnu aq] ST2) suo!l%?Jalu! luapuadapu+Q!suap 103 sploq qwqM tuaroaql s,.yy3las aqJ30 anl.yA Lq Buyajjv~zysv,au~uo~lna~/ uodvdvx
T ‘uoa~q~ ‘x
ZStJ
453
R. Alarcoq _I.Rapaport / Neutron inelastic scattering
U0Cokt.n')
30
60
21.7 hV CC3 Idrstwdl
40
120
is0
80
ecnl Fig. 5. Calculations done at 21.7 MeV for the coupling schemes CC2 (solid) and CC3 (dashed).
Finally, it is probably not a good assumption to assume that the inclusion of the 0: and 2: states has to be done in the framework of the vibrational model simply because the 0: (Ex= 3.35 MeV), 2: (E,=3.90)and 4: (E;, = 5.28 MeV) form a so-called quasi-beta band 14) which has a rotational character. The lack of experimental data for these states is the main obstacle to more complete calculations, i.e., to take into account vibrational and rotational states in the framework of one model. 4. Summary and coaciusioas The elastic and inelastic scattering from 40Ca were studied at 21.7 and 25.5 MeV using the WS and MIA approaches to calculate the vibrational model transition amplitudes for the excitation of the 3; (3.74 MeV) and 5; (4.49 MeV) collective states. The agreement with the data was good in both cases, but disagreement was found for the defo~ation lengths predicted by both approaches. When the normalized multipole moments QjO of the real potential were compared, it turned out that a reasonable agreement is found if the integration limit is between 2 and 7 fm
454
R Alarcon, J. Rapaport / Neutron inelastic scattering
which is the scattering sensitive region. When the O-10 fm range is used, then the &‘s calculated from the MIA approach reflect the shape anomalies of the MIA potentials at large radii. The coupling effect of the low-lying states 0: (3.35 MeV) and 2: (3.90 MeV) in the excitation of the 3; and 5; states was evaluated and it was found that in the framework of the vibrational model the 3; and 5; are merely one-step excitations. References 1) R. Alarcon, J. Rapaport and R.W. Finlay, Nucl. Phys. A462 (1987) 413 2) E. Friedman and C.J. Batty, Phys. Rev. Cl7 (1978) 34 3) Advanced methods in the evaluation of nuclear scattering data, ed. by H.J. Krappe and R. Lipperheide, Lecture Notes in Physics, vol. 236, (Springer, Berlin, 1985) 4) G.M. Honore, W. Tomow, C.R. Howell, R.S. Pedroni, R.C. Byrd, R.W. Walter and J.P. Delaroche, Phys. Rev. C (1986) 1129 5) T. Tamura, Rev. Mod. Phys. 37 (1965) 679 6) J. Raynal, code ECIS79, Saclay; IAEA Report SMR-918 (1972) 218 7) P.M. Endt and C. van der Leun, Nucl. Physl A310 (1978) 1 8) P.M. Endt, At. Data and Nucl. Data Tables 23 (1979) 3 9) R. Alarcon, Ph.D. dissertation, Ohio University (1985) 10) J.K. Hamilton and R.S. Mackintosh, J. Phys. 64 (1978) 557 11) G.R. Satchler, J. Math. Phys. 13 (1972) 1118 12) P.W.F. Alons, H.P. Blok, J.F.A. Van Hienen and J. Blok, Nucl. Phys. A367 (1981) 41 13) D.E. Bainum, R.W. Finlay, J. Rapaport and J.D. Carlson, Phys. Rev. Cl6 (1977) 1377 14) M. Sakai and A.C. Rester, At. Data and Nucl. Data Tables 20 (1977) 441
NEUTRON
INELASTIC SCAmERING 21.7 AND 25.5 MeV* R. ALARCON’ Ohio University, Received (Revised
and Athens,
FROM 40Ca AT
J. RAPAPORT OH 45701 USA
18 April 1986 22 July 1986)
Abstract: Differential cross sections for the scattering of neutrons from 40Ca for states up to 5 MeV excitation energy have been measured at 21.7 and 25.5 MeV. The analysis is done using a coupled channel formalism in terms of a vibration collective model. The results using a usual standard Woods-Saxon form factor for the transitions are compared with those obtained using form factors derived from a model-independent analysis to the elastic cross sections.
E
NUCLEAR
REACTIONS Ca(n, n’), E = 21.7, 25.5 MeV, ?Za levels deduced parameters, octupole moments, B(h). CC analysis.
deformation
1. Introduction In an accompanying paper ’ we present an analysis of nucleon elastic scattering from 40Ca between 11 and 48 MeV. One of the motivations was to study the effect of the traditional Woods-Saxon (WS) radial dependence of the optical model potentials by adopting a model-independent analysis (MIA). In this model, the potential shape is modified with the addition of spherical Fourier-Bessel (FB) functions.
The expansion
coefficients
are then adjusted
to provide
the lowest chi-
square. The inclusion of FB terms as proposed by Friedman and Batty ‘) provides a flexibility in the potential shape that leads to a substantial improvement in the scattering data and is being extended to probes other than nucleons ‘). The analysis usually assumes that a conventional OM analysis has been performed and then the FB terms are added and the expansion coefficients searched for. This procedure was followed in ref. ‘) under the assumption of a spherical OMP. In this paper we study the changes in the OMP parameters when coupling to the excited states in 4oCa are introduced (coupled channel calculations, CC), with the assumption of WS radial form factors. We also study the effects of using the MIA form factors to the shape of calculated distorted waves (DW) inelastic transitions. In a recent paper Honore et al. “) study the CC analysis of nucleon scattering from 40Ca up to 80 MeV. They analyzed an extended set of data including their own differential and analyzing powers for neutron scattering in the energy range ’ Supported in part by the National ’ Now at University 03759474/87/%03.50 0 (North-Holland Physics
of Illinois,
Science Foundation. Nuclear Physics Laboratory,
Elsevier Science Publishers Publishing Division)
B.V.
Champaign,
IL 61820.
446
R Alarcon, J. Rapaport / Neutron inelastic scattering
from 11 to 17 MeV. However, the experimental resolution was not adequate to separate the 3- and !!I-transitions. The work of Honore et al ‘) provides an excellent bibliography to previous studies of nucleon scattering from 4oCa. 2. Experimental
results
The experimental conditions under which the measurements were performed are described in detail in ref. ‘). In fig. 1 of ref. ‘), energy-converted backgroundsubtracted time-of-flight spectra are presented at En = 25.5 and 2 1.7 MeV. The spectra show the obtained resolution, about 450 keV (FWHM), enough to separate the 3and 5 transitions. The 2: state in 40Ca at E, = 3.904 MeV is 0.168 MeV above the 3; state. These two transitions were not resolved in the present experiment. The transition to the first excited state in 40Ca (E, = 3.35 MeV, J” = 0”) was not detected in these measurements.
3. Analysis of the inelastic scattering data The low-lying excited states of 40Ca up to 5 MeV excitation
energy are studied in this section under the framework of the vibrational model for nuclei of even mass 5)_The vibrational model considers shape changes produced by allowing the nuclear surface to be deformed. The nuclear radius R is expressed:
A Taylor expansion of the optical potential U(r, R) then gives to lowest order a non-spherical interaction term which is peaked at the nuclear in the operators (Ye@ surface (2) The operator ohP can be expressed in terms of the deformation parameters Ph and the phonon creation and annihilation operators (see ref. 6), eq. 32). From these phonon operators the wavefunctions of the vibrational states are constructed and used to calculate the matrix elements of eq. (2). The ground state is the no-phonon state from which phonon states of different multipola~ties are constructed by applying creation operators. When a WS shape is used for the optical potential U(r, R), then the radial part of eq. (2) is given by F,(r)=p,(dldr)[R,V(r)+
&W(r)1 t
(3)
where the potentials and nuclear radii have the usual definition ‘) and no deformation of the spin-orbit potential has been considered in eq. (3). For potentials with a non-WS shape, like the MIA potentials ‘), one can generalize eq. (3) by simply considering a deformation of the central potential of the same
R. Alarcon, J. Rapaport / Neutron inelastic scattering
form as the deformations
that distort
the radius
parameter
447
R (eq. (1)). Thus one
may write that K(r) where 6* are deformation
iAIWMdr)l,
= &(dldr)[AR&r,(r)+ lengths
to be determined
(4)
by fitting the experimental
data,
i.e., in the same way as the deformation parameters /3* of eq. (3) are found. The normalizing parameters AR and A1 represent the adjustment of the MIA potentials determined in elastic scattering and to be used in inelastic scattering. In the case of WS potentials this is accomplished by adjusting the potential depths in the inelastic scattering calculations. The aim is to perform CC calculations using the computer code ECIS79 [ref. “)I for the coupling schemes Ccl, CC2 and CC3 shown in fig. 1. The coupling scheme CC1 is studied using transition potentials given by eqs. (3) and (4) as external inputs into the code. For the WS potentials, searches on the real and imaginary potential depths are performed using as a starting set the best fits of table 1 [ref. ‘)I (this adjustment is mandatory because of the coupling of the g.s. to the inelastic states). Then a final search on the deformation parameters Ph is performed. In the case of the MIA potentials the search process is analogous. Thus it is possible to study the predictions for inelastic scattering using non-WS potential shapes and to study the possible consequences of ambiguities in their shapes. In the coupling schemes CC2 and CC3 the transitions correspond to the measured y-ray transition strengths 7P8). The purpose of studying these coupling schemes is to evaluate the effect (if any) of the remaining coupling in the measured cross sections for the 3; and 5; states. The measured neutron cross sections for the 3; state have been corrected for contributions due to the excitation of the 2: state at 3.90 MeV [ref. 9)]. The final optical model parameters obtained in the CC calculations are listed in table 1 (WS potentials), obtained
and in table 2 (MIA potentials);
with these two methods
is presented
“Ca
cc1 Fig. 1. Coupling
Couplmg
a comparison
of chi-squares
in table 3. The deformation
Schemes
cc3
cc2 schemes for the %a(n,
n’)?Za
reaction.
lengths
R. Alarcon, J. Rapaport / Neutron inelastic scattering
448
TABLE 1
WS optical model parameters for the 40Ca(n, n’) reaction Set
(h%,
va
V”
WD
83
Ps
rz,
(2)
elastic
21.7 25.5
47.0 45.3
0.0 0.5
7.30 7.40
cc1
21.7 25.5
45.3 43.63
0.0 0.5
6.09 6.07
0.33 0.32
0.23 0.23
1.34 1.32
0.93 0.93
cc2
21.7
45.62
0.0
5.88
0.33
0.23
1.34
0.93
cc3
21.7
44.98
0.0
5.04
0.33
0.23
1.34
0.93
Potential depths are in MeV, the geometrical parameters are those found in elastic scattering and are tabulated in table 1 (ref. I)), The spin orbit parameters were kept fixed at the values V, o = 5.90 MeV, r,, =l.l2fm, as0 =0.5Ofm.
TABLE 2
MIA optical mode1 parameters for the @Ca(n, n’) reaction
Set
S,, (MeV)
elastic
21.7 25.5
1.0 1.0
1.0 1.0
cc1
21.7 25.5
0.98 0.97
0.84 0.86
AR
AI
8, (fm)
& (fm)
1.17 1.00
0.82 0.74
TABLE 3
Values of chi-square per point x2/N, for the different coupled-channel calculations Set
E, (MeV)
gs.
3;
5;
ccl-ws Ccl-MIA ccl-ws Ccl-MIA ccz-ws cc3-ws
21.7 21.7 25.5 25.5 2.17 21.7
8.5 3.1 10.2 5.3 14.2 28.5
2.1 2.5 3.2 2.4 5.3 11.4
3.4 2.8 5.0 3.3 6.0 7.9
6, in table 1 are defined in the usual way, i.e., 8, = j3.,RR where RR is the radius of the reaI WS potential. Those of table 2 are the ones defined in eq. (4). The calculated cross sections (CC1 coupling scheme) using the two methods are compared to the experimental data at 21.7 MeV and 25.5 MeV in fig. 2. It can be observed that the MIA gives a slightly better fit to the elastic angular distribution at backward angles,
449
R. Alarcon. J. Rapaport / Neutron inelastic scattering
4oCo (n , n’)
Fig. 2. Neutron
2 I .7MeV
j
elastic and inelastic angular distributions. Results for the coupling MIA (solid) and WS (dashed) transition potentials.
scheme
CC1 using
and a slightly better fit to the 5; inelastic angular distribution in the same region. The 3; transition is better fit at larger angles using the WS potentials but the forward angles are better fit with the MIA potentials. The reduction in potential depths resulting when the CC1 approach is used instead of the spherical OMP is very similar for both the WS and MIA methods (tables 1 and 2). However, the deformation lengths are different. The reason for the discrepancy lies in the different potentials used in eqs. (3) and (4) to calculate the transition amplitudes Fh (r). In the case of a non-WS potential, it is inappropriate to compare quantities that depend strongly on the geometry of the optical potential. Instead it is more appropriate to compare the transition amplitudes FA (r) used in the WS and MIA methods. Fig. 3 shows the real (imaginary) F3(r) transition amplitudes as given by eqs. (3) and (4). It may be observed that these two methods give F3(r) values different only at very small r; however, the region sensitive to the neutron inelastic scattering is between 2 and 7 fm. This may be seen in fig. 4, which shows the ratio of the calculated total inelastic cross section between R and (R +0.5) [fm] to the total inelastic cross section calculated between R = 0 and R = 10 fm. This example is for the 3; state at 21.7 MeV neutron bombarding energy assuming the WS transition potentials. The calculation shows clearly that at the incident neutron energies here studied the interaction takes place mainly near the nuclear surface.
R. Alarcon, 1. Rapport / Neutron inelastic scattering
450 1
?a
(n,d)
‘OCa
3;state E,: 21.7 MeV
( n. d
3;state
-
En=25.5MeV MIA --_ws
0
2
4
6
_ ,; /
8
10
r (fm)
-5 4 0
2
4 r (fm)
I 10
8
6
Fig. 3. Real and imaginary MIA (solid) and WS (dashed) S(r) transition amplitudes. To compare method field:
deformation
of Hamilton
parameters
and Mackintosh
integral
in both methods,
lo) by using the moment
qhO = PA
and the volume
obtained
given by 5=&r
J J
The ratio of eqs. (5) and (6) defines
we follow
of the real vibrational
(5)
rA+’ Re F3(r) dr
r’V(r)
dr.
a “normalized
Qho= aolJ.
the
(6) moment”
Q, i.e., (7)
This normalized multipole moment was defined in ref. lo) for a rotational model, and by using Satchler’s theorem 11) it becomes equal to the multipole moment of the nuclear density.
451
R Alarcon, J. Rapaport / Neutron inelastic scatteting
'"Ca (n,d) 30
En= 21.7 MeV
3; state % of
Total
E,= 3.73 MeV Inelastic
0.5 fm
Cross Section
0;
bins
r (fm) Fig. 4. Ratio
of the 3; total
cross
section in radial intervals of 0.5 fm to the total calculated between 0 and 10 fm.
3;
cross section
The results obtained for the 3; state are presented in table 4 for two different integration limits: between 2 and 7 fm, and from 0.1 to 10 fm. The tabulated values show that at 21.7 MeV there is good agreement between the QSOvalues when they are calculated between 2-7 fm, but a discrepancy is found for the 0.1 to 10 fm range. This is due to the negative values of the MIA F,(r) at large radii (see fig. 3). At 25.5 MeV the agreement in the 2-7 fm range is not as good as in the 21.7 MeV case; also at 25.5 MeV the MIA &(r) gives an unusually large value of Q30 for the 0.1 to 10 fm range, likely due to the oscillation of &(r) at large r (see fig. 3). Also tabulated in table 4 are the B(E3) values obtained from B(E3) = (ZQJ0)‘e2 TABLE
Normalized
octupole
moments,
m
4
values for the g.s. + 3; transition
JBO E, WV)
Method
hi@j (e . fm3) 0.1-10 fm
(e. fm3) 2-7 fm
21.7
MIA ws
8.0* 0.7 8.2 f 0.7
160* 14 164+ 14
5.9*0.5 8.8*0.8
119* 10 176* 15
25.5
MIA ws
6.4+0.5 8.1 ho.7
127+11 162* 14
11.0* 1.0 8.6kO.7
220* 19 171*15
The EM value is that of ref. ‘). The quoted errors reflect lengths found in the CC1 calculation (tables 1 and 2).
only the uncertainty
in 40Ca
v@G (e . fm’) EM
132*4
of the deformation
wosytrdtuo:, ly%.queatu avow e alq!ssod aqeur pInoM uo~~a.~s!q$ III suo!~as sso.13 ~gse~au~30 sw3uIasnsaaur za)$aq ‘zafiaMoH *satyea ~33 aq%uI0.g iuaJag!p a3Bsa@uc pzwtqD??q w SUO~XXSSOJD~33 pue 233 aql30 sadeqs aqt osfv *arnaqDs8ugdno:, 133 aql II! palctIpa.td sanp?A aq$ 0%padsaJ ~J!M suo!yas SSOJ~a@~utlplaM.103 Js puz 1 E aql30 sanIeA aql30 uo!pnpoJdaJ ay~~o3 uaas aq ue3 q%noql swauIaaoJdwr I~XIIS -saws _Ig pue Jc aql30 uoy~pxa aql ‘03 alq!suodw ~C~U~XI.I aJE sassaDoJd dais-auo aql leql i?uge~pu! (z ‘%yaaS) 133 auIaqX aql ~I!M sf! ~00% se IOUaJe slpwal ayl wyl map s! 11 ‘(1 +i?y oqt! aas) (8 ‘3al u! pals!1 suo!~!sutw 11”30 asn Buppzur snql put? 233 u! alQ?)s;O aql Sa)vlOd.IOC?U! q%q~ ~33 sauIaq~s %ugdnoa aql ql!M s$ptsaJ aql a.n? (SaUg paqszp sa) u~oqs osjv ‘(saug pyos) 5 ‘8y u! ewp pwanrpadxa aql 01 paladuxo3 a.~?233 auraq% kIdno:, aq$ q$!M suog%~ SSOJSpav+wpz3 aqJ, ‘(1 aIqe1 u! a.nz s.xalaurr.wd ~XIIJaq)) Ougdno:, aJotu 30 uo!snpu! aq) IDagal 01 palsnfpv aJaM sqldap lagualod SM aql &IO *Zd 30 arqaA aq) u! auop stz~ yaleas ou snqi ‘awls Jz aql 103 aIqBIp5w ale wep ou wnu -nadxa )uasaJd aq$ UI *(EIguuaww uo$o.rd AaW OS-0~ 1.uo.13 pule %ugalws uo.wIau AaH 11 uioi3 SatyeA palnst?aw q$;M ~uauraa.& ~00% bran u! 01’0 = Zd .ta$auwzd uogzuuo3ap B SW@ q9tqM pasn uaaq seq ~3 OE**= fzg)“g anpz” e satye” (za)g aq$ ~013 s)uau~a~a x~rwu~ ~3 aql awzlnyw OJ. *auraqDs %ugdno:, 133 aq$ uy JOU suog~sue.~l ~3 0~1 saq aI.uaqDs %ugdno:, 233 aqJ, w3 fjz’p =:(sa)“x s! uo~l~Sue3~ Js +T% aq) .x03 pue tu3 LE-P= (~a)“‘)i 3;[ uogye~~ JE t-~% aql “03 tInsa au $,I luapuadap uogyw aq 01 uiklouq sj qXqM sn!peJ uop~suw aql awItya3 uci au0 ‘6“ sueazu $4~3 6q parnseam asoql q$!M sanp2A (?~)“a asaql Buyznba Ag *sanII?A (Q)ug ~0 sql8uam uo!l~suw~ xJst?lau! uoJmau aql al2fmIm 01 pasn ale (1 a1qel aaS) I\agbJL’TZ1~ pau??lqo ‘ff put! Ed s.xalauwxd u09?uL103ap aq&. ‘1 ‘%gu! umoqs s! 11puz 233 palIe:, s! aruaqDs ifuydno:, flugnsa3 aqi ‘(Aagq 06~ = ‘2) aws Jz aql 30 Daga aql ampowi! 03 SEM133 aUIaq% %ugdno~ aql 01 uogt?~yporu )s.~y aqL *&jawoa% SM aql pm s~alammd uoymto3ap aql lallaq auyap 01 pasn aq LIED0’0 aql axqs sIspua$od SM amd 1x10~3synm saledwo:, auo uaqkt In3asn os~a sr s!q~. ‘as83 luasald ay$ u! se sadeqs Iagua$od aq, u; sagn%~qure ilu~Apn$s~03 100%ahyisuas e aq OXswaas 0’0 aq%30 asn aq.& *I%yuaJod VIW aq$30 aage+ap aqz 8ugw dq pawpIcqe~ STJ! astwaq c1ow3 two3 DgszIaut aql~o3 pastwqua az A aZnzj 1%suo~yz~~pso au wep IwuaurPadxa aqt 0%$LJaq, ‘alojaraql ‘pue s[egualod VII/V aql30 adeqs aq$Ou!va$$% 61a8.y lnoql!M J a8.y le SUO~~Z~I~~SO IIEUISaql sampas lnq axxapuadap lapour ~pxus e sampow! u.136 30 smpw go-in:, B hapas lassaa-Japnod aql ~03 w3 01 30 srupm go-ltw ‘I?%u!sn (~ ‘3a.1r.q %u!Jallms ct!lwja .103pun03 alaM spxlualod asaw -saxws~p 1112 le pauyap 11aMspxlualod VIM %u!aoq 30 paau aql Moqs p aIqel30 s1Insa.Taq_L G‘ sanp?A ~r;ra pavodar aqj wq$ ~a%lal am sanp% (ES& palt?pmIm wasaId au -p a1qw u! papaga3 s! sg~_ *Q!suap ~0~~~s~~~~ aq) 30 sluaurom aIodf$Inur aql ueql xa%leI axe legualod paurrojap aql30 sluaurosu aIod -ynur pazgemou aql ‘paprqm! s! amapuadap-Qsuap 31 *(aiLmy:, snapnu aq] ST2) suo!l%?Jalu! luapuadapu+Q!suap 103 sploq qwqM tuaroaql s,.yy3las aqJ30 anl.yA Lq Buyajjv~zysv,au~uo~lna~/ uodvdvx
T ‘uoa~q~ ‘x
ZStJ
453
R. Alarcoq _I.Rapaport / Neutron inelastic scattering
U0Cokt.n')
30
60
21.7 hV CC3 Idrstwdl
40
120
is0
80
ecnl Fig. 5. Calculations done at 21.7 MeV for the coupling schemes CC2 (solid) and CC3 (dashed).
Finally, it is probably not a good assumption to assume that the inclusion of the 0: and 2: states has to be done in the framework of the vibrational model simply because the 0: (Ex= 3.35 MeV), 2: (E,=3.90)and 4: (E;, = 5.28 MeV) form a so-called quasi-beta band 14) which has a rotational character. The lack of experimental data for these states is the main obstacle to more complete calculations, i.e., to take into account vibrational and rotational states in the framework of one model. 4. Summary and coaciusioas The elastic and inelastic scattering from 40Ca were studied at 21.7 and 25.5 MeV using the WS and MIA approaches to calculate the vibrational model transition amplitudes for the excitation of the 3; (3.74 MeV) and 5; (4.49 MeV) collective states. The agreement with the data was good in both cases, but disagreement was found for the defo~ation lengths predicted by both approaches. When the normalized multipole moments QjO of the real potential were compared, it turned out that a reasonable agreement is found if the integration limit is between 2 and 7 fm
454
R Alarcon, J. Rapaport / Neutron inelastic scattering
which is the scattering sensitive region. When the O-10 fm range is used, then the &‘s calculated from the MIA approach reflect the shape anomalies of the MIA potentials at large radii. The coupling effect of the low-lying states 0: (3.35 MeV) and 2: (3.90 MeV) in the excitation of the 3; and 5; states was evaluated and it was found that in the framework of the vibrational model the 3; and 5; are merely one-step excitations. References 1) R. Alarcon, J. Rapaport and R.W. Finlay, Nucl. Phys. A462 (1987) 413 2) E. Friedman and C.J. Batty, Phys. Rev. Cl7 (1978) 34 3) Advanced methods in the evaluation of nuclear scattering data, ed. by H.J. Krappe and R. Lipperheide, Lecture Notes in Physics, vol. 236, (Springer, Berlin, 1985) 4) G.M. Honore, W. Tomow, C.R. Howell, R.S. Pedroni, R.C. Byrd, R.W. Walter and J.P. Delaroche, Phys. Rev. C (1986) 1129 5) T. Tamura, Rev. Mod. Phys. 37 (1965) 679 6) J. Raynal, code ECIS79, Saclay; IAEA Report SMR-918 (1972) 218 7) P.M. Endt and C. van der Leun, Nucl. Physl A310 (1978) 1 8) P.M. Endt, At. Data and Nucl. Data Tables 23 (1979) 3 9) R. Alarcon, Ph.D. dissertation, Ohio University (1985) 10) J.K. Hamilton and R.S. Mackintosh, J. Phys. 64 (1978) 557 11) G.R. Satchler, J. Math. Phys. 13 (1972) 1118 12) P.W.F. Alons, H.P. Blok, J.F.A. Van Hienen and J. Blok, Nucl. Phys. A367 (1981) 41 13) D.E. Bainum, R.W. Finlay, J. Rapaport and J.D. Carlson, Phys. Rev. Cl6 (1977) 1377 14) M. Sakai and A.C. Rester, At. Data and Nucl. Data Tables 20 (1977) 441