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Electron scattering at 180° from 181Ta
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985
E L E C T R O N S C A T T E R I N G A T 180 ° F R O M tStTa
M. N I S H I M U R A , D.W.L. S P R U N G Department of Physics, McMaster University, Hamilton, Ontario, Canada L8S 4M1
and E. M O Y A D E G U E R R A IEM, CSIC, Serrano 119, Madrid 28006, Spain and Departamento de Fisica Atomica y Nuclear, Universita Extremadura, Badajoz, Spain
Received 9 July 1985
Results of complete DWBA calculations of the differential cross section at 180° for elastic and inelastic electron scattering on 181Taare presented and compared with experimental data. The effects of the collective current and Coulomb distortion are investigated. It is shown that the Coulomb distortion must be properly taken into account to investigate low multipoles, and that experimental data in the low momentum transfer region may provide reliable information on the collective current.
In recent years some effort has been devoted to measuring transverse form factors of rare-earth rotational nuclei [1,2] as a means of obtaining information on the nature o f the collective rotational mode. Differential cross sections at 180 ° for elastic and inelastic scattering to the two first excited states of 181Ta were measured at the Bates Linear Accelerator Centre [ 1 ]. In ref. [ 1 ] the experimental data were compared to projected Hartree-Fock (PHF) calculations [3] of transverse form factors using the plane-wave Born approximation (PWBA) with the standard scale change o f q into qeff- Especially for inelastic scattering large disagreements were found between experiment and theory in the low~/region (q <~ 1.1 f m - 1 ). For this heavy rotational nucleus the l o w q region is the most sensitive to the actual form of the collective current, as well as to Coulomb distortion effects. In a previous paper [4] we calculated longitudinal contributions to the differential cross section at 180 ° on heavy deformed nuclei using the distorted-wave Born approximation (DWBA). We showed that at low q, the C2 contribution was as large or larger than the previously calculated transverse form factor [3] for the cases of inelastic scattering to the two first excited states of 181Ta, and hence should not be neglected. Since C2 and E2 multipoles interfere in DWBA, we concluded that complete DWBA calculations had to be done for meaningful comparison to experimental data. In this letter we present results of complete DWBA calculations using two extreme models - irrotational flow (IF) and rigid rotor (RR) - for the collective current. For 181Ta magnetic multipoles are dominated by the single particle (SP) contributions and are not very sensitive to the model used for the collective current. On the contrary transverse electric multipoles with ?~< 8 do not contain contributions from the SP current and are only sensitive to the collective rotational current~ It can be easily shown [5,6] that the RR and IF models provide, respectively, a lower and an upper bound on the E2 multipole when the model dependence of the moment of inertia is used. Predictions o f microscopic models (such as the cranking and PHF models) lie in general between these limits [7]. Hence, to investigate the dependence on the collective current it is sufficient here to focus on the two models mentioned above. The transverse electric and magnetic multipoles PXL of the nuclear transition current are given by [5,6] 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
235
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985
[i = (2I + 1)1/21
Pxx+l(r)='Ii{(IiKkOIIfK)[If(Zf + 1)-Ii(Ii. + 1)]pRh±l(r) + (--1)/i-K(/i -ICA2KIIfK)p2K±I(r)},
(1)
Phx(r)=ii[(liK~OJlfK){pKx(r ) + [kCA + 1)+/i(Ii + 1 ) - I f ( I f + 1)/~/2k(-h + 1)]pRx(r)} + (--1)li-K(Ii
- K)t2KIIfK)p2K(r)]
(2)
,
where I i = K = 7/2, and L, odd, = k -+ 1 (L, odd, = X) for transverse electric (magnetic) multipoles respectively. pR L are the multipoles of the collective current and pK(2K)are the multipoles of the SP current in the intrinsic state. The latter have been calculated from
pK L =
2K =(XKIP~.L "2K IX~-), Ph.L
(3)
using the Nilsson model for the unpaired proton wave function, XK, as in ref. [8]. Expanding this wave function in eigenstates o f J 2 one has
IXK) = ~ C/Inl/K) , I with n = O, l = 4, C7/2 = 0.993, C9/2 = 0.120 [9]. The SP current multipoles in eqs. (1), (2) are then •f K / f f pfk(r) = E C/iC/f(--ly - t -K
X~)
/iJf
2K
P~.L (r) -
E
liJf
(n/]f]J~k(r)]]nl]i)
0
(,f
• C/iC/f (-- l y f - ] i \.__
K
X
ilK)
2K -
(nl/f' I,oh L
'
(r)llnl/i) '
(4)
(5)
(6)
where the SP reduced matrix elements are (nyf! I/Sxx(r)lln//i) = + ix+l (eli/2mc)]f] i [(-1)x+K f/~/4~rk(k + 1)] (if ~/i -- i l•0) X {(gi + gf - ~k)(gi + gf + k + 1)/r + Up [XCA+ 1)/r + (gi + gf)(d/dr + 1/r)]}R2nl(r),
(7) (8)
with
D:(r) = - ( 1 / V ~ + i)(d/dr - ~/r), D~(r) = (1/V~)[d/dr + CA+ 1)/r] ,
(9)
x = (2/+ 1 ) ( / - / ) and/~p = 2.79. From eqs. (6) and (8) it can be seen that the only non-zero single particle (SP) transverse electric multipoles are those with X = 8 and these are proportional to C712C9/2. These multipoles give a contribution to the differential cross section for inelastic scattering that is negligible as compared to the contribution from the X = 7 multipole. In the numerical calculations a harmonic oscillator length parameter b = 2.2 fm has been used and standard center of mass and proton finite size corrections [3] have been taken into account. The collective multipoles for the rigid rotor (RR) and irrotational flow (IF) models are written as [5,6]:
(a ) Rigid rotor model R
iXrp~,(r)
PXh±l(r)=29RR ~ 2X
{ - ( x + 1)-112 } x_l/2 ,
pRx(r) = ( i X + l r ~ / 2 9 R R V ~ - k + 1) {(k/V~-k + 3)Px+l(r ) + [('h + 1)/2XVt2kZ-i-1]Ph_l(r)}. 236
(10) (11)
Volume 161B, number 4.5,6
PHYSICS LETTERS
31 October 1985
(b ) Irrotational flow model ,L=h+-1(r) = 8 L,h - 1(JAR0]2 c3IV)ah X/(2X + 1)] X(r/Ro)h- 1PF(r), p/h~hh(r)= [ih-lR0(2X + 1)]9IFVr~]
(12)
~a oth2phl(r)(r/Ro) h2-1(23,2 + 1)
hl ,h2
×~(X2+l)(2X2_l)(2Xl+1)(;2-1
X1 XI(X 1 0 01\0 where ph(r) are charge-density multipoles defined as
ph(r) = f d a Pd(r) yO(Q) ,
X2 -1
1
~k2
X
1 Xl}I '
(13)
(14)
with Pd(r) the intrinsic charge density of the deformed nucleus. For the numerical calculations we used the same parameterisation as in ref. [4]
Pd(r)=PF(r(l -
~
h=even ~2
axYO(~2)),
(15)
with OF(r) a spherically symmetric Fermi distribution. We take a m = 0t28h,2 so that P2(r) = --~2rOPF(r)[ar.Thus only the X = 1,2, 3 collective multipoles in eqs. (10)-(13) are different from zero. The monopole and quadrupole components of the charge density used here are as described in ref. [4]. In that reference the parameters c, a of the Fermi distribution were fitted to forward elastic scattering data,and ot2 was taken to fit the experimental quadrupole moment. We found that this model charge density satisfactorily reproduced the experimental data on inelastic scattering on 181Ta at 90 °. Actually, the model results are very similar to those of a density dependent Hartree-Fock (DDHF) calculation (see ref. [5]). The total differential cross section at 180 ° for elastic and inelastic scattering has been computed using our improved DWBA program based on eqs. (3)-(8) of ref. [4]. Results for the squared form factor, defined as IFI 2 = [4e2(1 + 2ei/Mtarget)/Z2et2] dO(el, 0 = rr)/dI2,
(16)
are presented in figs. 1--4, together with the experimental data of ref. [ 1] and with results of PWBA calculations. The results shown in fig. 1 are for inelastic scattering to the first excited state. They correspond to the use of the IF model for the collective current. In this figure, contributions from different multipoles in both DWBA and PWBA approximations are shown for comparison. At low q the difference between PWBA (E2) and DWBA (E2 + C2) is large. Because of this difference we have a factor of 3 enhancement for the DWBA result, compared to the PWBA calculation, at qeff = 0.5 f m - 1. DWBA and PWBA results are almost identical in the higher q region (q >~ 1.1 fm - 1 ) where SP magnetic multipoles are dominant. The latter are reduced by 20-30% when a HF wave function [3] is used for the odd particle, instead of the Nilsson model wave function used here, further improving the agreement with experiment. Total form factors for elastic and inelastic scattering corresponding to different models for the collective current are shown in figs. 2---4. Figs. 2 and 3 correspond to the IF and RR models, respectively, using the model moments of inertia [QIF (9]8~r)AMR2t~, QRR = ~AMR~(1 + ~ o t 2 ) ] . For elastic scattering there is no appreciable dependence on the collective model because of the dominance of the SP (magnetic) multipoles in the entire q region. An enhancement of the form factor at the M1 peak is found in the DWBA result as compared to the PWBA result. Comparison of the PWBA results for the 9]2 + and 1112+ transitions in figs. 2 and 3 shows a very large difference between IF and RR predictions at low q. This difference is somewhat attenuated when DWBA calculations are done, due to the C2 contribution. Nevertheless a sizeable difference still persists in the DWBA results corre=
237
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985
Irrotat~ onel Flo~
10 -3 10 -5
t0-'
E2 + C2
7/2 ÷
10 -~
(
X
10 ~)
9/2 +
!.,'2. 10 -5
10 -7
M7 i ~jt¢.
/
10-8 /i
t.~ i L
' I
10 -9 iI
10 -~
t "i
ii i t
i t i
/! / ;~
10 -7
i i i
O.
.5
1.0
1.5
qoff
i
2.0 (fro-')
2.5
3.0
10 -8 Fig. 1. Form factors for inelastic scattering to the 9/2 + state calculated with the IF model (see text). Individual mulfipoles are shown by dashed and dash-dotted lines for PWBAand DWBA respectively. Total form factors are shown by dotted and solid lines for PWBAand DWBArespectively.
10-' 11/2 + (
Fig. 2. Total form factors for elastic (7/2 +) and inelastic (9•2 + and 11/2 +) scattering corresponding to PWBA (dashed lines) and DWBA (solid lines) calculalions with the IF model.
10-'°o
.5
1.0
15
X
10-')
2.0
\
2.5
3.0
qoet (fro-')
sponding to the IF and RR models, especially so for the 11/2 + case where there is no M1 contribution. Finally fig. 4 shows the results o f the IF model with ~IF replaced by the experimental moment o f inertia 9ex p = [If(If + 1) - I i ( I i +
1)]/[2(Eif- E/i)] .
(17)
Similar DWBA results are obtained for the R R model when ~ R R is replaced by 9ex p. In either case the E2 amplitude is much smaller than the C2, which dominates the low q peak. This also applies to the DWBA results in fig. 3 resulting in very little difference between the DWBA form factors of figs. 3 and 4, despite noticeable differences between the corresponding PWBA form factors. In PWBA the PHF result [3] lies in between the two IF models o f figs. 2 and 4, and is almost the same order o f magnitude as C2 (see fig. 9 o f ref. [4] and dashed lines in figs. 2, 4). Hence, at low q, we expect DWBA form factors calculated with reliable microscopic models for the collective current to lie between the DWBA results of figs. 2 and 4. Since the main model dependence shows up in this q region, we may conclude by comparison o f the 238
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985 I r ~ o t a t i ona[ Flow
Rlgld Rotor
wlth experimental moment of Inert~a
10 -3
10 -3
10 -4
10-'
A /// '/'
10 -5
10 -~
10 -~
10 -~
10-'
10 -~
"~
7 / 2 ÷ ( X 10 ~)
]~
1 0 -8
10 -9
10 -~
11/2 ÷
@-,o
( X 10-')
\
i
,
i
i
,
.5
1.0
15
2.0
2.5
qefr (fm-')
Fig. 3. Same as fig. 2 with the RR model.
-' 11/2 ÷ ( X i0-') 3.0
i0-~o 0.
.5.
. 1.0.
.15 .
2.0 q .~f (fro-')
\
2.5
30
Fig. 4. Same as fig. 2 with 9IF replaced by 9exp (see text).
solidlines in figs. 2, 3 and 4 that there is still a quite sizeable model dependence for the transition to the 11/2 + state, in spite of the C2 contribution coming from Coulomb distortion. For the transition to the 9/2 + state we do not find detectable model dependence because of the dominant M1 single particle contribution. It is apparent that the DWBA results presented here agree much better with experimental data at low q than the PWBA results in ref. [1] for both elastic and inelastic scattering. For inelastic scattering the agreement is better for the IF model than for the RR model. At larger q values the dominant effect is due to the single particle current, and, as mentioned above, for the 7/2 + and 9/2 + transitions, the agreement with experimental data is improved using the single particle wave function o f DDHF calculations as in refs. [ 1,3]. 239
Volume 161B, number 4~5,6
PHYSICS LETTERS
31 October 1985
From the results obtained we can also draw the following general conclusions for electron scattering at 180 ° on heavy deformed nuclei. (1) Distortion effects lead to an enhancement o f the differential cross section at low q that cannot be accounted for by the standard procedure of replacing q by qeff in PWBA calculations. At larger q values (q ~ 1.1 fm - 1 ) this last procedure is reliable and it makes sense to use PWBA calculations for comparison with experimental data at larger momentum transfer. (2) The dependence on the model for the collective current shows up mainly in the differential cross section for inelastic scattering at low q, provided there is no dominant M1 SP contribution as is the case for the 11/2 + transition in 181Ta. (3) Clearer information on the collective current would be obtained by inelastic scattering to the ftrst excited states o f even--even rotational nuclei. Our resuits for 181Ta show that a large model dependence is to be expected for even--even rotors, even if the nuclear charge is large and DWBA calculations have to be done to treat C?~ and E~ multipoles on the same footing. We thank the Natural Sciences and Engineering Research Council of Canada for continued support under operating grant A-3198. E.M.G. is also indebted to the Comisi6n Asesora de Investigaci6n Cientifica y Tecnica for partial financial support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
240
F.N. Radet al., Phys. Rev. Lett. 45 (1980) 1758. R. Lindgren, private communication. E. Moya de Guerra and S. Kowalski, Phys. Rev. C22 (1980) 1308. M. Nishimura, E. Moya de Guerra and D.W.L. Spring, Nucl. Phys. A435 (1985) 523; see also P. Sarriguren and E. Moya de Guerra, Phys. Rev. C30 (1984) 2105. E. Moya de Guerra, to be published. E. Moya de Guerta, Ann. Phys. (NY) 118 (1980) 285. E. Moya de Guerta, Nucl. Phys. A366 (1981) 259; E. Wrist et al., Nucl. Phys. A402 (1983) 235. E. Moya de Guerra and A. Dieperink, Phys. Rev. C18 (1978) 1596. S.G. Nflsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) no. 16.
PHYSICS LETTERS
31 October 1985
E L E C T R O N S C A T T E R I N G A T 180 ° F R O M tStTa
M. N I S H I M U R A , D.W.L. S P R U N G Department of Physics, McMaster University, Hamilton, Ontario, Canada L8S 4M1
and E. M O Y A D E G U E R R A IEM, CSIC, Serrano 119, Madrid 28006, Spain and Departamento de Fisica Atomica y Nuclear, Universita Extremadura, Badajoz, Spain
Received 9 July 1985
Results of complete DWBA calculations of the differential cross section at 180° for elastic and inelastic electron scattering on 181Taare presented and compared with experimental data. The effects of the collective current and Coulomb distortion are investigated. It is shown that the Coulomb distortion must be properly taken into account to investigate low multipoles, and that experimental data in the low momentum transfer region may provide reliable information on the collective current.
In recent years some effort has been devoted to measuring transverse form factors of rare-earth rotational nuclei [1,2] as a means of obtaining information on the nature o f the collective rotational mode. Differential cross sections at 180 ° for elastic and inelastic scattering to the two first excited states of 181Ta were measured at the Bates Linear Accelerator Centre [ 1 ]. In ref. [ 1 ] the experimental data were compared to projected Hartree-Fock (PHF) calculations [3] of transverse form factors using the plane-wave Born approximation (PWBA) with the standard scale change o f q into qeff- Especially for inelastic scattering large disagreements were found between experiment and theory in the low~/region (q <~ 1.1 f m - 1 ). For this heavy rotational nucleus the l o w q region is the most sensitive to the actual form of the collective current, as well as to Coulomb distortion effects. In a previous paper [4] we calculated longitudinal contributions to the differential cross section at 180 ° on heavy deformed nuclei using the distorted-wave Born approximation (DWBA). We showed that at low q, the C2 contribution was as large or larger than the previously calculated transverse form factor [3] for the cases of inelastic scattering to the two first excited states of 181Ta, and hence should not be neglected. Since C2 and E2 multipoles interfere in DWBA, we concluded that complete DWBA calculations had to be done for meaningful comparison to experimental data. In this letter we present results of complete DWBA calculations using two extreme models - irrotational flow (IF) and rigid rotor (RR) - for the collective current. For 181Ta magnetic multipoles are dominated by the single particle (SP) contributions and are not very sensitive to the model used for the collective current. On the contrary transverse electric multipoles with ?~< 8 do not contain contributions from the SP current and are only sensitive to the collective rotational current~ It can be easily shown [5,6] that the RR and IF models provide, respectively, a lower and an upper bound on the E2 multipole when the model dependence of the moment of inertia is used. Predictions o f microscopic models (such as the cranking and PHF models) lie in general between these limits [7]. Hence, to investigate the dependence on the collective current it is sufficient here to focus on the two models mentioned above. The transverse electric and magnetic multipoles PXL of the nuclear transition current are given by [5,6] 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
235
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985
[i = (2I + 1)1/21
Pxx+l(r)='Ii{(IiKkOIIfK)[If(Zf + 1)-Ii(Ii. + 1)]pRh±l(r) + (--1)/i-K(/i -ICA2KIIfK)p2K±I(r)},
(1)
Phx(r)=ii[(liK~OJlfK){pKx(r ) + [kCA + 1)+/i(Ii + 1 ) - I f ( I f + 1)/~/2k(-h + 1)]pRx(r)} + (--1)li-K(Ii
- K)t2KIIfK)p2K(r)]
(2)
,
where I i = K = 7/2, and L, odd, = k -+ 1 (L, odd, = X) for transverse electric (magnetic) multipoles respectively. pR L are the multipoles of the collective current and pK(2K)are the multipoles of the SP current in the intrinsic state. The latter have been calculated from
pK L =
2K =(XKIP~.L "2K IX~-), Ph.L
(3)
using the Nilsson model for the unpaired proton wave function, XK, as in ref. [8]. Expanding this wave function in eigenstates o f J 2 one has
IXK) = ~ C/Inl/K) , I with n = O, l = 4, C7/2 = 0.993, C9/2 = 0.120 [9]. The SP current multipoles in eqs. (1), (2) are then •f K / f f pfk(r) = E C/iC/f(--ly - t -K
X~)
/iJf
2K
P~.L (r) -
E
liJf
(n/]f]J~k(r)]]nl]i)
0
(,f
• C/iC/f (-- l y f - ] i \.__
K
X
ilK)
2K -
(nl/f' I,oh L
'
(r)llnl/i) '
(4)
(5)
(6)
where the SP reduced matrix elements are (nyf! I/Sxx(r)lln//i) = + ix+l (eli/2mc)]f] i [(-1)x+K f/~/4~rk(k + 1)] (if ~/i -- i l•0) X {(gi + gf - ~k)(gi + gf + k + 1)/r + Up [XCA+ 1)/r + (gi + gf)(d/dr + 1/r)]}R2nl(r),
(7) (8)
with
D:(r) = - ( 1 / V ~ + i)(d/dr - ~/r), D~(r) = (1/V~)[d/dr + CA+ 1)/r] ,
(9)
x = (2/+ 1 ) ( / - / ) and/~p = 2.79. From eqs. (6) and (8) it can be seen that the only non-zero single particle (SP) transverse electric multipoles are those with X = 8 and these are proportional to C712C9/2. These multipoles give a contribution to the differential cross section for inelastic scattering that is negligible as compared to the contribution from the X = 7 multipole. In the numerical calculations a harmonic oscillator length parameter b = 2.2 fm has been used and standard center of mass and proton finite size corrections [3] have been taken into account. The collective multipoles for the rigid rotor (RR) and irrotational flow (IF) models are written as [5,6]:
(a ) Rigid rotor model R
iXrp~,(r)
PXh±l(r)=29RR ~ 2X
{ - ( x + 1)-112 } x_l/2 ,
pRx(r) = ( i X + l r ~ / 2 9 R R V ~ - k + 1) {(k/V~-k + 3)Px+l(r ) + [('h + 1)/2XVt2kZ-i-1]Ph_l(r)}. 236
(10) (11)
Volume 161B, number 4.5,6
PHYSICS LETTERS
31 October 1985
(b ) Irrotational flow model ,L=h+-1(r) = 8 L,h - 1(JAR0]2 c3IV)ah X/(2X + 1)] X(r/Ro)h- 1PF(r), p/h~hh(r)= [ih-lR0(2X + 1)]9IFVr~]
(12)
~a oth2phl(r)(r/Ro) h2-1(23,2 + 1)
hl ,h2
×~(X2+l)(2X2_l)(2Xl+1)(;2-1
X1 XI(X 1 0 01\0 where ph(r) are charge-density multipoles defined as
ph(r) = f d a Pd(r) yO(Q) ,
X2 -1
1
~k2
X
1 Xl}I '
(13)
(14)
with Pd(r) the intrinsic charge density of the deformed nucleus. For the numerical calculations we used the same parameterisation as in ref. [4]
Pd(r)=PF(r(l -
~
h=even ~2
axYO(~2)),
(15)
with OF(r) a spherically symmetric Fermi distribution. We take a m = 0t28h,2 so that P2(r) = --~2rOPF(r)[ar.Thus only the X = 1,2, 3 collective multipoles in eqs. (10)-(13) are different from zero. The monopole and quadrupole components of the charge density used here are as described in ref. [4]. In that reference the parameters c, a of the Fermi distribution were fitted to forward elastic scattering data,and ot2 was taken to fit the experimental quadrupole moment. We found that this model charge density satisfactorily reproduced the experimental data on inelastic scattering on 181Ta at 90 °. Actually, the model results are very similar to those of a density dependent Hartree-Fock (DDHF) calculation (see ref. [5]). The total differential cross section at 180 ° for elastic and inelastic scattering has been computed using our improved DWBA program based on eqs. (3)-(8) of ref. [4]. Results for the squared form factor, defined as IFI 2 = [4e2(1 + 2ei/Mtarget)/Z2et2] dO(el, 0 = rr)/dI2,
(16)
are presented in figs. 1--4, together with the experimental data of ref. [ 1] and with results of PWBA calculations. The results shown in fig. 1 are for inelastic scattering to the first excited state. They correspond to the use of the IF model for the collective current. In this figure, contributions from different multipoles in both DWBA and PWBA approximations are shown for comparison. At low q the difference between PWBA (E2) and DWBA (E2 + C2) is large. Because of this difference we have a factor of 3 enhancement for the DWBA result, compared to the PWBA calculation, at qeff = 0.5 f m - 1. DWBA and PWBA results are almost identical in the higher q region (q >~ 1.1 fm - 1 ) where SP magnetic multipoles are dominant. The latter are reduced by 20-30% when a HF wave function [3] is used for the odd particle, instead of the Nilsson model wave function used here, further improving the agreement with experiment. Total form factors for elastic and inelastic scattering corresponding to different models for the collective current are shown in figs. 2---4. Figs. 2 and 3 correspond to the IF and RR models, respectively, using the model moments of inertia [QIF (9]8~r)AMR2t~, QRR = ~AMR~(1 + ~ o t 2 ) ] . For elastic scattering there is no appreciable dependence on the collective model because of the dominance of the SP (magnetic) multipoles in the entire q region. An enhancement of the form factor at the M1 peak is found in the DWBA result as compared to the PWBA result. Comparison of the PWBA results for the 9]2 + and 1112+ transitions in figs. 2 and 3 shows a very large difference between IF and RR predictions at low q. This difference is somewhat attenuated when DWBA calculations are done, due to the C2 contribution. Nevertheless a sizeable difference still persists in the DWBA results corre=
237
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985
Irrotat~ onel Flo~
10 -3 10 -5
t0-'
E2 + C2
7/2 ÷
10 -~
(
X
10 ~)
9/2 +
!.,'2. 10 -5
10 -7
M7 i ~jt¢.
/
10-8 /i
t.~ i L
' I
10 -9 iI
10 -~
t "i
ii i t
i t i
/! / ;~
10 -7
i i i
O.
.5
1.0
1.5
qoff
i
2.0 (fro-')
2.5
3.0
10 -8 Fig. 1. Form factors for inelastic scattering to the 9/2 + state calculated with the IF model (see text). Individual mulfipoles are shown by dashed and dash-dotted lines for PWBAand DWBA respectively. Total form factors are shown by dotted and solid lines for PWBAand DWBArespectively.
10-' 11/2 + (
Fig. 2. Total form factors for elastic (7/2 +) and inelastic (9•2 + and 11/2 +) scattering corresponding to PWBA (dashed lines) and DWBA (solid lines) calculalions with the IF model.
10-'°o
.5
1.0
15
X
10-')
2.0
\
2.5
3.0
qoet (fro-')
sponding to the IF and RR models, especially so for the 11/2 + case where there is no M1 contribution. Finally fig. 4 shows the results o f the IF model with ~IF replaced by the experimental moment o f inertia 9ex p = [If(If + 1) - I i ( I i +
1)]/[2(Eif- E/i)] .
(17)
Similar DWBA results are obtained for the R R model when ~ R R is replaced by 9ex p. In either case the E2 amplitude is much smaller than the C2, which dominates the low q peak. This also applies to the DWBA results in fig. 3 resulting in very little difference between the DWBA form factors of figs. 3 and 4, despite noticeable differences between the corresponding PWBA form factors. In PWBA the PHF result [3] lies in between the two IF models o f figs. 2 and 4, and is almost the same order o f magnitude as C2 (see fig. 9 o f ref. [4] and dashed lines in figs. 2, 4). Hence, at low q, we expect DWBA form factors calculated with reliable microscopic models for the collective current to lie between the DWBA results of figs. 2 and 4. Since the main model dependence shows up in this q region, we may conclude by comparison o f the 238
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985 I r ~ o t a t i ona[ Flow
Rlgld Rotor
wlth experimental moment of Inert~a
10 -3
10 -3
10 -4
10-'
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i
,
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i
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.5
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-' 11/2 ÷ ( X i0-') 3.0
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.5.
. 1.0.
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30
Fig. 4. Same as fig. 2 with 9IF replaced by 9exp (see text).
solidlines in figs. 2, 3 and 4 that there is still a quite sizeable model dependence for the transition to the 11/2 + state, in spite of the C2 contribution coming from Coulomb distortion. For the transition to the 9/2 + state we do not find detectable model dependence because of the dominant M1 single particle contribution. It is apparent that the DWBA results presented here agree much better with experimental data at low q than the PWBA results in ref. [1] for both elastic and inelastic scattering. For inelastic scattering the agreement is better for the IF model than for the RR model. At larger q values the dominant effect is due to the single particle current, and, as mentioned above, for the 7/2 + and 9/2 + transitions, the agreement with experimental data is improved using the single particle wave function o f DDHF calculations as in refs. [ 1,3]. 239
Volume 161B, number 4~5,6
PHYSICS LETTERS
31 October 1985
From the results obtained we can also draw the following general conclusions for electron scattering at 180 ° on heavy deformed nuclei. (1) Distortion effects lead to an enhancement o f the differential cross section at low q that cannot be accounted for by the standard procedure of replacing q by qeff in PWBA calculations. At larger q values (q ~ 1.1 fm - 1 ) this last procedure is reliable and it makes sense to use PWBA calculations for comparison with experimental data at larger momentum transfer. (2) The dependence on the model for the collective current shows up mainly in the differential cross section for inelastic scattering at low q, provided there is no dominant M1 SP contribution as is the case for the 11/2 + transition in 181Ta. (3) Clearer information on the collective current would be obtained by inelastic scattering to the ftrst excited states o f even--even rotational nuclei. Our resuits for 181Ta show that a large model dependence is to be expected for even--even rotors, even if the nuclear charge is large and DWBA calculations have to be done to treat C?~ and E~ multipoles on the same footing. We thank the Natural Sciences and Engineering Research Council of Canada for continued support under operating grant A-3198. E.M.G. is also indebted to the Comisi6n Asesora de Investigaci6n Cientifica y Tecnica for partial financial support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
240
F.N. Radet al., Phys. Rev. Lett. 45 (1980) 1758. R. Lindgren, private communication. E. Moya de Guerra and S. Kowalski, Phys. Rev. C22 (1980) 1308. M. Nishimura, E. Moya de Guerra and D.W.L. Spring, Nucl. Phys. A435 (1985) 523; see also P. Sarriguren and E. Moya de Guerra, Phys. Rev. C30 (1984) 2105. E. Moya de Guerra, to be published. E. Moya de Guerta, Ann. Phys. (NY) 118 (1980) 285. E. Moya de Guerta, Nucl. Phys. A366 (1981) 259; E. Wrist et al., Nucl. Phys. A402 (1983) 235. E. Moya de Guerra and A. Dieperink, Phys. Rev. C18 (1978) 1596. S.G. Nflsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) no. 16.