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Spectroscopy of even tin isotopes by inelastic scattering of 24.5 MeV protons
Nuclear Physics A147 (1970) 326--368; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
SPECTROSCOPY
OF EVEN TIN ISOTOPES
BY I N E L A S T I C S C A T T E R I N G O F 24.5 M e V P R O T O N S O. BEER t, A. EL BEHAY tt, p. LOPATO, Y. TERRIEN, G. VALLOIS and KAMAL K. SETH ttt Service de Physique Nucldaire ~ Moyenne Eneryie, Centre d'Etudes Nucldaires de Saclay, France
Received 16 January 1970 Abstract: Elastic and inelastic scattering of 24.5 MeV protons by even isotopes of tin, i16, lxs, 12OSn and 122,124Sn' has been studied with an energy resolution better than 20 keV. Approximately 30 levels in each isotope have been resolved up to an excitation energy of 4 MeV. Elastic scattering angular distributions have been analysed to yield optical-model parameters and the inelastic scattering angular distributions have been analysed in the distorted wave Born approximation (DWBA) using collective form factors to determine l and fl~ corresponding to a large number of the transitions observed. The systematics of the states of each j~r are discussed and the results are compared to recent theoretical calculations. E I
I
NUCLEAR REACTIONS i16, llS. 12o.122,124Sn(p' p,), Ep = 24.5 MeV; measured tr(Ep,, 0). 116.11s, 120. Iz2. 124Sn deduced J, :r, fit. Enriched targets.
1. Introduction The a p p l i c a t i o n o f the techniques o f the BCS t h e o r y o f s u p e r c o n d u c t i v i t y l ) to nuclear physics p r o b l e m s z) has p r o v i d e d a p o w e r f u l t o o l for the study o f structure o f nuclei in which a large n u m b e r o f valence nucleons o c c u p y several shell-model orbits a n d for which the exact t r e a t m e n t o f the extra-core particles with residual forces is prohibitive, if n o t impossible. These techniques can be best tested in single closed shell nuclei. The tin isotopes, extending f r o m 1 lZSn to 124Sn offer a n excellent o p p o r tunity for detailed theoretical a n d e x p e r i m e n t a l investigations. The Z = 50 p r o t o n shell is closed a n d it has been experimentally shown t h a t a p p r e c i a b l e p r o t o n particlehole excitations d o n o t start until E* ~ 4 M e V [refs. 3, 4)]. T h u s below this energy one can assume t h a t states are f o r m e d only by excitations o f n e u t r o n s in the orbits 2d~, lg~, 3s½, 2d~ a n d lh~_. W i t h i n the f r a m e w o r k o f the BCS theory, Kisslinger a n d Sorensen 5) m a d e the initial simplifying a p p r o x i m a t i o n o f a p a i r i n g - p l u s - q u a d r u p o l e residual interaction. T h e y calculated p r o p o r t i e s o f low-lying states o f several single closed shell nuclei, including isotopes o f tin. Since then several theoretical a t t e m p t s have been m a d e to predict m o r e detailed p r o p e r t i e s o f tin isotopes. A r v i e u a n d colt Supported in part by Bundesministerium ftir wissenschaftliche Forschung. Present address: ISKP, Universit/it Bonn. tt Present address: Atomic Energy Establishment, Le Caire, UAR, Egypt. ttt Present and permanent address: Northwestern University, Evanston, Illinois, USA. 326
EVEN T I N ISOTOPES
327
laborators 6) and Kuo and Baranger 7) have used more conventional types of residual two quasiparticle interactions, both in Tamm-Dancoff approximation 6, 7) (TDA) and the random phase appioximation (RPA)6,7). More recently, Arvieu et al.'s work on even-A isotopes has been extended by Sawicki and collaborators s) to include four quasiparticle excitations (which would correspond to two-phonon excitations) using phenomenological 8) as well as realistic interactions 9) and by Clement and Baranger 10) to the explicit consideration of core excitations with the use of a realistic interaction. Thus considerable interest exists in the spectroscopy of tin nuclei. The existing information about even tin isotopes comes from a variety of experiments, but unfortunately most of them have investigated only a few low-lying excited states. This limitation arises partly from the nature of the decays and partly because of inadequate resolution in face of rapidly increasing level density, fl-decay experiments [refs. 1l-15)] from radioactive In and Sb nuclei and 7-ray correlations have provided m u c h of the detailed information on many low-lying states of 1~6,1t8, ~20Sn" Some additional information on these isotopes has been obtained by stripping x6, ~7), pickup 18), and two-neutron transfer reactions 19,20), Since the tin nuclei exhibit important collective properties, Coulomb excitation [refs. 21-z4)] and the inelastic scattering of electrons 2s), protons t8,26-31), deuterons 3z-34), alphas 35,36) and neutrons 37) provide powerful tools for the investigation of their structure. However, most of the existing experimental studies are also confined to detailed investigations of a few low-lying states. It was therefore felt that a systematic high resolution study of the even tin isotopes would be of considerable interest. In this paper we present the results of inelastic scattering of 24.5 MeV protons by the isotopes 116,118,12°,122,124Sn.
2. Experiment The experiments were done with the 24.5 MeV proton beam provided by the Saclay variable energy cyclotron. Fig. 1 shows the plan of the experimental arrangement. The extracted proton beam is focused on the entrance slit 0, and then analysed by a 60 ° magnet. The spread in energy of the beam causes it to be distributed along the plane LH. The quadrupole doublet forms an image L'H' on the target. The scattered protons are analysed by a 180 ° magnet with a spark chamber in its focal plane. In order to use allthe intensity of the beam without loss of energy resolution we employed the method of compensation of magnetic dispersions as described by Cohen 38). In this technique the spatial dispersion of a group of scattered protons in the focal plane of the 180 ° magnet depends (in first approximation) only on the width of the beam at the entrance slit O and is independent of the beam energy spread at the target. In other words the magnetic system is achromatic. As used in the present experiment the "spatial dispersion" was approximately 1.2 mm/10 keV for a central ray radius of 720 mm and a 30 ° inclined focal plane. The spark chamber has been described elsewhere in detail 39). It is triggered by a scintillation counter in a parallel position behind it, permitting the selection of pro-
328
o. BEER e t al.
tons, deuterons and tritons. The spark localization is obtained by the current division method due to Charpak et al. 40). In the present experiment further improvement in spark localization (i.e., better resolution) was achieved by dividing the position signal on-line (using a PDP-8 computer) by the sum signal. This modification also made the resolution independent of counting rates up to 200 cts/sec, which were frequently employed. For most spectra however we were limited b y beam intensity rather than by Cyclotron Anatysis Magnet (1OO*)
.,
Secondary
~
/ ,'
Target
(~'~"'~F
,
,,/
Hj
L .................. o - o
,'
................... ,.................
H
Spark Chamber and Scintittator
Ouadrupotes
Primary Anatysis Magnet (60*)
Fig. I. Experimental a r r a n g e m e n t . &
Sn 118(D.p'; E= 24.5 MeV
tm~ t'~ to •
'
x-~
= 35"
O
i
tD
, ~t~
-.r t-. -.r
~
tn ¢'4
o
,
I
4~)
,"
,
'
'
,
3.0
( - -
2~0
J
"-
1.23 Ex(MeV)
Fig. 2. A typical energy spectrum. T h e s p e c t r u m is for z l a s n ( p , p ' ) at 0 = 35 °. On-line division o f the position signal with the s u m signal was used. T h e energy resolution is a b o u t 15 keV. T h e n u m b e r s on p e a k s refer to the level n u m b e r s in table 5.
the spark chamber capability. The overall energy resolution was approximately 15 keV with on-line division and 25 keV without division, and depended also on target thickness, and angle of observation. Fig. 2 shows a typical spectrum in the case of 11aSh for which on-line division was utilized. The incident beam energy, 24.55 MeV was known only with a 50 keV accuracy, and the exact scattering angle only to within +_0.2 °. If the focal plane calibration were
EVENTIN ISOTOPES
329
independently and perfectly known, these uncertainties would lead to a relative uncertainty of _ 15 keV in the determination of the Q-value for inelastic scattering from a 4 MeV excited state in a Sn isotope. However, by calibrating the system (with the same beam and the same magnetic field) against elastic and inelastic scattering from 12C, which was present as a target impurity, one can achieve a calibration which is nearly independent of these uncertainties in the beam energy and the scattering angle. This method was used in determining the excitation energies from the present experiments. The absolute error for the excitation energies is estimated to be smaller than _ 10 keV. In the case of multiplets or weakly excited peaks the precision is not expected to be so good. TABLE 1 Isotopic composition (in ~ ) of targets used
116Sn aleSn 12°Sn 122Sn
llsS n
x16Sn
ttvSn
11SSn
0.12
92.64 0.37 0.16 0.40
2.87 0.42 0.10 0.38
0.43 33.3 32.92 33.1
0.30 0.55 0.54 0.62
0.02 0.27
t24Sn mixture 1 mixture 2 mixture 3
0.02
ltgSn
t20Sn
2.34 96.62 0.44 0.87
0.42 0.82 0.65 0.75
0.99 1.58 98.21 6.18
1.17 32.69 32.23 32.48
0.40 0.50 0.58 0.50
1.75 32.65 2.43 1.46
124Sn
Thickness (~g/cm2)
0.10 0.13 0.28 88.92
0.42 0.06 0.14 2.05
800 500 1000 1000
1.21 0.14 30.80 0.49
94.74 0.12 0.47 31.3
1000 500 500 700
~22Sn
The isotopically enriched targets were in the form of self-supporting foils. No attempt was made to estimate target thicknesses accurately. These thicknesses were however known to within ___30 ~o from the target evaporation conditions and weighting. Table 1 shows the target compositions. In order to measure accurately the variation of parameters/~2 and ]~3 in the Sn isotopes we also used three samples of mixtures of isotopes. Their constitution is indicated in the last three lines of table 1. For each isotope the elastic angular distribution was measured from 15 ° to 130° in steps of 5 °, and near maxima or minima in steps of 2.5 °. Inelastic angular distributions were measured from 20 ° to 120 ° in steps of 5 °. These distributions are shown in figs. 4-11.
3. Analysis of data 3.1. OPTICAL-MODEL ANALYSIS OF ELASTIC SCATTERING
Optical-model parameters are required for analysis of the inelastic scattering data by the DWBA method. Also since target thicknesses were known only to within + 30 ~ , optical-model analysis of elastic scattering was necessary to provide a more
330
o. BEERet aL
accurate absolute normalization of the cross sections. As is well known the optical potential has the form
V(r)=
Vc(rc)-V(eX+l)-l-iEW~-4Waa' drrd](e.,+ 1)_1 +(h/m~c) 2 V~o 1 d (e ..... + 1 ) - ~ . 1 ,
'rVr
where x = ( r - Ro)/ao, x' = ( r - R')/a', x .... = ( r - Rs.o.)/a .... with Ro = ro A~, e t c . . . , and the Coulomb potential Vc is that due for a uniformly charged sphere of radius Rc = rc A~. The optical-model parameters search was done using a modified version of the search code JIB 41). The program searches for the optimum parameters of the optical potential and the normalization constant K which minimize the quantity:
1 N [ o.,(o3-XOox.(O3 2. Several analyses of proton scattering from tin nuclei 30,31,42, 43) and several systematic studies over a wide range of nuclei 44-46) are available in the literature. We investigated several of these geometries. Two of these, the first due to J'arvis 42), who analysed 17.5 MeV elastic scattering only, and the second due to Satchler 43), who analysed elastic scattering 47), polarization 48), and total absorption cross section 49) at about 30 MeV, gave consistently better fits and lower Z2. The parameters based on Jarvis' geometry gave a factor two poorer Z 2 and led to a total absorption cross section of about 1433 mb for 12°Sn, which is about 11% lower than the experimental value of 1606_+80 mb obtained by extrapolating from the 28.5 MeV measurement [ref. 49)]. Satchler's geometry presented no such problems. We have therefore only investigated this geometry in detail. In the following we describe the results of our investigations starting from Satchler's average parameter set. This set has identical geometry for all real potentials and identical geometry for the imaginary potential but differs from all the previously consideredpotentials in having a finite volume absorption, Ws. We examined this aspect of Satchler's potentials first. In table 2 we show the result of keeping all parameters fixed and searching on V and Wo for a specified value of Ws. As illustrated in this table, in all cases the reaction cross section is now in excellent agreement with the measured value. However, the Z2 uniformly decreases for all isotopes as W~ goes to zero. The trend is very marked and appears quite convincing. Incidentally, it may be noted from table 2 that the quantity Wd+ 1.15 W~ remains constant to within 1% throughout all the best fits in table 2. On the basis of the results in table 2, it was decided to set W~ = 0, and search for optimum values of the geometry parameters. It was found that ro = rs.o. = 1.13 fm improves Z2 somewhat and when a search on V, Wd and a0 is made an interesting trend of increasing diffuseness with ( N - Z ) / A results. This trend is illustrated in
331
EVEN TIN ISOTOPES
TABLE 2 Optical-model p a r a m e t e r s search with Satchler's 4a) average geometry: rc -----1.20 fro, ro = rs.o. = 1.12 fm, ao = a,.o. = 0.75 fm, r ' = 1.33 fm, a' = 0.65 fro, Vs.o. = 6.1 M e V Isotope
116
118
120
122
124
W~ = 3.0 M e V V(MeV) Wa(MeV) ar (mb) Z2
58.70 7.51 1541 14.2
58.77 7.45 1556 8.5
V(MeV) Wa(MeV) cq(mb) g2
57.99 9.31 1558 5.9
58.04 9.29 1574 4.2
59.31 7.90 1586 5.9
59.54 7.88 1601 4.4
59.50 8.88 1641 10.0
58.73 9.57 1600 2.2
58.97 9.58 1616 1.15
59.01 10.47 1653 4.9
58.55 10.21 1604 1.55
58.79 10.15 1621 0.7
58.87 11.01 1656 4.1
58.37 10.69 1608 1.05
58.61 10.73 1625 0.4
58.75 11.52 1659 3.3
58.19 11.27 1611 0.85
58.44 11.30 1628 0.33
58.56 12.09 1662 2.8
W~ = 1.5 M e V
= 1.0 M e V V(MeV) Wa(MeV) ~rr(mb) Z2
57.76 9.90 1563 4.2
57.81 9.91 1579 3.3
V(MeV) Wa(MeV) tr, (nab)
57.55 10.51 1567 2.9
57.57 10.52 1584 2.7
W~ ~ 0.5 M e V
7,z
W, = O
V(MeV) Wa(MeV) tr,(mb) Zz
57.34 11.11 1571 1.9
57.35 11.13 1588 2.3 TABLE 3
(a) Best fit optical-model p a r a m e t e r s for Wsaz = 0, rc = 1.20 fro, ro = r~.o. = 1.13 fm, r ' ~ 1.33 fm, a ' = 0.65 fro, V,.o. = 6.1 MeV, a n d a~.o. = 0.75 fro. (b) S a m e p a r a m e t e r s as (a) except t h a t the v a r i a t i o n o f ao with N - Z was p a r a m e t e r i z e d as described in the text (a)
V Wa
ao aa Z2
11eSn
1laSh
57.05 10.64 0.715 1545 1.3
56.05 10.74 0.723 1568 2.2
120Sn 57.43 11.16 0.744 1609 0.8
122Sn
124Sn
57.61 11.26 0.749 1630 0.4
57.33 11.53 0.787 1695 2.5
57.35 11.26 1631 0.5
57.36 12.21 1675 2.4
(b) ao = 0 . 6 + 0 . 8 3 ( N - Z ) / A
V W'd era g2
56.87 10.63 1545 1.3
56.69 10.78 1571 2.3
57.27 11.08 1605 0.8
332
o. BEER et al.
table 3 and can be parameterized as ao = 0.60+ 0.83(N-Z)/A. The fits corresponding to this last set of parameters are shown in fig. 3. The agreement between the optical-model predictions and the experimental cross sections is generally excellent. This is also indicated in table 3 by the small values of Z2 for these fits. Our final param-
O-et]o'-c 17 ~ .
~ "+
t
116Sn
/'
..
+
,++
\j \+:,,
I ,'~,~j '~ ...x x
o .........
sio.........
;~ .........
'~o
Fig. 3. Elastic scattering fits for final optical-model parameters o f table 3(b) (elastic cross sections divided by Rutherford cross sections).
eters in table 3 are quite similar to those obtained by Makofske et al., in a reanalysis of their data a x). Our imaginary potential depths are in most cases identical to theirs, but our real potential well depths show somewhat less variation than theirs. This is no doubt due to the fact that we have put a large part of this dependence in the variation of a0 and they have kept it constant. This is in accordance with the ideas of Green-
EVEN TIN ISOTOPES
333
lees et al. 5 o), who stress the fact that the volume integral of the real Saxon potential J J A -- V A - 1 ~4 R o 3 [1 +~z2a2o/R2o]
is a constant equal to about 400 MeV" fm 3. The parameters of set 2 with variable " a " in table 3 give Jr/A = 400.6, 400.6, 406.0, 407.7, 409.0 MeV • fm a for the five tin isotopes. Both the absolute values and the trend is in excellent agreement with Greenlees' analysis so) of 30 MeV elastic scattering data with J / A = 4 1 2 + 2 2 MeV" fm 3 and 40 MeV data with Jr/A = 380+20 MeV" fm 3. 3.2. C R O S S - S E C T I O N N O R M A L I Z A T I O N
As mentioned earlier, sample thicknesses were only known to within -t-30 ~ . Accurate cross-section normalizations were therefore obtained from the optical-model fits to the elastic scattering cross sections. The reliability of these optical-model normalizations was tested as follows. F r o m our experiments with targets of accurately known isotopic mixtures, the ratios of inelastic cross-sections for the first 2 + states were determined to within +__5 ~ . By comparing these ratios with the relative differential cross sections for the same 2 + states obtained for the separated isotope targets, we could obtain ratios of the thicknesses of the separated isotope targets. This gave us the ratios of elastic scattering cross sections to within + 5 ~ . The same ratios could also be obtained from the independently normalized optical-model elastic scattering fits. It is gratifying to note that the ratios obtained in the two ways were identical within _ 5 ~ , i.e., the independent optical-model fits (with a given geometry) accurately reproduced the known ratios of elastic scattering cross sections. This gives us confidence that the absolute cross section normalization given by these optical-model fits is accurate to within +__15 ~ . It is possible, though we consider it highly improbable, that for all isotopes our seach program for the optical-model parameters has "locked in" on a non-physical X2 minimum and thus given us a uniformly wrong normalization *. In this case our absolute cross sections may have a higher error, but none of our conclusions regarding the variation of transition strengths from isotope to isotope will be altered. 3.3. D W B A A N A L Y S I S O F I N E L A S T I C S C A T T E R I N G
Differential cross sections were obtained for the inelastic scattering, except for the weakest levels or the very poorly resolved levels. These are shown in figs. 4 through 11. Integrated inelastic scattering cross sections were obtained as f130 °
O'tota1
=
27~
130 °
dO sin O[d~r(O)/da] ~ ~sTC2 Z 20 °
sin 0(da(0)/dO)
20 °
and these are indicated in tables 4-8. The true integrated cross sections from 0 = 0 ° to 180 ° are expected to be approximately l0 ~ larger than these. • Recently ( O a k Ridge N a t i o n a l L a b o r a t o r y R e p o r t , O R N L - 4 2 5 2 , 1968, u n p u b l i s h e d ) , Bertrand, Love, Baron, Percy a n d Dickens have reported elastic scattering cross sections for 20.6, 25.2, 30.6 a n d 36.2 M e V p r o t o n s on 12°Sn with 10 ~o absolute errors. O u r cross sections at 24.5 M e V are unif o r m l y ~ 20 ~ higher t h a n their cross sections at 25.2 MeV.
o. ~B~ERe t al.
334
(RP') Sn 116, Ep = 24,5 MeV '
~t~
1.29 MeV
23 ov
,
2.27 MeV
I
10; ~ /jJlriJ21111J
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, ~
2.64 MeV
10;
1021
2.79
,~
1Z,+1,(5-1 .03 MeV
MeV
I
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,
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(2+1 3.21MeV
2+
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3.40 MeV
/(~
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102
\ i lol
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(t, +) 3.50+3.51MeV
2+
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1~
i
: I [ I I I I f 1 I 1 I I 50 100
I
r
i
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sO
t
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I
f
loo
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e*cM Fig. 4. Inelastic scattering differential cross sections for states (with 1 ~ 5) in 116Sn. T h e experim e n t a l points are s h o w n with their error bars. T h e solid curves are D W B A predictions for final optical-model p a r a m e t e r s o f table 3(b) a n d G~ listed in table 4.
EVEN TIN I$oTOVES
335
(PP') Sn 118 . Ep = 24.5 MeV
~#~k2.2/'9+MeV
2+
~,
1.23 MW
3-+6"
2.33 MeV ~ ,
I l l I I I I l I l l t l l j l l l l i i l
,.~, T I J I I I I T J l l l l l J l [ l l i l l
2
U MeV
10-"
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10:
(,,. +)
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3.79
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1
50
[
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r
MeV
I
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t
100
0 CM o
Fig. 5. Inelastic scattering differential cross sections for states (with 1 ~ 5) in 11SSn. The experimental points are shown with their error bars. The solid curves are D W B A predictions for final optical-model parameters of table 3(b) and G, listed in table 5.
336
o. BEER et al. ( P P'} $n 120,Ep= 2/,.5 MeV
/\
2.
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1.17MeV
10:
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½
52.29 MeV
10 i
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eOcM Fig. 6. Inelastic scattering differential cross sections for states (with l _~ 5) in x2°Sn. The experimental points are shown with their error bars. The solid curves are D W B A predictions for final optical-model parameters of table 3(b) a n d G~ listed in table 6.
EVENTIN ISOTOPES
337
PP') Sn 122,Ep= 2/,.5 MeV
#
I i
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3.31MeV
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3.90
MeV
t
t r l t ~ l t l l l l l
(5-) 102
'.k+..,.
'+,
..........
Mixtureof t'~v~"~..~. exp.distributions
I
I
I
I
50
[
t
I
I
I
I(30
I
I
i
i
1
i
1
i
''~.~, i
50
I
t
I
i
I
I 1 T t l t l l I I 1 1
10(3
e~M Fig. 7. Inelastic scattering differential cross sections for states (with l ~ 5) in 122Sn. The experimental points are shown with their error bars. The solid curves are DWBA predictions for final optical-model parameters of table 3(b) and Gz listed in table 7.
338
o. ~E~ et aL
( P P ' ) Sn 124.Ep= 24.5 MeV
I 10~
~t.+-+-.+,++i J
i
~
[
i
i
~
/~
i
I
I ~°~ i
t
r
,
I
J
'
'
2+
/f
33.01 MeV
~
i
I I
r i
'
I
I I
!
i
i
I
I
I
2.71 M~
I II II II IJ I l Il ll il lt ll ll
i
I.+ 3.16 MeV
I ~
i I
I I
t I
I I
3.23 MeV
~ i
i I
l l l i l l i i 33.52 M~
4+
3.37 MeV
(4+1
~ tO~
.~
I i i l a i l l/,+l r I
i
'
32.61 MeV
*'~
i
~
" ~ t
3.42 MeV
i
llrliifIlii
i
i
I
t
50
i
i
I
I
I
100
I
I Ii It ll l l l i l l l l l i Jl ll ii l
i
I
f
I
J
,
I
50 e o ¢M
I
t
I
100
i
I
IIII
50
100
Fig. 8. Inelastic scattering differential cross sections for states (with l --< 5) in 124Sn. The experimental points are shown with their error bars. The solid curves are DWBA predictions for final optical-model parameters o f table 3(b) and G, listed in table 8.
EVEN T I N ISOTOPES
339
There is a large body of evidence which points to the fact that the main part of medium-energy inelastic scattering, at least for the strong transitions, comes from a direct interaction. One may therefore use DWBA for analysing inelastic scattering angular distributions as one does for particle transfer reactions except that the form factors in this case may be based on the collective model. In the collective model one attributes the low-lying states to rotations of a statically deformed shape (with deformation fit) or to oscillations in shape about a spherical mean (vibrations, with mean deformation fit)- One expects the optical potential to follow the shape of the nucleus and to be therefore correspondingly distorted. For convenience it is generally assumed that the distortion parameter fit is the same for all parts of the optical potential, and also that the potentials for the entrance and the exit channels are the same. The transition matrix elements are then completely determined by the radial derivatives of the optical potential which describes elastic scattering. The deformation parameter fl~ can be obtained by comparing the experimental differential cross sections with the DWBA predictions, since (dcr,(O)/df2)~xp = fl?(daJdO)DWB A. The DWBA calculations used to analyse the inelastic scattering angular distributions reported in this paper were made using our final optical-model parameters of table 3(b) in the code JULIE due to Bassel et al. $1). Sometimes in similar analysis only real form factors are used and Coulomb excitation is neglected. We have therefore examined and verified the importance of including Coulomb excitation (which arises from distortion of the Coulomb part) even at our relatively high energy (the effect is largest for l = 2 and for 0 <= 30°), and also the fact that the use of complex form factors (i.e., distortion of both V and W) is necessary. We have also investigated the effect of optical-model parameters on flz. It was found, for example, that for the best fit potentials based on Jarvis' geometry and Satchler's geometry, the peak cross sections differ only by about + 5 ~o, although the shape at back angles are often quite different. We have made several calculations using Tamura's code JPMA[N 52) for coupled channel analysis of the strong 2 + and 3- transitions, and concluded that, for the small values of fll involved in our analysis, the results of the coupled channel (CC) calculation do not differ sufficiently from the DWBA calculations to warrant the use of the CC method, which is certainly more time consuming. DWBA analysis of the differential inelastic cross sections yields two primary results, /-assignments and the transition strengths based on the one-phonon collective model, We discuss the /-assignments first. Angular momentum transfer 1 < 5. In figs. 4-8 our data for the differential cross sections are shown. Also shown there are the predictions of the DWBA calculations for the assumed/-transfers. For l < 5 there appears to be little doubt in the correctness of the/-transfer assignments. Very small changes in the shapes of the theoretical curves can be made by using different optical potentials, but the main features remain
2+
1.293
4+
4+
2+
4+
2+
7-
2.390
2.526
2.637
2.787
2.825
2.890
0.28
(8 + , 7 - )
(4 + )
(2 + )
2.977
3.027
3.074
0.3
1.4
0.17
2.941
1.0
0.85
2.5
3.9
1.1
7.0
4.0
1.6
0.35
4.1
1.7
2.9
8.5
5-
3.047
2.959
2.845
2.803
2.649
2.531
2.391
2.366
2.267
2.367
36.4
3-
2.269
15.0
2.108 2.224
1.762
1.291
E,
(2 + )
0.27
20.4
Gt
2.230
0.17
14.9
o'tot(mb)
(P, P') 24.5 MeV
2.110
1.755
jrr
E~
27) (P, P') 11MeV
This work
TABLE 4
3.04
2.90
2.78
2.53
2.27
1.28
Ex
4+
7-
4+
4+
3-
2+
j~r
(9, P') 55 MeV
t8)
5.2
3.2
5.5
4.2
34.5
18.5
Gz
2.87
2.60
2.27
2.24 3-
0+ 0+
1.76
2+
jlr
2.03
1.29
Ex
(p, p ' ) analog
29)
3.04
2.80
2.58
2.37
2.22
2.04
1.77
1.28
Ex
4+
(4 +)
4+(0 + )
5 - ( 4 +)
2+
0+
0+
2+
jrr
(p, d) (p, t) 55 MeV
18)
2.95
2.78
2.62
2.37
1+2+3 +
2+
0+
1 +2+3 +
2+
0+
2.23
0+
1.76
2+
J"
2.03
1.28
E,
(d, p) 15 MeV
16)
S u m m a r y o f experimental results for ~16Sn. Gz < 1 are not quoted
53)
2.913
2.777
2.393
2.368
2.268
2.00
1.293
E~
7-
6-
4+
5-
3-
2+
J"
(a, xn) 20-50 MeV
3.037
2.914
2.802
2.778
2.530
2.392
2.369
2.269
2.20
2.112
2.00
1.757
1.294
E~
fl-y
4+
7-
4+
6-
4+
4+
5-
3-
2+
2+
0+
2+
J"
ll-lS)
3)
E~
(3He, d)
t~
t.n t~
p
Y.
0.7
0.5
2+
(6 + )
2+
3.735
3.763
3,796
3.845
3.912
3,577
2+
2+
3.80
3 64
1-2-
1+2+3 +
1+2+3 + 3.232
(8 + )
3.22
(7-)
4.075
4,017
3.938
3.783
3.731
4.268
3.97
3.72
3.35
3.17
3.10
4.220
3.951
3.918
3,846
3,808
3.771
3,744
3,656
3,621
2+
4+
4.203
2.8
3.43
3.14
4,272
0.5
0.5
4.019
4.085 4.157
0.6
0.4
3.947
(5-)
0.8
3.686
0.4
1.1
0,3
3,654
0.45
0.85
3.632
4+
0.70
3.513
3.572
3.509
3,468
2.3
0.40
3,337
3,319
0.70
(4 + )
1.5
3,453
2+ (4 + )
3,404 3.436
0.72 0.35
3,230
3.499
(3-)
3.320 3,359
0.75 0.35
3,091
3,423
(2 + ) (8 + )
3,212 3.257
1.0
0.36 0.40
(6 +)
3.090
2.743
15
2.3 --
2.96
--
--
--
2.877
2.733
2.576
2.489
2.405
2.324
7-
4+
5-
2.73
2.67
2.54
2.49
2.32 2.38
4+
0.95
2.959
2.921
2.962
2.927
2.92
2.989
0.4
2.901
23
(2 + )
2.892
2.944
22
0.63
2.917
21
2+
2.86 2.89
2,896
19
20 2.881
2,74
2.49
2.33
2.81
2.769
2.728
2.671
2.485
2.398
2.321
2.84
2.6
4,5
2.4
31.0
4+
2.05
1.75
1.22
E,~
18
0.41
2.0
0.4
0.8
1.0
0.5
12.5
2.277
0+
0+
2+
j~r
17) (d, p) 11 MeV
17
(8 + )
4+
2+
2.686
14
16
7-
4+
2+
3- ~(5-)
2.583
2.497
2.409
2.328
13
12
11
10
9
8
7
6
1.1
1.759
1.233
Ex
2.283
2.285
+
~
5
1.99
1.74
1.22
Ex
20) (t, p) 12 MeV
2.064
4+
1.229 2.032
0.1
17.8
a4) (d, d') 15 MeV
4
2.050
13.2
3
2+
1.233
Ex
1.740
1
Gl
(p, p') 24.5 MeV
a(mb)
27, 28) (p, p') 11 MeV
This work
2
j~r
E~
TABLE 5
2,72
2.49
2.32
2.05
1.75
1.22
Ex
2+
0+
1+2+3 +
0+
0+
2+
j1r
(d, p) 15 MeV
16)
S u m m a r y of experimental results for 11SSn. Gz < 1 are not quoted 16)
2.72
2.47
2.30
2.03
1.74
1.22
Ex
(d, t) 15 MeV
53)
2.580
2.326
2.285
1.231
E~
7-
5-
4+
2+
j~r
(c¢, xn) 28-50 MeV
2.96
2.77
2.72
2.57
2.48 2.48
2.32
2.28
2.05
2.04
1.75
1.23
E,,
5-
4+
0+
2+
0+
2+
jTr
(4 + )
7-
0+ (4 + )
fT
II, 13-15)
.
.O
t.o
26
0.5
0.8
3.689
3.650
1.1
3.857
3.893
47
48
0.4
4.041
0.5
53
4.069
3.977
3.932
3.870
3.804
3.757
4.008
0.25
0.55
0.3
0.35
52
2+
4+
51
50
3.960
3.81t
49
3.789
45
46
3.746
(6 + )
0.4
3.581
4.02
3.91
3.89
3.79
3.70
3.57
3.575
1.9
3.55
3.555
3.52
3.47
3.38
3.520
44
3.715
42
4+
1.0
0.42
0.35
3.421
3.720
3.677
41
2+
(6 +, 7 - )
(4 +)
0.62
43
3.605
3.551
3.471
40
39
38
37
36
35
3.36 3.369
3.393
0.36
34
3.30
3.15
3.13
3.34
0+
3.05
3.362
(3 - )
3.136
3.055
33
3.06
32
0.34 3.332
3.260
3.320
0.65
31
(7-)
3.240
3.214
3.126
3.048
3.287
0.3
0.1
0.1
30
(8 + )
2+
29
28
3.241
3.148
25
27
3.073
3.113
24
4.04
3.91
3.79
3.70
3.06
1+2+3 +
3-, 4-
1 +2+3 +
1 +2 +3 +
4.01
3.89
3.80
3.67
3.04
3.058
0
Q ,-.-I
,z
4+
5-
2+
3-
(4 + )
(7-) (5-)
4+
(8 + )
2+
2.199
2,287
2,361
2.408
2.470
2,486 2,547
2.653
2,694
2.735
2.809
0.2
0.3
0.5
0.2
0.9 0.15
1.2
10.4
0.58
2.4
1.1
11.4
a(mb)
26.4
1.0
7,8
1.9
16.6
Gi
2.67
2.455
2.391
2.272
2.183
1.166
E~
4+
4+
3-
5-
4+
2+
J~
(p, p') 24.5 MeV
( 7 - , 8 + ) 0.15
2+
1.167
2.76
j~r
E~
,2) (p, p ' ) 17.8 M e V
This work
2.632
2.455
2.391
2.346
2,272
2.72
2.63
2.44
2.31
2.23
1.87 2.11
1.872 2.088 2.183
1.18
Ex
33)
(2 + )
(0 +)
3-
(0 +)
2+
J~
(d, d') 15 M e V
1.166
Ex
(p, p') 11 M e V
2~)
2,693
2.591
2.478
2.416
2.402
2.361
2.289
7-
5-
4+
0+
2.150 2.201
0+
2+
J~
1.881
1.174
Ex
(t, p) 13 M e V
20)
2.81
2.61
2.43
2.36
2.29
2.17
1.88 2.10
1.17
Ex
(d, p) 11 M e V
1~)
2.73
2.60
2.42
2.31
2.17
1.88
1.17
E~
1+2+3 +
0+
1+2+3 +
0+
0+
0+
2+
J~
(d, p) 15 M e V
~6)
TABLE 6 S u m m a r y o f experimental results for 12°Sn. Gt "< 1 are n o t q u o t e d s3)
2.490
2,280
2,180
1.171
E~
7-
5-
4+
2+
J~
(ct, x n ) 28-50 M e V
2.65
2.50
(2.46)
(2.36)
2.30
2.20
0.88) (2.12)
1.17
Ex
7-
5-
4+
2+
d~
fl, 7
I J. i , )
"~
r~
9
4~
0.3
0.25
0.45
0.25
0.5
3.962
3,995
4.080
4,190
3.986
3.926
3,860
0.25
2+
3.885
3.935
3.777 3.811
3.1
4.17
4.10
4.02
3.90
3.82
3.74
3.710
3,53
3,44
3.32
3.24
3.12
3,67
0.53
0.95
2.89 2.99
3,646
3.574
3,540
3.445
3.266
3.223
3.169
2.963
2+
(3-)
4+
4+
4+
3.45
3.17
3.06
3.796
0.75
3.664
1.6
6.0
2.1
2,830 2,919
3,840
0.35
3.592
(6 +, 7 - )
0.3
0.8
3,566
3-)
0.5
3.445
3,467
0.25
3,392
0.8
0.2
2 +)
(6 +, 7 - )
3.288
2.1
3,330
3,186
4+
4+
3.069
3.168
0.4
2,938
0.68
0.4
2 +
2,850
(2 + )
(2+)
3.818
3.780
3.593
3.475
3.183
2,852
2.84
4.19
4.06
3.99
3.94
3.88
3.80
3.73
3,71
3.66
3.60
3.58
3.55
3.47
3.39
3.29
3.21
3.16
3.00
2.94
4.15
4.03
3.87
3.70
3.56
3.23
2.94
2.84
1+2+3 +
1+2+3 +
1+2+3 +
1+2+3 +
1+2+3 +
1+2+3+
1+2+3 +
1+2+3 +
3.80
3,57
3.39
3.18
3.06
(4+)
0.-] 0
346
O. BEER et al. TABLE 7 S u m m a r y of experimental results for 122Sn. G~ < 1 are not quoted
Ex 1.143
This work
27)
34)
29)
26)
11)
(p, p') 24.5 MeV
(p, p') 11 MeV
(d, d') 15 MeV
analog
(p, p') 12 MeV
fl~,
jz" 2+
2.090
o-tot(mb)
Gl
10.0
15.0
0.16
Ex
E,~
~
1.142
1.15
+
2.103
Ex
jrr
Ex
Ex
1.14
2+
1.141
1.14
2+
2.08
(0 +) 2.14
(4 + )
2.142
4+
1.60
3.5
2.155
2.15
2.147
2.245
5-
2.7
8.7
2.239
2.25
2.247
2.328
4+
1.3
2.6
2.336
2.390
(7-)
1.2
2.412
2+
2.0
3.6
2.418
2.42
2.492
3-
8.9
22.6
2.496
2.50
2.556
(8 + )
2.260 2.34
2.30 2.41 --
0.25
2.684
0.30
2.750 0.14
3.038
2.86
0.15 (5-, 4 + )
0.3
2+
0.95
1.3
3.128
3.15
--
3.237
4+
1.3
4.2
3.236
3.26
--
3.283
3.33
--
3.313
4+
1.1
3.4
3.367
3-
0.6
3.457
(3 - )
0.3
3.478
(7-)
0.3
3.477
3.533
0.35
3.529
3.564
0.35
3.584
0.25
3.361 1.3
3.459 3.49 3.56 3.633
(4 + )
3.708
0.65 0.50
3.71 3.731
3.773 3.818
0.5 (6 + )
3.778
0.44 3.841
3.85
3.879 3.900
(2 + )
3.978 4.104 4.185
(0 +)
2.970
3.135
3.675
2.68 2.75 2.870
3.084
3-
0.49
2.654
2.976
1.1 0.75
(5-)
2.414
2.50
0.3 0.5
3.903
3.92
3.14
2.494
347
EVEN TIN ISOTOPES TABLE 8 S u m m a r y o f experimental results for 12"Sn. Gz < 1 are n o t q u o t e d This work (p, p ' ) 24.5 M e V Ex
jz,
atot(mb)
1.139 2.107
2+ 4+
9.0 1.65
2.130
(2 + )
0.8
27) (p, p ' ) ll MeV
34) (d, d') 15 M e V ~
29) analog
GI
Ex
Ex
14.0 4.9
1.132 2.105
1.13
2.133
2.13
2.12
+
.Ex 1.132
jz, 2+
26) (p, p ' ) 12 M e V E~ 1.131 2.101
2.199 2,213
5-+(4 +)
3.5
(9.9)
2.217
2.21
2.20
2.200
2.431
2.41
2.43
2.428
2.605
2.59
6 7
2.335
7-
1.35
8 9
2.435 2.455
2+ (8 + )
1.7 0.55
2.7
10
2.613
3-
7.6
19.6
11 12
2.713
(4 + )
0.84
1.8
2.85 2.900
+
0.34
2.879 2.952 2.988
2.98 3.13
--
3.215
3.19
+
3.35
--
15 16
3.009
3-
0.4
17
3.158
4+
0.34
1.0
18
3.232
2+
1.1
1.8
19
3.282
3.266
20
3.324
21
3.366
4+
1.55
5.4
3.359
22 23
3.416
4+
0.9
3.1
3.421
24
3.516
3-
0.67
1.5
2+
3.485
25
3.577
0.35
3.557
26
3.602
0.18
27 28
3.649
0.12
3.605 3.636
29
3.702
30
3.653 (6 +, 7 - )
0.75
3.698
3.759
0.56
31
3.794
0.27
3.746 3.775
32
3.827
0.27
3.806
33 34
3.879 3.909
(6 + ) 2+
0.8 0.9
35 36
4.019 4.159
(2 + ) 2+
0.5 0.5
37
4.281
(2 + )
38
4.350
0.5 0.6
2.59
3-
2.69
(0 +)
2.708
13 14
--
2.678
2.85
3.19
2.604
348
o. BEERet al.
the same. In particular the location o f the maxima, and more characteristically the angular position o f the first rapid fall-off o f the differential cross sections remain unchanged. As a matter o f fact it is quite surprising that g o o d fits are obtained even for the higher excited states for which no serious meaning can be attached to the single p h o n o n collective model form factors. The fits to these states emphasize what has '
'
'
'
I
J
i
i
,
1=7 ---
m3
exp.
distribution
of 2.89
+ ,
Sn 116 2 89 MeV {xO.2)
Tr-H44t.,"
;
of Sn 116
level
Sn 118 2.58 IO,eV ( x 0.5)
Sn
3~ 29
118 Mev
(x2)
~,\ ,,
Sn 122 3.48 N',eV {x,'. )
%
....
10-1
.,,
,
,
~
,
I
50
~
,
f
,
I
100
Sn 124 2.3" MeV (xLOO) ,
~
,
,
I
,
,
,
150
0%.
Fig. 9. Inelastic scattering differential cross sections for states corresponding to 1 = 7. The dashed curves correspond to the shape of angular distribution for the 2.890 MeV level in zleSn, known to have j~r = 7-. been often noted in particle transfer reactions, namely that the D W B A calculations reproduce the general features o f the angular distributions (especially the first maxim u m which is characteristic o f a given/-transfer) even though the detailed form factor for the state in question may be quite different from that assumed in the calculations. Angular momentum transfer l > 6. D W B A calculations were made for l = 6, 7 and 8 also. However, since no definite 6 + or 8 + assignments exist in the literature
(
tt
'
't'
,
,
50
I
~
,
~i\
~
~
~
0%.
tO0
I
_I L . '~>~+ TI !~+~'+- "..._!t!~.
~
~"~---~ H--.
h m'~-h. / ~'
~
t
'
,
i ',.._thi. ~
f.,
~,,
•
-+-+-+÷+h-, f~-i-H'-L~!~.. +"'~,'
~'+,
'
"-~'i I "~ t
"/L~'+'it
I
"~, ++~
"k,+ ,
"T,,&
. . . .
t f ~"
.--Lt+..
'-'~--a--L
~
Sn
118
~
~
(x 25.102)
3 . 7 0 MeV (,7.102 )
~o,~,
~
,,10~,
,22 ~v
so 3,~2
M~
I × ~o )
3,~
~o ,2o
3.72 MeV (xlO)
150
I
so 116
. . . .
3.5s Mev
'' [
•
/
Fig. 10. Inelastic scattering differential cross sections for states considered to be good candidates for the assignment j r = 6 +. F o r three levels (3.55 MeV o f ~taSn, 3.66 MeV o f zz°Sn and 3.70 MeV o f ~Z4Sn) an l = 7 transfer cannot be excluded. See also text.
G
D
102
'
10-I
10
-
-
-
t
I
--
. . . .
..
.
-
-
so
...
. -..I'~
" -
14o~',..
%,
"
t!~t[~L!~"~i T't't?t]
._
,~
t-t . .t . .t . t t t " ~- t-~- ~" t't'~t / t ""f'~ l ""* -
+-H+~-++4,
"
]4ft 4tf t,," 'r"t?
•
U_L~]]_tt t
.
........
. . . .
. . . .
,:8
2.81 MeV (x240)
15o
s . MeV 124 2.46 (xlO2)
Sn 122 2.56 MeV
Ix6)
Sn 120 .2.69 HeY
32,,x2,MoV
2.90 MeY
(xO.07)
~o.~
(x 0.03)
2.98 IVleV
Sn 116
of 2.56 levelof Sn 122
I
Fig. 11. Inelastic scattering differential cross sections for states considered to be good candidates for the assignment j~r = 8. For two levels (2.9S MeV of 116Sn and 2.81 MeV of izo Sn) an I = 7 transfer cannot be excluded.
G
10:
10~
I
~D
o. BEERet al.
350
no comparison could be made with experimental angular distribution to a k n o w n l = 6 or 8 transition. However, the first 7 - states in lt6Sn (2.890 MeV) and xaSSn (2.583 MeV) are well resolved and well established f r o m other experiments referenced in tables 4-8. In fig. 9 we show the observed angular distributions corresponding to these states. We notice that the two angular distributions are characterized by a sharp change in slope at 0 = 80 °. The D W B A predictions are not shown because they did not fit these angular distributions even qualitatively. In fig. 9 we also show angular distributions corresponding to the 3.29 MeV state in ltSSn, the 3.48 M e V state in 122Snand the 2.34 MeV state in 124Sn.As illustrated, their shapes are similar to those o f the transitions to the k n o w n 7 - states. We therefore assign these other levels as 7 also. These assignments must of course be regarded as tentative, although f r o m the systematics there seems little d o u b t that the 2.34 MeV state in tZ4Sn is indeed a 7 - . We mentioned earlier that the angular position of the first sharp decrease in a(O) appears to depend quite characteristically on/-transfer. F o r I = 4 this occurs at about 45-50 °, for l = 5 at about 55-60 ° and for 1 = 7 at a b o u t 75-80 °. We expect, qualitatively, that for l = 6 this should occur near 65-70 ° and for l = 8 near 85-90 °. Sample D W B A calculations bear out this expectation. Several states are found for which the measured angular distributions display these trends. The differential cross sections for the states with the break at 65-70 ° are shown in fig. 10. We tentatively assign these states as 6 +. In fig. 11 we display the observed differential cross sections for states which appear to be g o o d 8 + candidates, based on the criterion described above. As a matter o f fact it is n o t in all cases possible to discern unambiguously between 6 + and 7 - and between 7 - and 8 +. Transition strengths. Since no g o o d correspondence with D W B A calculations could be established for l _-> 6, we have derived transition strengths only for states with I < 5. These are listed in tables 4 through 8 in terms o f G~. Since different authors quote their results in different terms, we present below the relations between them *: B(EI) =
(3ZR'/4n)2fl]; G,-
B(El)s.p" - 2 l + 1 ~ 3 R t l 2 4re L3 + IJ '
B(E/) _ ,8~ B(El)s.p. flt2(s.p.) "
(1)
(2)
In particular for Sn isotopes G(l) = Kfl 2 where K = 995, 1023, 1083 and 1154 for l = 2, 3, 4 and 5, respectively. A t this stage it is worthwhile noting that analysis o f our data using collective f o r m factors, which correspond to o n e - p h o n o n excitations, is meaningful only for collective states such as the 21+ quadrupole and 3~- octupole vibrational states. It m a y be reasonable even for the 5~- states which show definite enhancement. But the use o f t Some authors 42) define B(EI) with an additional multiplier (2/+ 1)- ~ on the right-hand side. Barreau and Bellicard 2s) define/31 still differently. [fit (Barreau) = 9fl~z (above)/(2l+ 1)]. In all cases, however, quantities in single-particle units come out to be the same.
EVEN TIN ISOTOPES
351
these form factors is clearly not justified for weak 2 + states, most of the 4 + states, or 6 + or 8 + states. Some of these are quite pure 2 quasiparticle (q.p.) states. Some may involve excitations of four or more quasiparticles (corresponding to multiphonon excitation). In these cases G l values obtained by the DWBA analysis used here do not have much physical significance. They merely represent a convenient parameterization of the measured differential cross sections. In our tabulated results we quote G~ values only when they are larger than unity.
4. Experimental results We present the results of our experiments in tables 4-8, along with the relevant results from other experiments. The results are also summarized in fig. 12 where the length of the lines indicates Otot as defined earlier and the circles represent some of the 0 + states not seen in the present experiments but known from other work. 4.1. E X C I T A T I O N E N E R G I E S
Certain general features of the excitation energies and the nature of levels excited in our experiment may be noted from tables 4-8. To begin with, our excitation energies are, in general, in excellent agreement with those from other spectrograph experiments which were done with l0 to 20 keV resolution widths. These include Allan's (p, p') experiment for all Sn isotope s 27,2s), Norris and Moore's (d, p) experiments ~v) for 1laSh and 12°Sn, and Bjerregaard et al.'s (t, p) experiments 20) for 11SSn and ~2°Sn. One may also note that the excited 0 + states which are seen in (d, p), (t, p) and in some fl-? experiments are only weakly (if at all) populated in our experiments. Non-natural parity states are also very poorly (if at all) populated in our experiments. It may also be noted that our higher energy enables us to excite certain states not seen in low energy inelastic scattering experiments 27,28). These are generally the states with/-transfer > 6. 4.2. j~r A S S I G N M E N T S
We have assumed throughout that in our experiments only natural parity states, i.e., those with J[~] = /[re = ( - 1 ) z] are excited. ]'his assumption is certainly true for the stronger transitions, since spin-flip scattering as well as two-step excitation are expected to be weak and to have non-characteristic angular distributions, but it may not be completely true for the weaker transitions observed. Our belief in it is however strengthened by the fact that for most cases the agreement between the data and the non spin-flip DWBA predictions is quite good. Unfortunately only one definite unnatural parity assignment exists in the literature on Sn isotopes. A 6- state at 2.777 MeV in ~1 6 S n was assigned by radioactivity experiments 12). This state is not seen in our experiment. The 2.787_+0.010 MeV state that we observe has been definitely assigned as a positive parity state in both (d, p) and radioactivity experiments. We note, however, that the angular distributions for the 3.320 MeV state in 11aSn and
352
o . B~ER e t a L
the 3.392 MeV state in 12°Sn, both well resolved levels, are quite fiat and these may be non-natural parity states. For all the states populated in our experiment our J~ assignments are in agreement with those from f l - 7 experiment, whenever such assignments are available. Except
m
(5")-
m
(2+1-
4, • ( s ' ) 2"-4* = - -
4" ~
(~"
g.--
(6+A-) --
2"~
(3-)"
(6",~-)--
-
tO'l--
3"
(3")"
4"
"
3- ~
4"
(~-
"4* ~
2. -
-
-
2* --
(s-, 4.):
r--
,.
2':-
17-1--
(3")_---
~r0-
g-
:_
(,,~=
=
(S+,7-)-(4")-(37--"
¢=_
(e+)"
18+,7-)-"
2..
{
--
7- ~ 4---
(5"37;~_
14"1~
2"-
3"
0
0
•
, q)
3"..s; . . . . , --
(e") ~ 3"
,~)
' (~)
18~--
,
"
-)s"
.......
s~tA*)
s-
o
4"
(~)~ 0
2.
o 0
2"
"%.
2"
"%.
,,~,
"°s.
2"
,..,
2.
,q,
;%,
Fig. 12. S u m m a r y o f experimental results for 116. 118.12o. 12z. 124Sn" T h e lengths o f the lines represent the total cross section observed. Circles represent 0 + states n o t seen in the p r e s e n t experiments b u t reported by other authors.
for the first 2 + and 3- states J~ assignments from other inelastic scattering experiments are relatively rare. Our results are in agreement with two notable exceptions to this remark, namely the 17.8 MeV inelastic proton scattering from
EVEN T I N ISOTOPES
353
a2°Sn by Jarvis et al. 42), a n d the 55 MeV inelastic proton scattering from a a 6Sn by Yagi et al. 18). We generally disagree with the parity assignments from the (d, d') experiments 32, 34). This is not surprising since these assignments were made on the basis of comparison of a(0) at two angles only, based on a prescription due to Jolly 33) who used Blair's phase rule to justify this procedure. Particle transfer experiments are often quite selective. For this reason, and also because most of these experiments were done with relatively poorer resolution, it was found not possible to establish unambiguous correspondence between our levels and those seen in these experiments. A notable general feature of all the observed spectra is that whereas for the positive parity states, 2 + and 4 +, a greater number of states exists, for the negative parity states, 3 - , 5- and 7 - , no more than two (a third 3- state is occasionally suggested) states are found up to 4 MeV excitation. As will be seen later this is what is qualitatively expected on the basis of the valence orbits available for neutron excitation. The collective nature of the first 2 + and 3- states (hereafter referred as 2 + and 3]respectively) in tin isotopes is already well established from a number of experiments. As expected these states are exceptionally strongly excited, with ~tot ~ 12 mb. What is somewhat surprising however is the fact that the 5~- and 7~- states are also quite strongly excited with atot ~ 2.5 and ~ 1 mb respectively. These cross sections are factors 5 to 10 lower than those for the 2 + and 37 states, but are larger than those for most of the other 2 +, 3 - , 5- and 7- states. This is particularly significant if one considers the large angular momenta (5 and 7) involved. In contrast, nearly all 4 + states are more or less equally excited, with atot ~ 1-2 mb. 4.3. U N R E S O L V E D D O U B L E T S
In each of the figs. 4-8 the majority of J~ assignments are quite straightforward and unambiguous. For each isotope however some cases exist where doublets are suggested by width of experimental peaks, magnitude of cross sections, systematics of excitation energies, or shape of angular distributions. In the following we discuss some important cases among these. From fig. 12 we note that the 5~- energies decrease systematically and the 31 energies increase in the same manner. The two trend lines cross at 118Sn" Indeed only one state is seen at 2.328 MeV. The (p, p') angular distribution has mainly the shape of a l = 3 transition, although some evidence for the filling-in of the minima exists. The 5- assignment of the mixed-in state is supported by a number of experiments [refs. 2o,~2-a5,53)]. We have therefore analysed the transition as a 1 = 3+1 = 5 doublet, the cross-section decomposition being based on the cross sections of the 3[- and 5~- states in the adjoining tin isotopes, where the two states are well resolved. The "state" at 2.478 MeV in 12°Sn is found to be broad, suggesting an unresolved doublet. Indeed the trend lines of the 7~- and 4 + states suggest such a doublet. The observed angular distribution of the unresolved doublet (fig. 6) is also in agreement with this hypothesis.
354
o. BEERet al.
I n 122Sn a close d o u b l e t (with energy s e p a r a t i o n o f a b o u t 12 keV) at 2.412 keV is suggested by the shape o f the peak. The a n g u l a r d i s t r i b u t i o n indicates the definite presence o f the 2 + state. W e p r o p o s e that the other unresolved state is the 71, whose p o s i t i o n in this n e i g h b o r h o o d is indicated by the energy systematics. The m a g n i t u d e o f the integrated cross section in this case also indicates the presence o f 7~-. The 2.213 M e V " s t a t e " in lZ4Sn also a p p e a r s to be a doublet. The evidence is twofold. In A l l a n ' s high resolution (p, p ' ) experiments z7) two states at 2.199 a n d 2.217 M e V were seen. O u r single transition at 2.213 M e V m u s t c o n t a i n b o t h o f them. The essential features o f the observed a n g u l a r d i s t r i b u t i o n c o r r e s p o n d to that o f a 5 state a n d the m a i n c o n t r i b u t i o n is u n d o u b t e d l y due to the 57 state. The total cross sections for the 57 states in isotopes in which they are clearly resolved are 2.9 m b (116), 2.4 m b (120), a n d 2.7 m b (122). The transition at 2.213 M e V has a total cross section o f 3.5 mb. The higher cross section is also suggestive o f an unresolved group. W e t h i n k that the mixed-in level is a 4 +. This is consistent with the observed a n g u l a r d i s t r i b u t i o n as well as the t r e n d o f the 43 states. 4.4. THE 2 + STATES The first 2 + states a n d their collective n a t u r e in the isotopes o f tin are well k n o w n . O u r m a i n emphasis in r e g a r d to these states was to determine, as accurately as possible, the v a r i a t i o n offl2 o r G2 t h r o u g h these isotopes, in o r d e r to c o m p a r e with theoretical p r e d i c t i o n s as well as o t h e r e x p e r i m e n t a l results. This c o m p a r i s o n is especially i m p o r t a n t because it has often been stated t h a t the d e f o r m a t i o n p a r a m e t e r s o b t a i n e d by different kinds o f experiments should be different, since each o f t h e m is sensitive to a different aspect o f the nuclear d e f o r m a t i o n . TABLE 9 Summary of results for the Experiment
Ref.
116Sn
f12 Coulomb excitation b) reson, fluorescence c) electron scattering neutron scattering m-scattering deuteron scattering proton scattering
22) 2~) 25) a7) 36) as) a3) 42, 18, 55) 31) ours
~1s Sn
G2
0.135±0.003 18.0± 0.9 0.135-2_0.030 18.0± 4.2 0.1091 11.8± 1.8
f12
G2
0.136±0.003
18.4~0.9
0.13 ±0.04
12.4~ 2.5 16.8~10.3
0.10 ±0.02
11.5±2.3 9.9=]=2.4
0.133±0.010 0.143±0.004
18.5± 17.6:~ 2.6 20.4--+ 1.2
0.134±0.010 0.134±0.004
17.9 ~-2.6 17.8+__1.1
a) For all the above measurements, G2 = 995 •22. b) Measured quantity, B(E2). Computed f12 by relation f12 = B(E2)~:/(3ZRo2/4et) with Ro taken as the equi and a = 0.55 fm, according to the tabulation of Owens and Satchler. c) Measured quantity, half-life, Tz~.
EVEN TIN ISOTOPES
355
As was mentioned earlier, we have made measurements for the 2~- states with three samples of accurately known mixtures of tin isotopes. This enabled us to obtain the ratios of cross sections for the 2 + states between one isotope and another with errors less than + 5 ~o. Considering an additional uncertainty of +__5 ~o due to the choice of the optical-model parameters we get our relative elrors on G2 as ___7~ . As discussed earlier the absolute errors due to incorrect normalization of the absolute cross sections may be somewhat larger. We summarize the results of various experiments in table 9. It may be noted that comparison between various results is often not possible, either because of very large quoted errors or unspecified errors. By far the most accurate results are the Coulomb excitation B(E2) results of Stelson et al. z2). These results have quoted relative errors of + 2 ~ and absolute errors of +_5 ~o. The authors give/32 values also, based on eq. (1) of subsect. 3,3 with R taken as the uniform radius R = 1.20 A~ fm. However Owen and Satchler 54) have pointed out that in view of the Saxon form of the nuclear charge distribution a single uniform radius cannot be obtained for the different multipolarities. They have tabulated uniform radius equivalents for different values of 1 and different values of the half-way radius r, and the surface diffuseness a. We have converted the B(E2) values of Stelson et al. 2z) using the equivalent uniform radii tabulated by Owen and Satchler 54) for r = 1.075 fm and a -- 0.55 fm which are the appropriate values for tin nuclei according to electron scattering experiments. As seen in table 9 our results are in excellent agreement with the Coulomb excitation results of Stelson et al. 22), e v e n the largest difference (6 ~ for aa 6Sn ) being within the quoted errors. The only other results of comparable accuracy are the (p, p') results of Makofske et al. 3~). Once again the agreement is well within the quoted errors,
first 2 + states ") 120Sn
0.1304-0.003 0.098 4-
0.12 ± 0.12 4 - 0 . 0 2 0.12 40.12, 0.11 0.1194-0.010 0.129±0.004
1228n
16.94-0.8
0.1244-0.003
15.44-0.8
9.6 4-1.6
14.3--2.5 12.45:2.5 14.34-4.7 15 416.8, 14.3 14.1 ±2.4 16.64-1.0
124Sn
0.1144-0.003 0.098 ±
0.13 4 - 0 . 0 3
12.34-2.5 16.8--7.8
0.1124-0.007 0.122±0.004
12.5±1.6 15.0±1.1
13.04-0.7 9.6 i 1.6
14.1 ±2.8
0.1084-0.007 0.1194-0.004
11.6±1.6 14.0±1.0
valent square well radius which gives the same B(E2) as a Saxon-formcharge distribution with ro = 1.075fm
356
o. BEER et al.
although the results of Makofske et al. show a more rapid fall-off offl2 beyond 11SSn than either our results or those of Stelson et al. MeV
Me\
4
:
•
I
:
_I~'- •
•
I
.,
-./
t
MeV
l
l
4-
•
/o/
I
4-
0
0
0
i
f
s ~ / / ..~/ / ~j'*" o / //o ~// / /
•
r
3" •
•
•
2-
1
........
I
/,
"--:t
11G 1,8
Exp:• 3-
I ,I
, ,o(?
I
I
I
I
Me
~, ,
,
,
4 +
•(4+1
ds ,;8 ,~,o ;22 124s. (b)
,20 ,22 124Sn
(a) MeV
Exp:•
• (3-1
t
~l~ .'8
MeV.
4- o
,20' ,.'
' 1245n
(c) f
t
i
t
4
I •
•
•
o
o
o
x
×
3-
2-
2-
J Exp:e 5•(5-) ]20 122 124Sn
(d)
" Exp:e 7• (7-) 1'16 11'8 120 12~2 124Sn
(e)
Exp : X [8+1
l
®(7.8"1
I
• 16+1 o(G+~7 -) 8 +ThtoreL;col i 116 118 1120 1122 1245n
(f)
Fig. 13. Comparison of experimental excitation energy trends for each jTr with results o f several theoretical calculations. The solid curves correspond to calculations of Arvieu 6) the dashed curves correspond to calculations of E. Baranger.
357
EVEN TIN ISOTOPES
The electron scattering G2 values are consistently about 35 ~o lower than our results as well as all other entries in the table. They have quoted errors of about +__17 ~o. It is not clear whether these errors are relative or absolute. In any case no other entries indicate a significant variation of f12 from experiment to experiment, and the variation, if it exists, must be considered to be smaller than + 15 ~o. The energy systematics of the 2+ states as well as all other 2 + states is displayed in fig. 13(a). The smooth behaviour of the 2~+ state is quite obvious. It is difficult to attribute much systematics to the higher-lying 2 + states. As a matter of fact additional 2 + states, other than those shown in fig. 12 has been suggested by some experiments. These are indicated in the tables 4-8. Telegdi and Gell-Mann 5 6) and Lane and Pandelbury 57) have introduced a model independent energy weighted sum rule which follows from the charge independence of the nuclear interaction. It may be assumed that the low-lying vibrational states correspond to the oscillations in which the neutrons and protons move in phase (T = 0). For such T = 0 states, the sum rule limit (SRL) is L G , E ' = h z Z 2 1_~ l ( 2 l + 1 ) ( 3 + l ) 2 , i l SMp A R~,
l > 1,
(3)
where REt is the equivalent uniform radius for the multipolarity l, Z is the atomic number of the target and Mp is the mass of a proton. Lane 5s) considers that a good criterion of collectiveness is that a given transition "exhausts at least a fair fraction (say >~ 5 /o/o ) of a sum rule". The 2 + transitions do meet this test successfully. They exhaust 7.3 ~ , 6.3 ~o, 5.7 ~ , 5.1 ~o and 4.9 ~ of SRL for tin 116, 118, 120, 122 and 124 respectively. The remaining 2 + transitions together add up to an almost equal amount, bringing up the total fractions of the SRL observed to 12 ~ , 12 ~ , 9 ~ , 12 ~o and 13 ~o respectively. These numbers once again bear out an almost uniform degree of collectivity in the tin isotopes. 4.5. THE 3- STATES In contrast to the many 2 + states no more than two or three 3- states have been identified in each isotope. The 3~- states are already well known from other experiments. As fig. 13(b) illustrates, the 3~- excitation energies rise monotonically in energy. The cross section corresponding to them as well as G3 decreases slowly with increasing number of neutrons. Once again our main interest concerning 3~- states was to determine the variation of their transition strength in the tin isotopes. The normalization obtained by our mixed target experiments leads us to G3 values with relative errors ranging from 9 to 13 ~ . Our results are listed in table 10 together with results from other experiments. Unlike the 2 + states, for the 3[- states accurate values from Coulomb excitation experiments are not available because of serious experimental problems. The Coulomb excitation results of Alkhazov et al. 23), though a factor of three lower than the earlier results of Hansen and Nathan 24) are still 30 }~o, 13 ~ , 3 ~o, 100 ~ and 100 ~ higher than ours. Agreement between our results and those of Ma-
358
o. BEERet al. TABLE 10 Summary of results for Experiment
Ref.
~~6Sn f13
Coulomb excitation b) electron scattering neutron scattering cz-scattering deuteron scattering proton scattering
23) 25) 37) 36) 3s) 33) 42, ~s, 55) 3t) ours
xxs Sn G3
/53
G3
0.182±
48 34 ~5
35
0.15 ±0.03
12.3 :k 2.5 23 ___9
0.15 zk0.03
0.185±0.010 0.188±0.008
34.5 ± 35 ±4 36.4-E3
0.168±0.14 29 ~5 0.174±0.11 ~) 31.44-4 c)
12.5 ~ 2.5 23 ±9
a) G3 = 1023 /~32. b) Measured quantity, B(E3) • B(E3)s.p. = 0.1393Ro 6 with Ro taken as the equivalent square well radius which the tabulation of Owens and Satchler). c) Obtained by subtracting the contribution due to the unresolved 5 - state, assumed to be approximately kofske et al. 32) is within __+3 ~o, well within the q u o t e d errors. U n l i k e the 2 + states, the a g r e e m e n t between o u r results for the 3~- states a n d the electron scattering results is also excellent. I n terms o f the sum rule o f e q . (3), the 3~" states exhaust 7.6 ~o, 6.8 ~ , 6.2 ~ , 5.4 9/0 a n d 5.3 ~ o f the S R L for tin 116, 118, 120, 122 a n d 124 respectively a n d meet the collectivity criterion o f L a n e ss). The r e m a i n i n g 3 - states a d d u p to a p p r o x i m a t e l y 1 ~ a d d i t i o n a l c o n t r i b u t i o n to the S R L , the t o t a l fractions o f the S R L observed being 8.8 ~ , 7.1 ~ , 6.8 9/00, 6.2 ~ a n d 6.2 ~ respectively. 4.6. THE 4 + STATES The observed 4 + states are illustrated in fig. 13(c). The 4 + states are numerous, like the 2 + states a n d unlike the 3 - o r 5 - states. The first 4 + states follow the t r e n d o f s m o o t h l y decreasing energy, m u c h like the 2+ states. A s a m a t t e r o f fact the 4~- a n d 4+ states always occur at energies a p p r o x i m a t e l y twice the energy o f the first 2 + states. We r e t u r n to this p o i n t later. The first 2 + a n d 3 - states were m o r e t h a n 10 times stronger t h a n any other states o f the same J~. This is n o t true for the 4 + states. M o s t o f the 4 + states c a r r y m o r e o r less the same strength (within a factor o f two). A l s o their n u m b e r s in each i s o t o p e are essentially the same. The s u m m e d t r a n s i t i o n strength is essentially the same (within + 10 ~ ) in all isotopes. 4.7. THE 5- STATES N o m o r e than two 5 - states have been identified in each isotope. The identification o f the first 5~- states is certain, a n d it is illustrated in fig. 13(d). The t r a n s i t i o n strength
EVEN TIN ISOTOPES
359
the first 3 - states a) 120Sn 33
122Sn G3
124Sn
f13
G~
27
f13
42
0.172± 0.17 !
30 30
0.14 4-0.03
20
4-9
0.14 40.17, 0,14
20 30
i :E7, 20
0.1594-0.020
26
4-7
0,161 =/:0.008
26.44-2.4
42
±4 4-7
0.133±
11.0i2,2
11.1 4-2.2 0.14 =t_0.03
20
G3
18
~:3
10.74-2.2
4-9
0.1524-0.014
23.64-4.5
0.133 ~:0.020
18.14-5.4
0.1494-0.008
22.6=k2.2
0.138+0.007
19.6±2.0
gives the s a m e B(E3) as a S a x o n f o r m charge distribution with ro = 1.075 f m a n d a = 0.55 f m (according to the s a m e as in t h e other even tin isotopes.
for the 5~- states in all isotopes is about 8 s.p.u. The assignment of the second 5- state in all cases is less certain. The second 5- state is generally a factor 5 to 10 weaker than the first. 4.8. T H E 6 + , 7 - , 8 + S T A T E S
In fig. 13 we indicate the trends for these states. The assignments for the (6 + ) and (8 +) states are rather uncertain, although the (8 +) states show a good systematic trend. The 7~- states show a well established trend, and although we cannot quote G 7 values, we note that the cross sections corresponding to these increase slowly from 1'6Sn to 124Sn.
5. Discussion
In this section we describe briefly the different theoretical attempts which have been made in order to understand the low-lying states of even tin isotopes, and compare our experimental results with the predictions of these calculations. For nuclei with large number of particles outside assumed closed cores, exact shellmodel calculations with residual interactions are prohibitive in spite of recent developments of powerful computational techniques 59, 60). For tin nuclei even if the doubly magic core N = Z = 50 is considered completely inert, the nearly 20 valence neutrons in the five subshells 2d~, lg~, 3s~, 2d~ and lh.~ present a secular problem of the order of 101 o x 101 o. In order to make the problem manageable major simplifications about the residual nuclear interaction must be made. The first of these was the realization
360
o. BEER et al.
that a major part of configuration mixing can be taken into account by the BCS pairing which arises from the short range part of the nuclear force and which also produces the characteristic energy gap. All attempts to calculate the spectra and other properties of tin isotopes do indeed have this simplification in common. The first step is always to transform the problem from that between particles to that between "quasiparticles" in order to take account of the pairing correlations. The singleparticle energies are replaced by their counterparts, the single quasiparticle energies which are defined as Ej =
v /..
2d 5/22
3.
2,
1.
~
I
5n116
2
d
I
Sn118
~
I
Sn120
2
I
Sn122
I
Sn124
Fig. /4. Quasiparticle energies Ej according to Arvieu 6).
where 2 is the Fermi energy and Aj is the so called "energy gap". For a doubly even nucleus the excited states are made up of excitations of two, four, six, etc. quasiparticles. If the quasiparticles are considered non-interacting, the energies of quasiparticle states can be obtained simply by summing single quasiparticle energies. For example for a two quasiparticle state of spin J, the excitation energy above the 0 + ground state is given by
E(jl,j2, J)-Eg.s.
(0 +) =
EjI+Ej2.
The quasiparticles of course interact, but certain qualitative aspects of the excitation spectra follow just from the simple model of non-interacting quasiparticles.
EVEN TIN ISOTOPES
361
In fig. 14 we show the trend of quasiparticle energies according to Arvieu 6). We note that the order of the quasiparticle levels, and their A-dependence is completely different from that for the single-particle levels. We note, for example, that the d~ and gk quasiparticle energies increase rapidly with A, while the s~, h~, d~ energies change quite slowly. We expect therefore that the low-lying states will be dominated by ½, ~ and - ~ quasiparticles, with states involving -~ and ½ quasiparticles following at higher energies. Between the two kinds of states there may even develop a region of low level density, particularly for higher A. Indeed in 122Sn and 1248nsuch a region is observed between 2.6 and 3.2 MeV. Since, of the five quasiparticle levels available, only the hq has negative parity we also expect far more positive parity states. We have already remarked on this feature of our data. We expect the 8 + states to be almost pure two quasiparticle 2E~ states. Their energy should therefore decrease smoothly with increasing A. (As a matter of fact, the energies are approximately equal to 2()~-), see fig. 14.) For any other J~ more than one configuration is possible: nine 2 + states, seven 4 + states, three 6 + states, two 3 - states, f o u r 5- states and three 7 - states are predicted in this simple model. One would expect the 2 +, 4 +, 6~- 5[- and 7~- states to decrease in energy with increasing A. The 3 - states require one of the quasiparticles to be in the ~ or ~ orbit and we may therefore expect that 3[- states will rise in energy with increasing A. It is somewhat surprising that all these qualitative predictions are indeed very confirmed by the data. In any serious calculation the interaction between the quasiparticles must, of course, be taken into account. This interaction, which arises from the long-range part of the nuclear force, can be specified phenomenologically or it may be obtained from the realistic nucleon-nucleon interaction. Also the interaction can be treated in one of the numerous approximate ways. Many calculations of even tin isotopes have been reported in the literature. Each differs from the others in one or more of these ways. We summarize these calculations in table 11. The first four calculations, numbers 1,2, 3 and4in table 11, use a very simple phenomenological interaction. They all assume inert N = Z = 50 core and a configuration space of five shell-model states for the neutrons. Core excitation is not considered and in all, but ref. 8,), excitations are limited to two quasiparticles. The fits to the 2 + and 3~" excitation energies are quite good. This is not surprising since the parameters of the interaction are chosen to optimize these fits. The transition rates predicted (table 12) do not agree as well with the data, but even that agreement can be mocked up by an appropriate choice of the neutron effective charge. In view of the availability of calculations using more conventional residual interactions, we do not discuss the above calculations any further. In fig. 13 we have not shown the results ofYoshida [ref. 61)] and Veje [ref. 64)] who obtain reasonable fits to the trend of the 3[- states by introducing an octupole-octupole residual interaction whose parameters are adjusted for the best fit to the experimental data. In table 12, we tabulate the B(E3) values obtained in refs. 61,64) mainly because not many other complete predictions are available.
362
o. BErRet
al.
TABLE 11
Summary of theoretical calculations Authors
Ref.
5) 61) 62) 8a) 6) 7) sb,¢) lo) 9.,b,c) 66)
Residual interaction
pairing+quadrupole pairing + quadrupole-}- octupole pairing+octupole pairing+delta function Gaussian with exchange Gaussian with exchange Gaussian with exchange realistic-Tabakin realistic-Tabakin-Yale Gaussian with exchange
Inert core N= Z = 50 50 50 50 50 50 50 28 50 50
Number of active subshells
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Kisslinger, Sorensen Yoshida Veje Ottaviani e t a L Arvieu et al. Kuo Trieste Group Clement, Baranger Trieste Group Gillet e t aL
5 5 5 5 5 5 5 12.}.12 tt) 5 5
*) tt) *) **)
STDA denotes "second Tamm-Dancoff approximation" which permits inclusion of 4 q.p. excitations. 12 neutron and 12 proton subshells above N = Z -- 28 were included. Core excitation taken into account by means of "core polarization" renormalization. One particle-one hole excitations of the core were included.
The second g r o u p o f calculations, numbers 5, 6 and 7 in table 11, use somewhat more conventional residual interactions. Once again N = Z --- 50 inert core was assumed and only a 5 sub-shell neutron configuration space was considered. Gaussian interaction with appropriate exchange terms was first used by Arvieu 6) who also verified that there are not any major differences between T D A and R P A results. The theoretical results o f Arvieu are shown in fig. 13. K u o 7) optimized single-particle energies and force constants and obtained somewhat better fits than Arvieu using RPA. Both these calculations considered only two quasiparticle excitations. The Trieste group [ref. 8b, c)] included four quasiparticle contributions in their calculations with Gaussian interactions, but f o u n d that the predictions were extremely interaction dependent. For this reason we will only refer to their latest work using realisticinteractions 9). It supersedes their earlier work. Both Arvieu and Salusti 6d) and K u o 7) have also calculated B(E2) values for the 2+ -~ 0 transition. These are listed in table 12. In all the above calculations phenomenological forces were used and their range, depth, and other parameters, and to a certain extent the input single-particle energies (s.p.e.) were adjusted to give the best fits to the experimental data. As illustrated in fig. 13, these calculations often produced quite g o o d fits to the energies o f the lowlying states. The Z 2 search for best fit parameters leads to an "effective" interaction which masks the effect o f those aspects which are a p r i o r i neglected in the formulation o f the problem. F o r this reason these calculations do not shed any light on the importance o f four quasiparticle excitations, or effects of core excitation, etc. These effects, which may be quite large, can be meaningfully investigated only when the parameters o f the interaction are fixed, such as when one uses realistic forces (i.e., those which fit nucleon-nucleon scattering data). Calculations using realistic inter-
363
EVEN TIN ISOTOPES
of even tin isotopes 50 core excitation
4 quasiparticle excitation
N=Z=
yes
yes yes yes *) yes **)
yes
Predictions
Approximation used
Adiabatic TDA TDA STDA t TDA RPA STDA t TDA STDA t RPA
21 + energies; B(E2) 2 +, 3- energies; B(E2), B(E3) 3- energies; B(E3) 0 +, 2 +, 4 + energies 0 +, 2 +, 4 ÷, 5-, 7- energies; B(E2) 0 +, 2 +, 4 +, 5-, 7- energies; B(E2) 0 +, 2 +, 4 +, 3-, 5-, 7- energies 0 +, 2 +, 4 +, 6 +, 3-, 5-, 7- energies; B(E2) 0 ÷, 2 +, 4 +, 3-, 5-, 6-, 7- energies; B(E2) 2 +, 3-, energies; B(E2), B ( E 3 )
a c t i o n s h a v e r e c e n t l y b e e n r e p o r t e d b y b o t h the T r i e s t e g r o u p 9) a n d t h e P i t t s b u r g h g r o u p 10). T h e m a i n t h r u s t o f t h e w o r k o f t h e T r i e s t e g r o u p 9) has b e e n to e x a m i n e t h e i m p o r t a n c e o f f o u r q u a s i p a r t i c l e effects, w h e r e a s the c a l c u l a t i o n s o f C l e m e n t a n d B a r a n g e r lO) h a v e c o n c e n t r a t e d m a i n l y o n t h e i m p o r t a n c e o f c o r e e x c i t a t i o n s . W e i l l u s t r a t e s o m e o f t h e results o f t h e s e c a l c u l a t i o n s in fig. 15, a n d discuss t h e m briefly below.
TABLE 12 Transition strengths for even isotopes of tin Gl
G2 21 ÷ ~ 0
Ga 3~- --->0
A
This expt.
116 118 120 122 124
20.44-1.2 17.84-1.1 16.64-1.0 15.04-1.1 14.04-1.0
116 118 120 122 124
36.44-3.0 31.04-4.0 26.44-2.4 21.64-2.2 19.6±2.0
KS [ref. 5)]
Y [ref. 61)]
15.5 16.8 15.9 13.9 10.9
8.9 8.1 8.5 8.3 7.2 12.3
Theoretical predictions a) V AS K CB [ref. 6,)] [ref. 6a)l [ref. 7)] [ref. 16)] 5.0 6.1 7.1 7.2 6.5 32.0 34.4 36.5 38.6 40.4
9.3 9.6 8.2 8.2 6.8
2.5
1.8
GRSW [ref. 9)]
GGR [ref. 66)]
6.2 --> 7.5
0.43 0.48 0.47 0.57 0.48 20.0 21.3 19.8 15.8 9.8
~) The values of G~ quoted in this table are for effective charge = 1.0e for neutrons except for the calculation V of ref. 64) where eoff = 0.Se for neutrons and protons, and calculation CB and G G R in which no effective charge was used. B(E/) from theoretical calculations were transformed into G~ values by using R = 1.2 A4".
364
o. BEERet al.
In the new calculations o f the Trieste group, G m i t r o e t al. 9b) examine four quasiparticle c o n t r i b u t i o n s (in w h a t they call second T a m m - D a n c o f f a p p r o x i m a t i o n , S T D A ) a n d take core excitations into a c c o u n t a p p r o x i m a t e l y by using Bertsch 62),
1165n
MeV 3-!--
7-
..8* 0÷ 6+
3.1-4,+
4+
(6.71
"
-
~
,~-
!,8,*)
o+
,
62+
I
/
2.9-
) 4+
\
/ ~ ~,+ 1 i A~
4+ (5,9)
,r'~6-/-5-(2.7,, ;[ -"
2.7-
6-(3.6) "G" [4.2)
/
' I /" I 0+(16 O) I // ' 4 + ( ~ /
. 5=mmr~
/ I
2+ 7-
/
/
I
5, - -
2+
/
] 4'~
I
t
/
! \,
" ) f,,- 2+f 7'5)
2+
I
, ' ~ '
5-
5-(2.6)
, 7-
I
~,/ ,/',,
2.5
,~,-..,,
2+
~
6+
/
//
.4.+.
/
"-'a.~',
~.
4.
3"
2..~ \
2+
',
0+
~1I 2+
2.1
/~\
0+ O+ 2
0+
0+(5.7) --
0
3"
/
~L~J(4.4.)-'.
0+ 2+
-.
2+
/ I /
/
2+
0+
0+(36.2)
0+
0 4.`•
0 +
TDA
STDA
Exp
N3~PI
NI-PI
ALZETTA eL
=I. (Ref. Sd)
CLEMENT ~nd E.BARANGER(Ref.IO)
Fig. 15. The experimental spectrum of lZ6Sn and results of calculations with realistic forces. The first 3- state is predicted above 3.3 MeV in both TDA and STDA calculations ofref. 9d) and is not shown. The numbers in parentheses next to excitation energies refer to four quasiparticle components. K u o a n d Brown 63) c o r e - p o l a r i z a t i o n r e n o r m a l i z a t i o n . The m a i n p a r t o f these calculations was confined to l l6Sn for which several choices o f s.p.e, were e x a m i n e d
EVEN TIN ISOTOPES
365
with both Tabakin and Yale-Shakin interactions, with core excitations through N = Z = 28 as well as N = Z = 8 being included. Recently the Trieste group has used an improved inverse-gap-equation (I.G.E.) method to determine the input parammeters 9a). This way of specifying input parameters may be the most preferable one, but it is still not free of uncertainties, as discussed in detail in ref. 9d). When the I.G.E. method was used the results for the Tabakin and Yale-Shakin interaction were found to be generally very similar for all cases. It was found that the inclusion of four quasiparticle configurations depresses all energies including the ground state energy, so that the excitation as measured from the ground state do not differ significantly between TDA and STDA calculations. B(E2, 2 + ~ 0 +) predictions were found to be actually 30 to 40 % poorer. As illustrated in fig. 15, where results of calculations using the LG.E. method are shown, no significant improvements in fits to the higher 0 +, 2 + and 4 + states were obtained in STDA ,. At the present it is difficult to appreciate the possible improvements obtained in STDA calculations as long as there persist the larger uncertainties due to the choice of input parameters and due to the extent to which core polarization corrections are made. It may however be noted at this point that the discrepancies between experimental 4 + and 4 + states and the two quasiparticle predictions for the same (see fig. 13c) are maximum for 1 2 4 S n and minimum for 116Sn. Also in 124Sn the quasiparticle energies are higher (see fig. 14), so that two quasiparticle excitations overlap considerably with four quasiparticle excitations. It may therefore be expected the four quasiparticle admixtures will be maximum in a~4Sn, and that a STDA calculation for aZ4Sn would be far more instructive than the a a 6 S n calculation made so far. Clement and Barange~ xo) start with the assumption that for most of the states of interest four quasiparticle effects are unimportant. Having neglected four quasiparticle components, Clement and Baranger were able to take a more detailed account of core-polarization by considering the N = Z = 28 core inert, and explicitly considering the next 12 sub-shells as being active for both neutrons and protons. The importance of excitation of the N = Z = 50 core was dramatized by the result that for the 3~ states only 35 % of the wave function was found to be from the usual fivelevel valence neutron space. For the 2 + states the corresponding fraction was 72 %. This explains why 2~- energies are reasonably reproduced in calculations which do not take account of core-excitation, such as those of Arvieu 6) and Kuc 7), whereas 3~- energies are not obtained even approximately correctly. "[he results of Clement and Baranger are shown in fig. 13 and also in fig. 15. In fig. 15 we have presented their results for two different choices of single-particle energies N1-P1 (the same as in fig. 13) and N3-P1 (with neutron single-particle energies being different). From these * G m i t r o et al. 9b) have p o i n t e d out, t h a t the g r o u n d state shift is connected with //4o, which is left o u t in S T D A calculations for j~r :/~ 0 +. If the g r o u n d state shift is ignored (as p r o p o s e d in ref. 9)) the S T D A energy eigenvalues as n o t e d in fig. 15 s h o u l d be lowered by 206 keV. F o r m o s t o f the positive parity states the a g r e e m e n t with experimental excitation energies would t h e n be considerably ameliorated. T h e fit to m o s t o f the negative parity states would however become worse.
366
o. BEERe t
al.
two results it is apparent that the choice of single-particle energies can cause quite large changes in the spectra, and that this represents the largest uncertainty in the present calculations. In the calculations of Clement and Baranger 1o) proton excitations were explicitly considered and it was hoped that B(E2) values could be obtained without use of effective charges. The calculated B(E2) values were found to be a factor eight smaller than the measured values, indicating that deeper excitations of the core play an important part in the enhancement. Similar conclusions about the importance of the lf~ subshell were also reached by Gmitro e t al. 9b). Gillet, Giraud and Rho 66) have recently taken a somewhat different approach to the question of core excitation. In a RPA calculation in the two quasiparticle subspace they have included one particle-one hole excitations of the N = Z -- 50 core. In these calculations single-particle energies and gaussian force parameters were determined by the inverse gap method. The main thrust of the work was to try to reproduce experimental transition rates without use of effective charges. As indicated in table 12, reasonable agreement with experimental values of G 3 was obtained, but G2 values obtained were factors 30 to 40 smaller than the experimental values. These results, in conjunction with those of Clement and Baranger lo), indicate that while core excitations for the 31 states are indeed mostly of the lp-lh type, for the 2~" states more complicated configurations play a dominant role. In summarizing the present status of theoretical calculations we can state safely that while the use of realistic interactions has reduced the number of adjustable parameters, the uncertainty in the input single-particle energies remains a serious problem. Core excitations have been found to have an important effect in determining the energies of not only the collective 2~- and 3~- states but also the higher lying states, and it appears that calculations must at least include shells as deep as the 1~ for both neutrons and protons in order to achieve reasonable agreements in transition strengths, without use of effective charges. In the calculations reported so far for 116Sn four quasiparticle effects have been found to be generally small (they may be larger for 124Sn, but no STDA calculations for 124Sn have so far been reported), and no 2 + or 4 + states lend themselves to a two-phonon description, although the experimental energy systematics of 4 + states is highly suggestive of such an interpretation. We wish to thank Drs. J. Thirion and J. Saudinos for their continued interest in the present investigations. We are grateful to M. Laspalles for work on the improvement of the performance of the spark chamber and to the cyclotron crew for the excellent performance of the accelerator. The authors are also thankful to Dr. E. Baranger for communicating the results of her calculations prior to their publication. One of us (K.K.S.) also wishes to thank Drs. A. Messiah and E. Cotton for their hospitality during the very profitable year of his stay at Saclay.
EVEN TIN ISOTOPES
367
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368 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66)
o. BEER et aL R. K. Jolly, Phys. Rev. 139 (1965) B318 Y. S. Kim and B. L. Cohen, Phys. Rev. 142 (1966) 788 N. Baron, R. F. Leonard, J. L. Need, W. G. Stewart and V. A. Madsen, Phys. Rev. 146 (1966) 861 G. Bruge, Thesis, Orsay (1968); G. Bruge e t al., Nucl. Phys. A146 (1970) 593 P. H. Stelson, R. L. Robinson, H. J. Kim, J. Rappaport and G. R. Satchler, Nucl. Phys. 68 (1965) 97 B. L. Cohen, Rev. Sci. Instr. 30 (1959) 415 J. Saudinos, G. Vallois and C. Laspalles, Nucl. Instr. 46 (1967) 229; O. Beer, C. Laspalles, Y. Terrien and G. Vallois, Rev. Phys. Appl. 4 (1969) 305 G. Charpak, L. Massonet and J. Favier, Nucl. Instr. 24 (1963) 501 F. G. Perey, private communication O. N. Jarvis, B. G. Harvey, D. L. Hendrie and J. Mahoney, Nucl. Phys. h102 (1967) 625 G. R. Satchler, Nucl. Phys. A82 (1967) 273 F. G. Perey, Phys. Rev. 131 (1963) 745 L. Rosen, J. G. Beery, A. S. Goldhaber and E. H. Auerbach, Ann. of Phys. 34 (1965) 96 F. D. Becchetti and G. W. Greenlees, Phys. Rev. 182 (1969) 1190 B. W. Ridley and J. F. Turner, Nucl. Phys. 58 (1964) 497 R. M. Craig, J. C. Dove, G. W. Greenlees, J. S. Lilley, J. Lowe and P. C. Rowe, Nuch Phys. 58 (1964) 515 J. F. Turner, B. W. Ridley, P. E. Cavanagh, G. A. Gard and A. H. Hardacre, Nucl. Phys. 58 (1964) 5O9 G . W . Greenlees, G. J. Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115 R. H. Bassel, R. M. Drisko and G. R. Satchler, Oak Ridge National Laboratory Report, ORNL3240 (1962) and private communication T. Tamura, Oak Ridge National Laboratory Report, ORNL-4152 (1967) and private communication T. Yamazaki and G. T. Ewan, Nucl. Phys. A134 (1969) 81; Ch. Chang, G. B. Hageman and T. Yamazaki, Nucl. Phys. A134 (1969) 110 L. W. Owen and G. R. Satchler, Nucl. Phys. 51 (1964) 155; Oak Ridge National Laboratory Report, ORNL-3525 (1963) unpublished S. A. Fulling and G. R. Satchler, Nucl. Phys. A l l l (1968) 81 V. L. Telegdi and M. Gell-Mann, Phys. Rev. 91 (1953) 169 A. M. Lane and E. Pendlebury, Nucl. Phys. 15 (1960) 39 A. M. Lane, in Nuclear theory (Benjamin, New York, 1964) p. 80 S. Cohen, R. D. Lawson, M. H. Mac Farlane, S. P. Pandya and M. Soga, Phys. Rev. 160 (1967) 903 J. B. French, E. C. Halbert, J. B. McGrory and S. S. M. Wong, in Advances in nuclear physics, vol. 3, eds. M. Baranger and E. Vogt (Plenum Press, N.Y., 1969) to be published S. Yoshida, Nucl. Phys. 38 (1962) 380 G. F. Bertsch, Nucl. Phys. 74 (1965) 234 T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40; A92 (1967) 481 C. J. Veje, Mat. Fys. Medd. Dan. Vid. Selsk. 35 (1966) no. 1 T. H. Curtis, R. A. Eisenstein, D. W. Madsen and C. K. Bockelman, Phys. Rev. 184 (1969) 1162 V. Gillet, B. Giraud and M. Rho, Phys. Rev. 178 (1969) 1695
SPECTROSCOPY
OF EVEN TIN ISOTOPES
BY I N E L A S T I C S C A T T E R I N G O F 24.5 M e V P R O T O N S O. BEER t, A. EL BEHAY tt, p. LOPATO, Y. TERRIEN, G. VALLOIS and KAMAL K. SETH ttt Service de Physique Nucldaire ~ Moyenne Eneryie, Centre d'Etudes Nucldaires de Saclay, France
Received 16 January 1970 Abstract: Elastic and inelastic scattering of 24.5 MeV protons by even isotopes of tin, i16, lxs, 12OSn and 122,124Sn' has been studied with an energy resolution better than 20 keV. Approximately 30 levels in each isotope have been resolved up to an excitation energy of 4 MeV. Elastic scattering angular distributions have been analysed to yield optical-model parameters and the inelastic scattering angular distributions have been analysed in the distorted wave Born approximation (DWBA) using collective form factors to determine l and fl~ corresponding to a large number of the transitions observed. The systematics of the states of each j~r are discussed and the results are compared to recent theoretical calculations. E I
I
NUCLEAR REACTIONS i16, llS. 12o.122,124Sn(p' p,), Ep = 24.5 MeV; measured tr(Ep,, 0). 116.11s, 120. Iz2. 124Sn deduced J, :r, fit. Enriched targets.
1. Introduction The a p p l i c a t i o n o f the techniques o f the BCS t h e o r y o f s u p e r c o n d u c t i v i t y l ) to nuclear physics p r o b l e m s z) has p r o v i d e d a p o w e r f u l t o o l for the study o f structure o f nuclei in which a large n u m b e r o f valence nucleons o c c u p y several shell-model orbits a n d for which the exact t r e a t m e n t o f the extra-core particles with residual forces is prohibitive, if n o t impossible. These techniques can be best tested in single closed shell nuclei. The tin isotopes, extending f r o m 1 lZSn to 124Sn offer a n excellent o p p o r tunity for detailed theoretical a n d e x p e r i m e n t a l investigations. The Z = 50 p r o t o n shell is closed a n d it has been experimentally shown t h a t a p p r e c i a b l e p r o t o n particlehole excitations d o n o t start until E* ~ 4 M e V [refs. 3, 4)]. T h u s below this energy one can assume t h a t states are f o r m e d only by excitations o f n e u t r o n s in the orbits 2d~, lg~, 3s½, 2d~ a n d lh~_. W i t h i n the f r a m e w o r k o f the BCS theory, Kisslinger a n d Sorensen 5) m a d e the initial simplifying a p p r o x i m a t i o n o f a p a i r i n g - p l u s - q u a d r u p o l e residual interaction. T h e y calculated p r o p o r t i e s o f low-lying states o f several single closed shell nuclei, including isotopes o f tin. Since then several theoretical a t t e m p t s have been m a d e to predict m o r e detailed p r o p e r t i e s o f tin isotopes. A r v i e u a n d colt Supported in part by Bundesministerium ftir wissenschaftliche Forschung. Present address: ISKP, Universit/it Bonn. tt Present address: Atomic Energy Establishment, Le Caire, UAR, Egypt. ttt Present and permanent address: Northwestern University, Evanston, Illinois, USA. 326
EVEN T I N ISOTOPES
327
laborators 6) and Kuo and Baranger 7) have used more conventional types of residual two quasiparticle interactions, both in Tamm-Dancoff approximation 6, 7) (TDA) and the random phase appioximation (RPA)6,7). More recently, Arvieu et al.'s work on even-A isotopes has been extended by Sawicki and collaborators s) to include four quasiparticle excitations (which would correspond to two-phonon excitations) using phenomenological 8) as well as realistic interactions 9) and by Clement and Baranger 10) to the explicit consideration of core excitations with the use of a realistic interaction. Thus considerable interest exists in the spectroscopy of tin nuclei. The existing information about even tin isotopes comes from a variety of experiments, but unfortunately most of them have investigated only a few low-lying excited states. This limitation arises partly from the nature of the decays and partly because of inadequate resolution in face of rapidly increasing level density, fl-decay experiments [refs. 1l-15)] from radioactive In and Sb nuclei and 7-ray correlations have provided m u c h of the detailed information on many low-lying states of 1~6,1t8, ~20Sn" Some additional information on these isotopes has been obtained by stripping x6, ~7), pickup 18), and two-neutron transfer reactions 19,20), Since the tin nuclei exhibit important collective properties, Coulomb excitation [refs. 21-z4)] and the inelastic scattering of electrons 2s), protons t8,26-31), deuterons 3z-34), alphas 35,36) and neutrons 37) provide powerful tools for the investigation of their structure. However, most of the existing experimental studies are also confined to detailed investigations of a few low-lying states. It was therefore felt that a systematic high resolution study of the even tin isotopes would be of considerable interest. In this paper we present the results of inelastic scattering of 24.5 MeV protons by the isotopes 116,118,12°,122,124Sn.
2. Experiment The experiments were done with the 24.5 MeV proton beam provided by the Saclay variable energy cyclotron. Fig. 1 shows the plan of the experimental arrangement. The extracted proton beam is focused on the entrance slit 0, and then analysed by a 60 ° magnet. The spread in energy of the beam causes it to be distributed along the plane LH. The quadrupole doublet forms an image L'H' on the target. The scattered protons are analysed by a 180 ° magnet with a spark chamber in its focal plane. In order to use allthe intensity of the beam without loss of energy resolution we employed the method of compensation of magnetic dispersions as described by Cohen 38). In this technique the spatial dispersion of a group of scattered protons in the focal plane of the 180 ° magnet depends (in first approximation) only on the width of the beam at the entrance slit O and is independent of the beam energy spread at the target. In other words the magnetic system is achromatic. As used in the present experiment the "spatial dispersion" was approximately 1.2 mm/10 keV for a central ray radius of 720 mm and a 30 ° inclined focal plane. The spark chamber has been described elsewhere in detail 39). It is triggered by a scintillation counter in a parallel position behind it, permitting the selection of pro-
328
o. BEER e t al.
tons, deuterons and tritons. The spark localization is obtained by the current division method due to Charpak et al. 40). In the present experiment further improvement in spark localization (i.e., better resolution) was achieved by dividing the position signal on-line (using a PDP-8 computer) by the sum signal. This modification also made the resolution independent of counting rates up to 200 cts/sec, which were frequently employed. For most spectra however we were limited b y beam intensity rather than by Cyclotron Anatysis Magnet (1OO*)
.,
Secondary
~
/ ,'
Target
(~'~"'~F
,
,,/
Hj
L .................. o - o
,'
................... ,.................
H
Spark Chamber and Scintittator
Ouadrupotes
Primary Anatysis Magnet (60*)
Fig. I. Experimental a r r a n g e m e n t . &
Sn 118(D.p'; E= 24.5 MeV
tm~ t'~ to •
'
x-~
= 35"
O
i
tD
, ~t~
-.r t-. -.r
~
tn ¢'4
o
,
I
4~)
,"
,
'
'
,
3.0
( - -
2~0
J
"-
1.23 Ex(MeV)
Fig. 2. A typical energy spectrum. T h e s p e c t r u m is for z l a s n ( p , p ' ) at 0 = 35 °. On-line division o f the position signal with the s u m signal was used. T h e energy resolution is a b o u t 15 keV. T h e n u m b e r s on p e a k s refer to the level n u m b e r s in table 5.
the spark chamber capability. The overall energy resolution was approximately 15 keV with on-line division and 25 keV without division, and depended also on target thickness, and angle of observation. Fig. 2 shows a typical spectrum in the case of 11aSh for which on-line division was utilized. The incident beam energy, 24.55 MeV was known only with a 50 keV accuracy, and the exact scattering angle only to within +_0.2 °. If the focal plane calibration were
EVENTIN ISOTOPES
329
independently and perfectly known, these uncertainties would lead to a relative uncertainty of _ 15 keV in the determination of the Q-value for inelastic scattering from a 4 MeV excited state in a Sn isotope. However, by calibrating the system (with the same beam and the same magnetic field) against elastic and inelastic scattering from 12C, which was present as a target impurity, one can achieve a calibration which is nearly independent of these uncertainties in the beam energy and the scattering angle. This method was used in determining the excitation energies from the present experiments. The absolute error for the excitation energies is estimated to be smaller than _ 10 keV. In the case of multiplets or weakly excited peaks the precision is not expected to be so good. TABLE 1 Isotopic composition (in ~ ) of targets used
116Sn aleSn 12°Sn 122Sn
llsS n
x16Sn
ttvSn
11SSn
0.12
92.64 0.37 0.16 0.40
2.87 0.42 0.10 0.38
0.43 33.3 32.92 33.1
0.30 0.55 0.54 0.62
0.02 0.27
t24Sn mixture 1 mixture 2 mixture 3
0.02
ltgSn
t20Sn
2.34 96.62 0.44 0.87
0.42 0.82 0.65 0.75
0.99 1.58 98.21 6.18
1.17 32.69 32.23 32.48
0.40 0.50 0.58 0.50
1.75 32.65 2.43 1.46
124Sn
Thickness (~g/cm2)
0.10 0.13 0.28 88.92
0.42 0.06 0.14 2.05
800 500 1000 1000
1.21 0.14 30.80 0.49
94.74 0.12 0.47 31.3
1000 500 500 700
~22Sn
The isotopically enriched targets were in the form of self-supporting foils. No attempt was made to estimate target thicknesses accurately. These thicknesses were however known to within ___30 ~o from the target evaporation conditions and weighting. Table 1 shows the target compositions. In order to measure accurately the variation of parameters/~2 and ]~3 in the Sn isotopes we also used three samples of mixtures of isotopes. Their constitution is indicated in the last three lines of table 1. For each isotope the elastic angular distribution was measured from 15 ° to 130° in steps of 5 °, and near maxima or minima in steps of 2.5 °. Inelastic angular distributions were measured from 20 ° to 120 ° in steps of 5 °. These distributions are shown in figs. 4-11.
3. Analysis of data 3.1. OPTICAL-MODEL ANALYSIS OF ELASTIC SCATTERING
Optical-model parameters are required for analysis of the inelastic scattering data by the DWBA method. Also since target thicknesses were known only to within + 30 ~ , optical-model analysis of elastic scattering was necessary to provide a more
330
o. BEERet aL
accurate absolute normalization of the cross sections. As is well known the optical potential has the form
V(r)=
Vc(rc)-V(eX+l)-l-iEW~-4Waa' drrd](e.,+ 1)_1 +(h/m~c) 2 V~o 1 d (e ..... + 1 ) - ~ . 1 ,
'rVr
where x = ( r - Ro)/ao, x' = ( r - R')/a', x .... = ( r - Rs.o.)/a .... with Ro = ro A~, e t c . . . , and the Coulomb potential Vc is that due for a uniformly charged sphere of radius Rc = rc A~. The optical-model parameters search was done using a modified version of the search code JIB 41). The program searches for the optimum parameters of the optical potential and the normalization constant K which minimize the quantity:
1 N [ o.,(o3-XOox.(O3 2. Several analyses of proton scattering from tin nuclei 30,31,42, 43) and several systematic studies over a wide range of nuclei 44-46) are available in the literature. We investigated several of these geometries. Two of these, the first due to J'arvis 42), who analysed 17.5 MeV elastic scattering only, and the second due to Satchler 43), who analysed elastic scattering 47), polarization 48), and total absorption cross section 49) at about 30 MeV, gave consistently better fits and lower Z2. The parameters based on Jarvis' geometry gave a factor two poorer Z 2 and led to a total absorption cross section of about 1433 mb for 12°Sn, which is about 11% lower than the experimental value of 1606_+80 mb obtained by extrapolating from the 28.5 MeV measurement [ref. 49)]. Satchler's geometry presented no such problems. We have therefore only investigated this geometry in detail. In the following we describe the results of our investigations starting from Satchler's average parameter set. This set has identical geometry for all real potentials and identical geometry for the imaginary potential but differs from all the previously consideredpotentials in having a finite volume absorption, Ws. We examined this aspect of Satchler's potentials first. In table 2 we show the result of keeping all parameters fixed and searching on V and Wo for a specified value of Ws. As illustrated in this table, in all cases the reaction cross section is now in excellent agreement with the measured value. However, the Z2 uniformly decreases for all isotopes as W~ goes to zero. The trend is very marked and appears quite convincing. Incidentally, it may be noted from table 2 that the quantity Wd+ 1.15 W~ remains constant to within 1% throughout all the best fits in table 2. On the basis of the results in table 2, it was decided to set W~ = 0, and search for optimum values of the geometry parameters. It was found that ro = rs.o. = 1.13 fm improves Z2 somewhat and when a search on V, Wd and a0 is made an interesting trend of increasing diffuseness with ( N - Z ) / A results. This trend is illustrated in
331
EVEN TIN ISOTOPES
TABLE 2 Optical-model p a r a m e t e r s search with Satchler's 4a) average geometry: rc -----1.20 fro, ro = rs.o. = 1.12 fm, ao = a,.o. = 0.75 fm, r ' = 1.33 fm, a' = 0.65 fro, Vs.o. = 6.1 M e V Isotope
116
118
120
122
124
W~ = 3.0 M e V V(MeV) Wa(MeV) ar (mb) Z2
58.70 7.51 1541 14.2
58.77 7.45 1556 8.5
V(MeV) Wa(MeV) cq(mb) g2
57.99 9.31 1558 5.9
58.04 9.29 1574 4.2
59.31 7.90 1586 5.9
59.54 7.88 1601 4.4
59.50 8.88 1641 10.0
58.73 9.57 1600 2.2
58.97 9.58 1616 1.15
59.01 10.47 1653 4.9
58.55 10.21 1604 1.55
58.79 10.15 1621 0.7
58.87 11.01 1656 4.1
58.37 10.69 1608 1.05
58.61 10.73 1625 0.4
58.75 11.52 1659 3.3
58.19 11.27 1611 0.85
58.44 11.30 1628 0.33
58.56 12.09 1662 2.8
W~ = 1.5 M e V
= 1.0 M e V V(MeV) Wa(MeV) ~rr(mb) Z2
57.76 9.90 1563 4.2
57.81 9.91 1579 3.3
V(MeV) Wa(MeV) tr, (nab)
57.55 10.51 1567 2.9
57.57 10.52 1584 2.7
W~ ~ 0.5 M e V
7,z
W, = O
V(MeV) Wa(MeV) tr,(mb) Zz
57.34 11.11 1571 1.9
57.35 11.13 1588 2.3 TABLE 3
(a) Best fit optical-model p a r a m e t e r s for Wsaz = 0, rc = 1.20 fro, ro = r~.o. = 1.13 fm, r ' ~ 1.33 fm, a ' = 0.65 fro, V,.o. = 6.1 MeV, a n d a~.o. = 0.75 fro. (b) S a m e p a r a m e t e r s as (a) except t h a t the v a r i a t i o n o f ao with N - Z was p a r a m e t e r i z e d as described in the text (a)
V Wa
ao aa Z2
11eSn
1laSh
57.05 10.64 0.715 1545 1.3
56.05 10.74 0.723 1568 2.2
120Sn 57.43 11.16 0.744 1609 0.8
122Sn
124Sn
57.61 11.26 0.749 1630 0.4
57.33 11.53 0.787 1695 2.5
57.35 11.26 1631 0.5
57.36 12.21 1675 2.4
(b) ao = 0 . 6 + 0 . 8 3 ( N - Z ) / A
V W'd era g2
56.87 10.63 1545 1.3
56.69 10.78 1571 2.3
57.27 11.08 1605 0.8
332
o. BEER et al.
table 3 and can be parameterized as ao = 0.60+ 0.83(N-Z)/A. The fits corresponding to this last set of parameters are shown in fig. 3. The agreement between the optical-model predictions and the experimental cross sections is generally excellent. This is also indicated in table 3 by the small values of Z2 for these fits. Our final param-
O-et]o'-c 17 ~ .
~ "+
t
116Sn
/'
..
+
,++
\j \+:,,
I ,'~,~j '~ ...x x
o .........
sio.........
;~ .........
'~o
Fig. 3. Elastic scattering fits for final optical-model parameters o f table 3(b) (elastic cross sections divided by Rutherford cross sections).
eters in table 3 are quite similar to those obtained by Makofske et al., in a reanalysis of their data a x). Our imaginary potential depths are in most cases identical to theirs, but our real potential well depths show somewhat less variation than theirs. This is no doubt due to the fact that we have put a large part of this dependence in the variation of a0 and they have kept it constant. This is in accordance with the ideas of Green-
EVEN TIN ISOTOPES
333
lees et al. 5 o), who stress the fact that the volume integral of the real Saxon potential J J A -- V A - 1 ~4 R o 3 [1 +~z2a2o/R2o]
is a constant equal to about 400 MeV" fm 3. The parameters of set 2 with variable " a " in table 3 give Jr/A = 400.6, 400.6, 406.0, 407.7, 409.0 MeV • fm a for the five tin isotopes. Both the absolute values and the trend is in excellent agreement with Greenlees' analysis so) of 30 MeV elastic scattering data with J / A = 4 1 2 + 2 2 MeV" fm 3 and 40 MeV data with Jr/A = 380+20 MeV" fm 3. 3.2. C R O S S - S E C T I O N N O R M A L I Z A T I O N
As mentioned earlier, sample thicknesses were only known to within -t-30 ~ . Accurate cross-section normalizations were therefore obtained from the optical-model fits to the elastic scattering cross sections. The reliability of these optical-model normalizations was tested as follows. F r o m our experiments with targets of accurately known isotopic mixtures, the ratios of inelastic cross-sections for the first 2 + states were determined to within +__5 ~ . By comparing these ratios with the relative differential cross sections for the same 2 + states obtained for the separated isotope targets, we could obtain ratios of the thicknesses of the separated isotope targets. This gave us the ratios of elastic scattering cross sections to within + 5 ~ . The same ratios could also be obtained from the independently normalized optical-model elastic scattering fits. It is gratifying to note that the ratios obtained in the two ways were identical within _ 5 ~ , i.e., the independent optical-model fits (with a given geometry) accurately reproduced the known ratios of elastic scattering cross sections. This gives us confidence that the absolute cross section normalization given by these optical-model fits is accurate to within +__15 ~ . It is possible, though we consider it highly improbable, that for all isotopes our seach program for the optical-model parameters has "locked in" on a non-physical X2 minimum and thus given us a uniformly wrong normalization *. In this case our absolute cross sections may have a higher error, but none of our conclusions regarding the variation of transition strengths from isotope to isotope will be altered. 3.3. D W B A A N A L Y S I S O F I N E L A S T I C S C A T T E R I N G
Differential cross sections were obtained for the inelastic scattering, except for the weakest levels or the very poorly resolved levels. These are shown in figs. 4 through 11. Integrated inelastic scattering cross sections were obtained as f130 °
O'tota1
=
27~
130 °
dO sin O[d~r(O)/da] ~ ~sTC2 Z 20 °
sin 0(da(0)/dO)
20 °
and these are indicated in tables 4-8. The true integrated cross sections from 0 = 0 ° to 180 ° are expected to be approximately l0 ~ larger than these. • Recently ( O a k Ridge N a t i o n a l L a b o r a t o r y R e p o r t , O R N L - 4 2 5 2 , 1968, u n p u b l i s h e d ) , Bertrand, Love, Baron, Percy a n d Dickens have reported elastic scattering cross sections for 20.6, 25.2, 30.6 a n d 36.2 M e V p r o t o n s on 12°Sn with 10 ~o absolute errors. O u r cross sections at 24.5 M e V are unif o r m l y ~ 20 ~ higher t h a n their cross sections at 25.2 MeV.
o. ~B~ERe t al.
334
(RP') Sn 116, Ep = 24,5 MeV '
~t~
1.29 MeV
23 ov
,
2.27 MeV
I
10; ~ /jJlriJ21111J
"~ r ~1 I J l l l IT'I 'e I F I J l l ~ l J [ l l 2.53 MeV
, ~
2.64 MeV
10;
1021
2.79
,~
1Z,+1,(5-1 .03 MeV
MeV
I
,-o _ / ~
,
,,i j ', ', ', ', ~ ', ', (3-) 3.32 MeV
(2+1 3.21MeV
2+
/~
3.40 MeV
/(~
3.80 MeV
102
\ i lol
I0
Illllll~llJI
I [ I J ~ I I/,+r J
(t, +) 3.50+3.51MeV
2+
1oi i i ~ l ~ l r l l l
50
1~
i
: I [ I I I I f 1 I 1 I I 50 100
I
r
i
I
sO
t
I
I
I
f
loo
I
I
e*cM Fig. 4. Inelastic scattering differential cross sections for states (with 1 ~ 5) in 116Sn. T h e experim e n t a l points are s h o w n with their error bars. T h e solid curves are D W B A predictions for final optical-model p a r a m e t e r s o f table 3(b) a n d G~ listed in table 4.
EVEN TIN I$oTOVES
335
(PP') Sn 118 . Ep = 24.5 MeV
~#~k2.2/'9+MeV
2+
~,
1.23 MW
3-+6"
2.33 MeV ~ ,
I l l I I I I l I l l t l l j l l l l i i l
,.~, T I J I I I I T J l l l l l J l [ l l i l l
2
U MeV
10-"
I
/,+
i
i
+
I
i
i
f
i
J
I
1
,
I
,
,
,
,
11
I
t
I
J
I
I
l
I
l
[ i
I
i
2.so M.v
I
I
f
}
I
I
l
I
l
I
i
i
i
i
J
I
[
i
I
i
i
i
10 2 ~ . 9 2 2 "
2.~
I
- - DWBA 3........ DWBA3% 5-
t
I
z 9h'.0v
MeV
10:
(,,. +)
13-1
+5 ,:;+
102
3.47 MeV
MeV
°E,,
+
ll[[llIIIll
1
l l J T I l l l l l 50
IiItlt,tEllt
=',;1:lltltl
I t l l l l l l I l l i l J [ l l l i l l
I_ 100
I
I
I l l l l I l l l l l&*z
3.68 MeV
I
I
I
I
I
I
50
I
I
100
3.79
I
I
I
I
I
I
1
50
[
I
r
MeV
I
I
I
t
100
0 CM o
Fig. 5. Inelastic scattering differential cross sections for states (with 1 ~ 5) in 11SSn. The experimental points are shown with their error bars. The solid curves are D W B A predictions for final optical-model parameters of table 3(b) and G, listed in table 5.
336
o. BEER et al. ( P P'} $n 120,Ep= 2/,.5 MeV
/\
2.
4 +
1.17MeV
10:
2.20 MeV
½
52.29 MeV
10 i
,
,
I
ITS '
I
I
I
I
I
I
( =
I
~ ' ' '2+' i ' '
'/IZ~
3-
2.36 Ne¥
1 J
I i
I ,
I I
I I
I I
I =
f J
I I
I i
I i
4+ + 72.47+2.49 MeV
103
exp. distributions
""~"i"
io2 i ,
h
i
I
i
I
I
i
l
I
,
I
4÷
~II~+
3.07 MW
r
I
I
t
I
('~
, , , f , , , l l i [ l l l l J i l
F
2.72+MeV
.
.
.
I
=
=
I
s
I
2/v
.
of ....
,
i
~d
I
I
I
I
I
I
I
I
I
I
I
I
I
4+ 3.19 MeV
~ ,
IO~
I I I l l l l l l l l l I l l l l l ~ l l l l
t i
I I
i I
I I
r '
I I
I ~
; '
I i
I '
; '
4" 88 MeV
f
l
t
l
l
l
l
l
l
l
l
f
I
I
I
I
t
I
50
I
I
I
IO0
I
i
I
I
I
I
I
50
I
I
I
I
I
100
eOcM Fig. 6. Inelastic scattering differential cross sections for states (with l _~ 5) in x2°Sn. The experimental points are shown with their error bars. The solid curves are D W B A predictions for final optical-model parameters of table 3(b) a n d G~ listed in table 6.
EVENTIN ISOTOPES
337
PP') Sn 122,Ep= 2/,.5 MeV
#
I i
i s
t i
) I
I i
I i
I i
i i
I j
i J
-
4+
"
I 1
I~
,
I
t~
I0~
i
I
,7-+
i
.5"
i
i
l
,
i
i
t
I I
i I
I
i
l i
2÷
I.~......% ...........Mixtureof two
~ + ~"3;,. T ~,.~,
exp. distributions =
lo' -
I
I
t
I
t
I
i
I
I
1 =
t I
t i
l I
I i
z
I
I
t
I
t
~
i
l
~
]
i
i I
i I
i i
i l
I i
l I
t
I
t
,
I
t
I t
I I
j
4
L
,
,
]
i
i
t
(s;~,+) 3.08
MeV
t l
i I
10 , I
~ l
~ I
i l
I
I
I I
i I
I I
I I
I I
-
: /;+
'~
] t l l l t l l l l l t L I I t i l l I L i i t
1 i
33.3?MeV
3.31MeV
(3-) 3.46 ~V
/. lo
-
,
i
,
i
I
i
l
•
I
I
I
~
(z.+)
•- ~
t
I
3.68 + 3.71MeV
1
t
i
I
,
t
1031
t
t
I
i
12+)+
(~+1
3.90
MeV
t
t r l t ~ l t l l l l l
(5-) 102
'.k+..,.
'+,
..........
Mixtureof t'~v~"~..~. exp.distributions
I
I
I
I
50
[
t
I
I
I
I(30
I
I
i
i
1
i
1
i
''~.~, i
50
I
t
I
i
I
I 1 T t l t l l I I 1 1
10(3
e~M Fig. 7. Inelastic scattering differential cross sections for states (with l ~ 5) in 122Sn. The experimental points are shown with their error bars. The solid curves are DWBA predictions for final optical-model parameters of table 3(b) and Gz listed in table 7.
338
o. ~E~ et aL
( P P ' ) Sn 124.Ep= 24.5 MeV
I 10~
~t.+-+-.+,++i J
i
~
[
i
i
~
/~
i
I
I ~°~ i
t
r
,
I
J
'
'
2+
/f
33.01 MeV
~
i
I I
r i
'
I
I I
!
i
i
I
I
I
2.71 M~
I II II II IJ I l Il ll il lt ll ll
i
I.+ 3.16 MeV
I ~
i I
I I
t I
I I
3.23 MeV
~ i
i I
l l l i l l i i 33.52 M~
4+
3.37 MeV
(4+1
~ tO~
.~
I i i l a i l l/,+l r I
i
'
32.61 MeV
*'~
i
~
" ~ t
3.42 MeV
i
llrliifIlii
i
i
I
t
50
i
i
I
I
I
100
I
I Ii It ll l l l i l l l l l i Jl ll ii l
i
I
f
I
J
,
I
50 e o ¢M
I
t
I
100
i
I
IIII
50
100
Fig. 8. Inelastic scattering differential cross sections for states (with l --< 5) in 124Sn. The experimental points are shown with their error bars. The solid curves are DWBA predictions for final optical-model parameters o f table 3(b) and G, listed in table 8.
EVEN T I N ISOTOPES
339
There is a large body of evidence which points to the fact that the main part of medium-energy inelastic scattering, at least for the strong transitions, comes from a direct interaction. One may therefore use DWBA for analysing inelastic scattering angular distributions as one does for particle transfer reactions except that the form factors in this case may be based on the collective model. In the collective model one attributes the low-lying states to rotations of a statically deformed shape (with deformation fit) or to oscillations in shape about a spherical mean (vibrations, with mean deformation fit)- One expects the optical potential to follow the shape of the nucleus and to be therefore correspondingly distorted. For convenience it is generally assumed that the distortion parameter fit is the same for all parts of the optical potential, and also that the potentials for the entrance and the exit channels are the same. The transition matrix elements are then completely determined by the radial derivatives of the optical potential which describes elastic scattering. The deformation parameter fl~ can be obtained by comparing the experimental differential cross sections with the DWBA predictions, since (dcr,(O)/df2)~xp = fl?(daJdO)DWB A. The DWBA calculations used to analyse the inelastic scattering angular distributions reported in this paper were made using our final optical-model parameters of table 3(b) in the code JULIE due to Bassel et al. $1). Sometimes in similar analysis only real form factors are used and Coulomb excitation is neglected. We have therefore examined and verified the importance of including Coulomb excitation (which arises from distortion of the Coulomb part) even at our relatively high energy (the effect is largest for l = 2 and for 0 <= 30°), and also the fact that the use of complex form factors (i.e., distortion of both V and W) is necessary. We have also investigated the effect of optical-model parameters on flz. It was found, for example, that for the best fit potentials based on Jarvis' geometry and Satchler's geometry, the peak cross sections differ only by about + 5 ~o, although the shape at back angles are often quite different. We have made several calculations using Tamura's code JPMA[N 52) for coupled channel analysis of the strong 2 + and 3- transitions, and concluded that, for the small values of fll involved in our analysis, the results of the coupled channel (CC) calculation do not differ sufficiently from the DWBA calculations to warrant the use of the CC method, which is certainly more time consuming. DWBA analysis of the differential inelastic cross sections yields two primary results, /-assignments and the transition strengths based on the one-phonon collective model, We discuss the /-assignments first. Angular momentum transfer 1 < 5. In figs. 4-8 our data for the differential cross sections are shown. Also shown there are the predictions of the DWBA calculations for the assumed/-transfers. For l < 5 there appears to be little doubt in the correctness of the/-transfer assignments. Very small changes in the shapes of the theoretical curves can be made by using different optical potentials, but the main features remain
2+
1.293
4+
4+
2+
4+
2+
7-
2.390
2.526
2.637
2.787
2.825
2.890
0.28
(8 + , 7 - )
(4 + )
(2 + )
2.977
3.027
3.074
0.3
1.4
0.17
2.941
1.0
0.85
2.5
3.9
1.1
7.0
4.0
1.6
0.35
4.1
1.7
2.9
8.5
5-
3.047
2.959
2.845
2.803
2.649
2.531
2.391
2.366
2.267
2.367
36.4
3-
2.269
15.0
2.108 2.224
1.762
1.291
E,
(2 + )
0.27
20.4
Gt
2.230
0.17
14.9
o'tot(mb)
(P, P') 24.5 MeV
2.110
1.755
jrr
E~
27) (P, P') 11MeV
This work
TABLE 4
3.04
2.90
2.78
2.53
2.27
1.28
Ex
4+
7-
4+
4+
3-
2+
j~r
(9, P') 55 MeV
t8)
5.2
3.2
5.5
4.2
34.5
18.5
Gz
2.87
2.60
2.27
2.24 3-
0+ 0+
1.76
2+
jlr
2.03
1.29
Ex
(p, p ' ) analog
29)
3.04
2.80
2.58
2.37
2.22
2.04
1.77
1.28
Ex
4+
(4 +)
4+(0 + )
5 - ( 4 +)
2+
0+
0+
2+
jrr
(p, d) (p, t) 55 MeV
18)
2.95
2.78
2.62
2.37
1+2+3 +
2+
0+
1 +2+3 +
2+
0+
2.23
0+
1.76
2+
J"
2.03
1.28
E,
(d, p) 15 MeV
16)
S u m m a r y o f experimental results for ~16Sn. Gz < 1 are not quoted
53)
2.913
2.777
2.393
2.368
2.268
2.00
1.293
E~
7-
6-
4+
5-
3-
2+
J"
(a, xn) 20-50 MeV
3.037
2.914
2.802
2.778
2.530
2.392
2.369
2.269
2.20
2.112
2.00
1.757
1.294
E~
fl-y
4+
7-
4+
6-
4+
4+
5-
3-
2+
2+
0+
2+
J"
ll-lS)
3)
E~
(3He, d)
t~
t.n t~
p
Y.
0.7
0.5
2+
(6 + )
2+
3.735
3.763
3,796
3.845
3.912
3,577
2+
2+
3.80
3 64
1-2-
1+2+3 +
1+2+3 + 3.232
(8 + )
3.22
(7-)
4.075
4,017
3.938
3.783
3.731
4.268
3.97
3.72
3.35
3.17
3.10
4.220
3.951
3.918
3,846
3,808
3.771
3,744
3,656
3,621
2+
4+
4.203
2.8
3.43
3.14
4,272
0.5
0.5
4.019
4.085 4.157
0.6
0.4
3.947
(5-)
0.8
3.686
0.4
1.1
0,3
3,654
0.45
0.85
3.632
4+
0.70
3.513
3.572
3.509
3,468
2.3
0.40
3,337
3,319
0.70
(4 + )
1.5
3,453
2+ (4 + )
3,404 3.436
0.72 0.35
3,230
3.499
(3-)
3.320 3,359
0.75 0.35
3,091
3,423
(2 + ) (8 + )
3,212 3.257
1.0
0.36 0.40
(6 +)
3.090
2.743
15
2.3 --
2.96
--
--
--
2.877
2.733
2.576
2.489
2.405
2.324
7-
4+
5-
2.73
2.67
2.54
2.49
2.32 2.38
4+
0.95
2.959
2.921
2.962
2.927
2.92
2.989
0.4
2.901
23
(2 + )
2.892
2.944
22
0.63
2.917
21
2+
2.86 2.89
2,896
19
20 2.881
2,74
2.49
2.33
2.81
2.769
2.728
2.671
2.485
2.398
2.321
2.84
2.6
4,5
2.4
31.0
4+
2.05
1.75
1.22
E,~
18
0.41
2.0
0.4
0.8
1.0
0.5
12.5
2.277
0+
0+
2+
j~r
17) (d, p) 11 MeV
17
(8 + )
4+
2+
2.686
14
16
7-
4+
2+
3- ~(5-)
2.583
2.497
2.409
2.328
13
12
11
10
9
8
7
6
1.1
1.759
1.233
Ex
2.283
2.285
+
~
5
1.99
1.74
1.22
Ex
20) (t, p) 12 MeV
2.064
4+
1.229 2.032
0.1
17.8
a4) (d, d') 15 MeV
4
2.050
13.2
3
2+
1.233
Ex
1.740
1
Gl
(p, p') 24.5 MeV
a(mb)
27, 28) (p, p') 11 MeV
This work
2
j~r
E~
TABLE 5
2,72
2.49
2.32
2.05
1.75
1.22
Ex
2+
0+
1+2+3 +
0+
0+
2+
j1r
(d, p) 15 MeV
16)
S u m m a r y of experimental results for 11SSn. Gz < 1 are not quoted 16)
2.72
2.47
2.30
2.03
1.74
1.22
Ex
(d, t) 15 MeV
53)
2.580
2.326
2.285
1.231
E~
7-
5-
4+
2+
j~r
(c¢, xn) 28-50 MeV
2.96
2.77
2.72
2.57
2.48 2.48
2.32
2.28
2.05
2.04
1.75
1.23
E,,
5-
4+
0+
2+
0+
2+
jTr
(4 + )
7-
0+ (4 + )
fT
II, 13-15)
.
.O
t.o
26
0.5
0.8
3.689
3.650
1.1
3.857
3.893
47
48
0.4
4.041
0.5
53
4.069
3.977
3.932
3.870
3.804
3.757
4.008
0.25
0.55
0.3
0.35
52
2+
4+
51
50
3.960
3.81t
49
3.789
45
46
3.746
(6 + )
0.4
3.581
4.02
3.91
3.89
3.79
3.70
3.57
3.575
1.9
3.55
3.555
3.52
3.47
3.38
3.520
44
3.715
42
4+
1.0
0.42
0.35
3.421
3.720
3.677
41
2+
(6 +, 7 - )
(4 +)
0.62
43
3.605
3.551
3.471
40
39
38
37
36
35
3.36 3.369
3.393
0.36
34
3.30
3.15
3.13
3.34
0+
3.05
3.362
(3 - )
3.136
3.055
33
3.06
32
0.34 3.332
3.260
3.320
0.65
31
(7-)
3.240
3.214
3.126
3.048
3.287
0.3
0.1
0.1
30
(8 + )
2+
29
28
3.241
3.148
25
27
3.073
3.113
24
4.04
3.91
3.79
3.70
3.06
1+2+3 +
3-, 4-
1 +2+3 +
1 +2 +3 +
4.01
3.89
3.80
3.67
3.04
3.058
0
Q ,-.-I
,z
4+
5-
2+
3-
(4 + )
(7-) (5-)
4+
(8 + )
2+
2.199
2,287
2,361
2.408
2.470
2,486 2,547
2.653
2,694
2.735
2.809
0.2
0.3
0.5
0.2
0.9 0.15
1.2
10.4
0.58
2.4
1.1
11.4
a(mb)
26.4
1.0
7,8
1.9
16.6
Gi
2.67
2.455
2.391
2.272
2.183
1.166
E~
4+
4+
3-
5-
4+
2+
J~
(p, p') 24.5 MeV
( 7 - , 8 + ) 0.15
2+
1.167
2.76
j~r
E~
,2) (p, p ' ) 17.8 M e V
This work
2.632
2.455
2.391
2.346
2,272
2.72
2.63
2.44
2.31
2.23
1.87 2.11
1.872 2.088 2.183
1.18
Ex
33)
(2 + )
(0 +)
3-
(0 +)
2+
J~
(d, d') 15 M e V
1.166
Ex
(p, p') 11 M e V
2~)
2,693
2.591
2.478
2.416
2.402
2.361
2.289
7-
5-
4+
0+
2.150 2.201
0+
2+
J~
1.881
1.174
Ex
(t, p) 13 M e V
20)
2.81
2.61
2.43
2.36
2.29
2.17
1.88 2.10
1.17
Ex
(d, p) 11 M e V
1~)
2.73
2.60
2.42
2.31
2.17
1.88
1.17
E~
1+2+3 +
0+
1+2+3 +
0+
0+
0+
2+
J~
(d, p) 15 M e V
~6)
TABLE 6 S u m m a r y o f experimental results for 12°Sn. Gt "< 1 are n o t q u o t e d s3)
2.490
2,280
2,180
1.171
E~
7-
5-
4+
2+
J~
(ct, x n ) 28-50 M e V
2.65
2.50
(2.46)
(2.36)
2.30
2.20
0.88) (2.12)
1.17
Ex
7-
5-
4+
2+
d~
fl, 7
I J. i , )
"~
r~
9
4~
0.3
0.25
0.45
0.25
0.5
3.962
3,995
4.080
4,190
3.986
3.926
3,860
0.25
2+
3.885
3.935
3.777 3.811
3.1
4.17
4.10
4.02
3.90
3.82
3.74
3.710
3,53
3,44
3.32
3.24
3.12
3,67
0.53
0.95
2.89 2.99
3,646
3.574
3,540
3.445
3.266
3.223
3.169
2.963
2+
(3-)
4+
4+
4+
3.45
3.17
3.06
3.796
0.75
3.664
1.6
6.0
2.1
2,830 2,919
3,840
0.35
3.592
(6 +, 7 - )
0.3
0.8
3,566
3-)
0.5
3.445
3,467
0.25
3,392
0.8
0.2
2 +)
(6 +, 7 - )
3.288
2.1
3,330
3,186
4+
4+
3.069
3.168
0.4
2,938
0.68
0.4
2 +
2,850
(2 + )
(2+)
3.818
3.780
3.593
3.475
3.183
2,852
2.84
4.19
4.06
3.99
3.94
3.88
3.80
3.73
3,71
3.66
3.60
3.58
3.55
3.47
3.39
3.29
3.21
3.16
3.00
2.94
4.15
4.03
3.87
3.70
3.56
3.23
2.94
2.84
1+2+3 +
1+2+3 +
1+2+3 +
1+2+3 +
1+2+3 +
1+2+3+
1+2+3 +
1+2+3 +
3.80
3,57
3.39
3.18
3.06
(4+)
0.-] 0
346
O. BEER et al. TABLE 7 S u m m a r y of experimental results for 122Sn. G~ < 1 are not quoted
Ex 1.143
This work
27)
34)
29)
26)
11)
(p, p') 24.5 MeV
(p, p') 11 MeV
(d, d') 15 MeV
analog
(p, p') 12 MeV
fl~,
jz" 2+
2.090
o-tot(mb)
Gl
10.0
15.0
0.16
Ex
E,~
~
1.142
1.15
+
2.103
Ex
jrr
Ex
Ex
1.14
2+
1.141
1.14
2+
2.08
(0 +) 2.14
(4 + )
2.142
4+
1.60
3.5
2.155
2.15
2.147
2.245
5-
2.7
8.7
2.239
2.25
2.247
2.328
4+
1.3
2.6
2.336
2.390
(7-)
1.2
2.412
2+
2.0
3.6
2.418
2.42
2.492
3-
8.9
22.6
2.496
2.50
2.556
(8 + )
2.260 2.34
2.30 2.41 --
0.25
2.684
0.30
2.750 0.14
3.038
2.86
0.15 (5-, 4 + )
0.3
2+
0.95
1.3
3.128
3.15
--
3.237
4+
1.3
4.2
3.236
3.26
--
3.283
3.33
--
3.313
4+
1.1
3.4
3.367
3-
0.6
3.457
(3 - )
0.3
3.478
(7-)
0.3
3.477
3.533
0.35
3.529
3.564
0.35
3.584
0.25
3.361 1.3
3.459 3.49 3.56 3.633
(4 + )
3.708
0.65 0.50
3.71 3.731
3.773 3.818
0.5 (6 + )
3.778
0.44 3.841
3.85
3.879 3.900
(2 + )
3.978 4.104 4.185
(0 +)
2.970
3.135
3.675
2.68 2.75 2.870
3.084
3-
0.49
2.654
2.976
1.1 0.75
(5-)
2.414
2.50
0.3 0.5
3.903
3.92
3.14
2.494
347
EVEN TIN ISOTOPES TABLE 8 S u m m a r y o f experimental results for 12"Sn. Gz < 1 are n o t q u o t e d This work (p, p ' ) 24.5 M e V Ex
jz,
atot(mb)
1.139 2.107
2+ 4+
9.0 1.65
2.130
(2 + )
0.8
27) (p, p ' ) ll MeV
34) (d, d') 15 M e V ~
29) analog
GI
Ex
Ex
14.0 4.9
1.132 2.105
1.13
2.133
2.13
2.12
+
.Ex 1.132
jz, 2+
26) (p, p ' ) 12 M e V E~ 1.131 2.101
2.199 2,213
5-+(4 +)
3.5
(9.9)
2.217
2.21
2.20
2.200
2.431
2.41
2.43
2.428
2.605
2.59
6 7
2.335
7-
1.35
8 9
2.435 2.455
2+ (8 + )
1.7 0.55
2.7
10
2.613
3-
7.6
19.6
11 12
2.713
(4 + )
0.84
1.8
2.85 2.900
+
0.34
2.879 2.952 2.988
2.98 3.13
--
3.215
3.19
+
3.35
--
15 16
3.009
3-
0.4
17
3.158
4+
0.34
1.0
18
3.232
2+
1.1
1.8
19
3.282
3.266
20
3.324
21
3.366
4+
1.55
5.4
3.359
22 23
3.416
4+
0.9
3.1
3.421
24
3.516
3-
0.67
1.5
2+
3.485
25
3.577
0.35
3.557
26
3.602
0.18
27 28
3.649
0.12
3.605 3.636
29
3.702
30
3.653 (6 +, 7 - )
0.75
3.698
3.759
0.56
31
3.794
0.27
3.746 3.775
32
3.827
0.27
3.806
33 34
3.879 3.909
(6 + ) 2+
0.8 0.9
35 36
4.019 4.159
(2 + ) 2+
0.5 0.5
37
4.281
(2 + )
38
4.350
0.5 0.6
2.59
3-
2.69
(0 +)
2.708
13 14
--
2.678
2.85
3.19
2.604
348
o. BEERet al.
the same. In particular the location o f the maxima, and more characteristically the angular position o f the first rapid fall-off o f the differential cross sections remain unchanged. As a matter o f fact it is quite surprising that g o o d fits are obtained even for the higher excited states for which no serious meaning can be attached to the single p h o n o n collective model form factors. The fits to these states emphasize what has '
'
'
'
I
J
i
i
,
1=7 ---
m3
exp.
distribution
of 2.89
+ ,
Sn 116 2 89 MeV {xO.2)
Tr-H44t.,"
;
of Sn 116
level
Sn 118 2.58 IO,eV ( x 0.5)
Sn
3~ 29
118 Mev
(x2)
~,\ ,,
Sn 122 3.48 N',eV {x,'. )
%
....
10-1
.,,
,
,
~
,
I
50
~
,
f
,
I
100
Sn 124 2.3" MeV (xLOO) ,
~
,
,
I
,
,
,
150
0%.
Fig. 9. Inelastic scattering differential cross sections for states corresponding to 1 = 7. The dashed curves correspond to the shape of angular distribution for the 2.890 MeV level in zleSn, known to have j~r = 7-. been often noted in particle transfer reactions, namely that the D W B A calculations reproduce the general features o f the angular distributions (especially the first maxim u m which is characteristic o f a given/-transfer) even though the detailed form factor for the state in question may be quite different from that assumed in the calculations. Angular momentum transfer l > 6. D W B A calculations were made for l = 6, 7 and 8 also. However, since no definite 6 + or 8 + assignments exist in the literature
(
tt
'
't'
,
,
50
I
~
,
~i\
~
~
~
0%.
tO0
I
_I L . '~>~+ TI !~+~'+- "..._!t!~.
~
~"~---~ H--.
h m'~-h. / ~'
~
t
'
,
i ',.._thi. ~
f.,
~,,
•
-+-+-+÷+h-, f~-i-H'-L~!~.. +"'~,'
~'+,
'
"-~'i I "~ t
"/L~'+'it
I
"~, ++~
"k,+ ,
"T,,&
. . . .
t f ~"
.--Lt+..
'-'~--a--L
~
Sn
118
~
~
(x 25.102)
3 . 7 0 MeV (,7.102 )
~o,~,
~
,,10~,
,22 ~v
so 3,~2
M~
I × ~o )
3,~
~o ,2o
3.72 MeV (xlO)
150
I
so 116
. . . .
3.5s Mev
'' [
•
/
Fig. 10. Inelastic scattering differential cross sections for states considered to be good candidates for the assignment j r = 6 +. F o r three levels (3.55 MeV o f ~taSn, 3.66 MeV o f zz°Sn and 3.70 MeV o f ~Z4Sn) an l = 7 transfer cannot be excluded. See also text.
G
D
102
'
10-I
10
-
-
-
t
I
--
. . . .
..
.
-
-
so
...
. -..I'~
" -
14o~',..
%,
"
t!~t[~L!~"~i T't't?t]
._
,~
t-t . .t . .t . t t t " ~- t-~- ~" t't'~t / t ""f'~ l ""* -
+-H+~-++4,
"
]4ft 4tf t,," 'r"t?
•
U_L~]]_tt t
.
........
. . . .
. . . .
,:8
2.81 MeV (x240)
15o
s . MeV 124 2.46 (xlO2)
Sn 122 2.56 MeV
Ix6)
Sn 120 .2.69 HeY
32,,x2,MoV
2.90 MeY
(xO.07)
~o.~
(x 0.03)
2.98 IVleV
Sn 116
of 2.56 levelof Sn 122
I
Fig. 11. Inelastic scattering differential cross sections for states considered to be good candidates for the assignment j~r = 8. For two levels (2.9S MeV of 116Sn and 2.81 MeV of izo Sn) an I = 7 transfer cannot be excluded.
G
10:
10~
I
~D
o. BEERet al.
350
no comparison could be made with experimental angular distribution to a k n o w n l = 6 or 8 transition. However, the first 7 - states in lt6Sn (2.890 MeV) and xaSSn (2.583 MeV) are well resolved and well established f r o m other experiments referenced in tables 4-8. In fig. 9 we show the observed angular distributions corresponding to these states. We notice that the two angular distributions are characterized by a sharp change in slope at 0 = 80 °. The D W B A predictions are not shown because they did not fit these angular distributions even qualitatively. In fig. 9 we also show angular distributions corresponding to the 3.29 MeV state in ltSSn, the 3.48 M e V state in 122Snand the 2.34 MeV state in 124Sn.As illustrated, their shapes are similar to those o f the transitions to the k n o w n 7 - states. We therefore assign these other levels as 7 also. These assignments must of course be regarded as tentative, although f r o m the systematics there seems little d o u b t that the 2.34 MeV state in tZ4Sn is indeed a 7 - . We mentioned earlier that the angular position of the first sharp decrease in a(O) appears to depend quite characteristically on/-transfer. F o r I = 4 this occurs at about 45-50 °, for l = 5 at about 55-60 ° and for 1 = 7 at a b o u t 75-80 °. We expect, qualitatively, that for l = 6 this should occur near 65-70 ° and for l = 8 near 85-90 °. Sample D W B A calculations bear out this expectation. Several states are found for which the measured angular distributions display these trends. The differential cross sections for the states with the break at 65-70 ° are shown in fig. 10. We tentatively assign these states as 6 +. In fig. 11 we display the observed differential cross sections for states which appear to be g o o d 8 + candidates, based on the criterion described above. As a matter o f fact it is n o t in all cases possible to discern unambiguously between 6 + and 7 - and between 7 - and 8 +. Transition strengths. Since no g o o d correspondence with D W B A calculations could be established for l _-> 6, we have derived transition strengths only for states with I < 5. These are listed in tables 4 through 8 in terms o f G~. Since different authors quote their results in different terms, we present below the relations between them *: B(EI) =
(3ZR'/4n)2fl]; G,-
B(El)s.p" - 2 l + 1 ~ 3 R t l 2 4re L3 + IJ '
B(E/) _ ,8~ B(El)s.p. flt2(s.p.) "
(1)
(2)
In particular for Sn isotopes G(l) = Kfl 2 where K = 995, 1023, 1083 and 1154 for l = 2, 3, 4 and 5, respectively. A t this stage it is worthwhile noting that analysis o f our data using collective f o r m factors, which correspond to o n e - p h o n o n excitations, is meaningful only for collective states such as the 21+ quadrupole and 3~- octupole vibrational states. It m a y be reasonable even for the 5~- states which show definite enhancement. But the use o f t Some authors 42) define B(EI) with an additional multiplier (2/+ 1)- ~ on the right-hand side. Barreau and Bellicard 2s) define/31 still differently. [fit (Barreau) = 9fl~z (above)/(2l+ 1)]. In all cases, however, quantities in single-particle units come out to be the same.
EVEN TIN ISOTOPES
351
these form factors is clearly not justified for weak 2 + states, most of the 4 + states, or 6 + or 8 + states. Some of these are quite pure 2 quasiparticle (q.p.) states. Some may involve excitations of four or more quasiparticles (corresponding to multiphonon excitation). In these cases G l values obtained by the DWBA analysis used here do not have much physical significance. They merely represent a convenient parameterization of the measured differential cross sections. In our tabulated results we quote G~ values only when they are larger than unity.
4. Experimental results We present the results of our experiments in tables 4-8, along with the relevant results from other experiments. The results are also summarized in fig. 12 where the length of the lines indicates Otot as defined earlier and the circles represent some of the 0 + states not seen in the present experiments but known from other work. 4.1. E X C I T A T I O N E N E R G I E S
Certain general features of the excitation energies and the nature of levels excited in our experiment may be noted from tables 4-8. To begin with, our excitation energies are, in general, in excellent agreement with those from other spectrograph experiments which were done with l0 to 20 keV resolution widths. These include Allan's (p, p') experiment for all Sn isotope s 27,2s), Norris and Moore's (d, p) experiments ~v) for 1laSh and 12°Sn, and Bjerregaard et al.'s (t, p) experiments 20) for 11SSn and ~2°Sn. One may also note that the excited 0 + states which are seen in (d, p), (t, p) and in some fl-? experiments are only weakly (if at all) populated in our experiments. Non-natural parity states are also very poorly (if at all) populated in our experiments. It may also be noted that our higher energy enables us to excite certain states not seen in low energy inelastic scattering experiments 27,28). These are generally the states with/-transfer > 6. 4.2. j~r A S S I G N M E N T S
We have assumed throughout that in our experiments only natural parity states, i.e., those with J[~] = /[re = ( - 1 ) z] are excited. ]'his assumption is certainly true for the stronger transitions, since spin-flip scattering as well as two-step excitation are expected to be weak and to have non-characteristic angular distributions, but it may not be completely true for the weaker transitions observed. Our belief in it is however strengthened by the fact that for most cases the agreement between the data and the non spin-flip DWBA predictions is quite good. Unfortunately only one definite unnatural parity assignment exists in the literature on Sn isotopes. A 6- state at 2.777 MeV in ~1 6 S n was assigned by radioactivity experiments 12). This state is not seen in our experiment. The 2.787_+0.010 MeV state that we observe has been definitely assigned as a positive parity state in both (d, p) and radioactivity experiments. We note, however, that the angular distributions for the 3.320 MeV state in 11aSn and
352
o . B~ER e t a L
the 3.392 MeV state in 12°Sn, both well resolved levels, are quite fiat and these may be non-natural parity states. For all the states populated in our experiment our J~ assignments are in agreement with those from f l - 7 experiment, whenever such assignments are available. Except
m
(5")-
m
(2+1-
4, • ( s ' ) 2"-4* = - -
4" ~
(~"
g.--
(6+A-) --
2"~
(3-)"
(6",~-)--
-
tO'l--
3"
(3")"
4"
"
3- ~
4"
(~-
"4* ~
2. -
-
-
2* --
(s-, 4.):
r--
,.
2':-
17-1--
(3")_---
~r0-
g-
:_
(,,~=
=
(S+,7-)-(4")-(37--"
¢=_
(e+)"
18+,7-)-"
2..
{
--
7- ~ 4---
(5"37;~_
14"1~
2"-
3"
0
0
•
, q)
3"..s; . . . . , --
(e") ~ 3"
,~)
' (~)
18~--
,
"
-)s"
.......
s~tA*)
s-
o
4"
(~)~ 0
2.
o 0
2"
"%.
2"
"%.
,,~,
"°s.
2"
,..,
2.
,q,
;%,
Fig. 12. S u m m a r y o f experimental results for 116. 118.12o. 12z. 124Sn" T h e lengths o f the lines represent the total cross section observed. Circles represent 0 + states n o t seen in the p r e s e n t experiments b u t reported by other authors.
for the first 2 + and 3- states J~ assignments from other inelastic scattering experiments are relatively rare. Our results are in agreement with two notable exceptions to this remark, namely the 17.8 MeV inelastic proton scattering from
EVEN T I N ISOTOPES
353
a2°Sn by Jarvis et al. 42), a n d the 55 MeV inelastic proton scattering from a a 6Sn by Yagi et al. 18). We generally disagree with the parity assignments from the (d, d') experiments 32, 34). This is not surprising since these assignments were made on the basis of comparison of a(0) at two angles only, based on a prescription due to Jolly 33) who used Blair's phase rule to justify this procedure. Particle transfer experiments are often quite selective. For this reason, and also because most of these experiments were done with relatively poorer resolution, it was found not possible to establish unambiguous correspondence between our levels and those seen in these experiments. A notable general feature of all the observed spectra is that whereas for the positive parity states, 2 + and 4 +, a greater number of states exists, for the negative parity states, 3 - , 5- and 7 - , no more than two (a third 3- state is occasionally suggested) states are found up to 4 MeV excitation. As will be seen later this is what is qualitatively expected on the basis of the valence orbits available for neutron excitation. The collective nature of the first 2 + and 3- states (hereafter referred as 2 + and 3]respectively) in tin isotopes is already well established from a number of experiments. As expected these states are exceptionally strongly excited, with ~tot ~ 12 mb. What is somewhat surprising however is the fact that the 5~- and 7~- states are also quite strongly excited with atot ~ 2.5 and ~ 1 mb respectively. These cross sections are factors 5 to 10 lower than those for the 2 + and 37 states, but are larger than those for most of the other 2 +, 3 - , 5- and 7- states. This is particularly significant if one considers the large angular momenta (5 and 7) involved. In contrast, nearly all 4 + states are more or less equally excited, with atot ~ 1-2 mb. 4.3. U N R E S O L V E D D O U B L E T S
In each of the figs. 4-8 the majority of J~ assignments are quite straightforward and unambiguous. For each isotope however some cases exist where doublets are suggested by width of experimental peaks, magnitude of cross sections, systematics of excitation energies, or shape of angular distributions. In the following we discuss some important cases among these. From fig. 12 we note that the 5~- energies decrease systematically and the 31 energies increase in the same manner. The two trend lines cross at 118Sn" Indeed only one state is seen at 2.328 MeV. The (p, p') angular distribution has mainly the shape of a l = 3 transition, although some evidence for the filling-in of the minima exists. The 5- assignment of the mixed-in state is supported by a number of experiments [refs. 2o,~2-a5,53)]. We have therefore analysed the transition as a 1 = 3+1 = 5 doublet, the cross-section decomposition being based on the cross sections of the 3[- and 5~- states in the adjoining tin isotopes, where the two states are well resolved. The "state" at 2.478 MeV in 12°Sn is found to be broad, suggesting an unresolved doublet. Indeed the trend lines of the 7~- and 4 + states suggest such a doublet. The observed angular distribution of the unresolved doublet (fig. 6) is also in agreement with this hypothesis.
354
o. BEERet al.
I n 122Sn a close d o u b l e t (with energy s e p a r a t i o n o f a b o u t 12 keV) at 2.412 keV is suggested by the shape o f the peak. The a n g u l a r d i s t r i b u t i o n indicates the definite presence o f the 2 + state. W e p r o p o s e that the other unresolved state is the 71, whose p o s i t i o n in this n e i g h b o r h o o d is indicated by the energy systematics. The m a g n i t u d e o f the integrated cross section in this case also indicates the presence o f 7~-. The 2.213 M e V " s t a t e " in lZ4Sn also a p p e a r s to be a doublet. The evidence is twofold. In A l l a n ' s high resolution (p, p ' ) experiments z7) two states at 2.199 a n d 2.217 M e V were seen. O u r single transition at 2.213 M e V m u s t c o n t a i n b o t h o f them. The essential features o f the observed a n g u l a r d i s t r i b u t i o n c o r r e s p o n d to that o f a 5 state a n d the m a i n c o n t r i b u t i o n is u n d o u b t e d l y due to the 57 state. The total cross sections for the 57 states in isotopes in which they are clearly resolved are 2.9 m b (116), 2.4 m b (120), a n d 2.7 m b (122). The transition at 2.213 M e V has a total cross section o f 3.5 mb. The higher cross section is also suggestive o f an unresolved group. W e t h i n k that the mixed-in level is a 4 +. This is consistent with the observed a n g u l a r d i s t r i b u t i o n as well as the t r e n d o f the 43 states. 4.4. THE 2 + STATES The first 2 + states a n d their collective n a t u r e in the isotopes o f tin are well k n o w n . O u r m a i n emphasis in r e g a r d to these states was to determine, as accurately as possible, the v a r i a t i o n offl2 o r G2 t h r o u g h these isotopes, in o r d e r to c o m p a r e with theoretical p r e d i c t i o n s as well as o t h e r e x p e r i m e n t a l results. This c o m p a r i s o n is especially i m p o r t a n t because it has often been stated t h a t the d e f o r m a t i o n p a r a m e t e r s o b t a i n e d by different kinds o f experiments should be different, since each o f t h e m is sensitive to a different aspect o f the nuclear d e f o r m a t i o n . TABLE 9 Summary of results for the Experiment
Ref.
116Sn
f12 Coulomb excitation b) reson, fluorescence c) electron scattering neutron scattering m-scattering deuteron scattering proton scattering
22) 2~) 25) a7) 36) as) a3) 42, 18, 55) 31) ours
~1s Sn
G2
0.135±0.003 18.0± 0.9 0.135-2_0.030 18.0± 4.2 0.1091 11.8± 1.8
f12
G2
0.136±0.003
18.4~0.9
0.13 ±0.04
12.4~ 2.5 16.8~10.3
0.10 ±0.02
11.5±2.3 9.9=]=2.4
0.133±0.010 0.143±0.004
18.5± 17.6:~ 2.6 20.4--+ 1.2
0.134±0.010 0.134±0.004
17.9 ~-2.6 17.8+__1.1
a) For all the above measurements, G2 = 995 •22. b) Measured quantity, B(E2). Computed f12 by relation f12 = B(E2)~:/(3ZRo2/4et) with Ro taken as the equi and a = 0.55 fm, according to the tabulation of Owens and Satchler. c) Measured quantity, half-life, Tz~.
EVEN TIN ISOTOPES
355
As was mentioned earlier, we have made measurements for the 2~- states with three samples of accurately known mixtures of tin isotopes. This enabled us to obtain the ratios of cross sections for the 2 + states between one isotope and another with errors less than + 5 ~o. Considering an additional uncertainty of +__5 ~o due to the choice of the optical-model parameters we get our relative elrors on G2 as ___7~ . As discussed earlier the absolute errors due to incorrect normalization of the absolute cross sections may be somewhat larger. We summarize the results of various experiments in table 9. It may be noted that comparison between various results is often not possible, either because of very large quoted errors or unspecified errors. By far the most accurate results are the Coulomb excitation B(E2) results of Stelson et al. z2). These results have quoted relative errors of + 2 ~ and absolute errors of +_5 ~o. The authors give/32 values also, based on eq. (1) of subsect. 3,3 with R taken as the uniform radius R = 1.20 A~ fm. However Owen and Satchler 54) have pointed out that in view of the Saxon form of the nuclear charge distribution a single uniform radius cannot be obtained for the different multipolarities. They have tabulated uniform radius equivalents for different values of 1 and different values of the half-way radius r, and the surface diffuseness a. We have converted the B(E2) values of Stelson et al. 2z) using the equivalent uniform radii tabulated by Owen and Satchler 54) for r = 1.075 fm and a -- 0.55 fm which are the appropriate values for tin nuclei according to electron scattering experiments. As seen in table 9 our results are in excellent agreement with the Coulomb excitation results of Stelson et al. 22), e v e n the largest difference (6 ~ for aa 6Sn ) being within the quoted errors. The only other results of comparable accuracy are the (p, p') results of Makofske et al. 3~). Once again the agreement is well within the quoted errors,
first 2 + states ") 120Sn
0.1304-0.003 0.098 4-
0.12 ± 0.12 4 - 0 . 0 2 0.12 40.12, 0.11 0.1194-0.010 0.129±0.004
1228n
16.94-0.8
0.1244-0.003
15.44-0.8
9.6 4-1.6
14.3--2.5 12.45:2.5 14.34-4.7 15 416.8, 14.3 14.1 ±2.4 16.64-1.0
124Sn
0.1144-0.003 0.098 ±
0.13 4 - 0 . 0 3
12.34-2.5 16.8--7.8
0.1124-0.007 0.122±0.004
12.5±1.6 15.0±1.1
13.04-0.7 9.6 i 1.6
14.1 ±2.8
0.1084-0.007 0.1194-0.004
11.6±1.6 14.0±1.0
valent square well radius which gives the same B(E2) as a Saxon-formcharge distribution with ro = 1.075fm
356
o. BEER et al.
although the results of Makofske et al. show a more rapid fall-off offl2 beyond 11SSn than either our results or those of Stelson et al. MeV
Me\
4
:
•
I
:
_I~'- •
•
I
.,
-./
t
MeV
l
l
4-
•
/o/
I
4-
0
0
0
i
f
s ~ / / ..~/ / ~j'*" o / //o ~// / /
•
r
3" •
•
•
2-
1
........
I
/,
"--:t
11G 1,8
Exp:• 3-
I ,I
, ,o(?
I
I
I
I
Me
~, ,
,
,
4 +
•(4+1
ds ,;8 ,~,o ;22 124s. (b)
,20 ,22 124Sn
(a) MeV
Exp:•
• (3-1
t
~l~ .'8
MeV.
4- o
,20' ,.'
' 1245n
(c) f
t
i
t
4
I •
•
•
o
o
o
x
×
3-
2-
2-
J Exp:e 5•(5-) ]20 122 124Sn
(d)
" Exp:e 7• (7-) 1'16 11'8 120 12~2 124Sn
(e)
Exp : X [8+1
l
®(7.8"1
I
• 16+1 o(G+~7 -) 8 +ThtoreL;col i 116 118 1120 1122 1245n
(f)
Fig. 13. Comparison of experimental excitation energy trends for each jTr with results o f several theoretical calculations. The solid curves correspond to calculations of Arvieu 6) the dashed curves correspond to calculations of E. Baranger.
357
EVEN TIN ISOTOPES
The electron scattering G2 values are consistently about 35 ~o lower than our results as well as all other entries in the table. They have quoted errors of about +__17 ~o. It is not clear whether these errors are relative or absolute. In any case no other entries indicate a significant variation of f12 from experiment to experiment, and the variation, if it exists, must be considered to be smaller than + 15 ~o. The energy systematics of the 2+ states as well as all other 2 + states is displayed in fig. 13(a). The smooth behaviour of the 2~+ state is quite obvious. It is difficult to attribute much systematics to the higher-lying 2 + states. As a matter of fact additional 2 + states, other than those shown in fig. 12 has been suggested by some experiments. These are indicated in the tables 4-8. Telegdi and Gell-Mann 5 6) and Lane and Pandelbury 57) have introduced a model independent energy weighted sum rule which follows from the charge independence of the nuclear interaction. It may be assumed that the low-lying vibrational states correspond to the oscillations in which the neutrons and protons move in phase (T = 0). For such T = 0 states, the sum rule limit (SRL) is L G , E ' = h z Z 2 1_~ l ( 2 l + 1 ) ( 3 + l ) 2 , i l SMp A R~,
l > 1,
(3)
where REt is the equivalent uniform radius for the multipolarity l, Z is the atomic number of the target and Mp is the mass of a proton. Lane 5s) considers that a good criterion of collectiveness is that a given transition "exhausts at least a fair fraction (say >~ 5 /o/o ) of a sum rule". The 2 + transitions do meet this test successfully. They exhaust 7.3 ~ , 6.3 ~o, 5.7 ~ , 5.1 ~o and 4.9 ~ of SRL for tin 116, 118, 120, 122 and 124 respectively. The remaining 2 + transitions together add up to an almost equal amount, bringing up the total fractions of the SRL observed to 12 ~ , 12 ~ , 9 ~ , 12 ~o and 13 ~o respectively. These numbers once again bear out an almost uniform degree of collectivity in the tin isotopes. 4.5. THE 3- STATES In contrast to the many 2 + states no more than two or three 3- states have been identified in each isotope. The 3~- states are already well known from other experiments. As fig. 13(b) illustrates, the 3~- excitation energies rise monotonically in energy. The cross section corresponding to them as well as G3 decreases slowly with increasing number of neutrons. Once again our main interest concerning 3~- states was to determine the variation of their transition strength in the tin isotopes. The normalization obtained by our mixed target experiments leads us to G3 values with relative errors ranging from 9 to 13 ~ . Our results are listed in table 10 together with results from other experiments. Unlike the 2 + states, for the 3[- states accurate values from Coulomb excitation experiments are not available because of serious experimental problems. The Coulomb excitation results of Alkhazov et al. 23), though a factor of three lower than the earlier results of Hansen and Nathan 24) are still 30 }~o, 13 ~ , 3 ~o, 100 ~ and 100 ~ higher than ours. Agreement between our results and those of Ma-
358
o. BEERet al. TABLE 10 Summary of results for Experiment
Ref.
~~6Sn f13
Coulomb excitation b) electron scattering neutron scattering cz-scattering deuteron scattering proton scattering
23) 25) 37) 36) 3s) 33) 42, ~s, 55) 3t) ours
xxs Sn G3
/53
G3
0.182±
48 34 ~5
35
0.15 ±0.03
12.3 :k 2.5 23 ___9
0.15 zk0.03
0.185±0.010 0.188±0.008
34.5 ± 35 ±4 36.4-E3
0.168±0.14 29 ~5 0.174±0.11 ~) 31.44-4 c)
12.5 ~ 2.5 23 ±9
a) G3 = 1023 /~32. b) Measured quantity, B(E3) • B(E3)s.p. = 0.1393Ro 6 with Ro taken as the equivalent square well radius which the tabulation of Owens and Satchler). c) Obtained by subtracting the contribution due to the unresolved 5 - state, assumed to be approximately kofske et al. 32) is within __+3 ~o, well within the q u o t e d errors. U n l i k e the 2 + states, the a g r e e m e n t between o u r results for the 3~- states a n d the electron scattering results is also excellent. I n terms o f the sum rule o f e q . (3), the 3~" states exhaust 7.6 ~o, 6.8 ~ , 6.2 ~ , 5.4 9/0 a n d 5.3 ~ o f the S R L for tin 116, 118, 120, 122 a n d 124 respectively a n d meet the collectivity criterion o f L a n e ss). The r e m a i n i n g 3 - states a d d u p to a p p r o x i m a t e l y 1 ~ a d d i t i o n a l c o n t r i b u t i o n to the S R L , the t o t a l fractions o f the S R L observed being 8.8 ~ , 7.1 ~ , 6.8 9/00, 6.2 ~ a n d 6.2 ~ respectively. 4.6. THE 4 + STATES The observed 4 + states are illustrated in fig. 13(c). The 4 + states are numerous, like the 2 + states a n d unlike the 3 - o r 5 - states. The first 4 + states follow the t r e n d o f s m o o t h l y decreasing energy, m u c h like the 2+ states. A s a m a t t e r o f fact the 4~- a n d 4+ states always occur at energies a p p r o x i m a t e l y twice the energy o f the first 2 + states. We r e t u r n to this p o i n t later. The first 2 + a n d 3 - states were m o r e t h a n 10 times stronger t h a n any other states o f the same J~. This is n o t true for the 4 + states. M o s t o f the 4 + states c a r r y m o r e o r less the same strength (within a factor o f two). A l s o their n u m b e r s in each i s o t o p e are essentially the same. The s u m m e d t r a n s i t i o n strength is essentially the same (within + 10 ~ ) in all isotopes. 4.7. THE 5- STATES N o m o r e than two 5 - states have been identified in each isotope. The identification o f the first 5~- states is certain, a n d it is illustrated in fig. 13(d). The t r a n s i t i o n strength
EVEN TIN ISOTOPES
359
the first 3 - states a) 120Sn 33
122Sn G3
124Sn
f13
G~
27
f13
42
0.172± 0.17 !
30 30
0.14 4-0.03
20
4-9
0.14 40.17, 0,14
20 30
i :E7, 20
0.1594-0.020
26
4-7
0,161 =/:0.008
26.44-2.4
42
±4 4-7
0.133±
11.0i2,2
11.1 4-2.2 0.14 =t_0.03
20
G3
18
~:3
10.74-2.2
4-9
0.1524-0.014
23.64-4.5
0.133 ~:0.020
18.14-5.4
0.1494-0.008
22.6=k2.2
0.138+0.007
19.6±2.0
gives the s a m e B(E3) as a S a x o n f o r m charge distribution with ro = 1.075 f m a n d a = 0.55 f m (according to the s a m e as in t h e other even tin isotopes.
for the 5~- states in all isotopes is about 8 s.p.u. The assignment of the second 5- state in all cases is less certain. The second 5- state is generally a factor 5 to 10 weaker than the first. 4.8. T H E 6 + , 7 - , 8 + S T A T E S
In fig. 13 we indicate the trends for these states. The assignments for the (6 + ) and (8 +) states are rather uncertain, although the (8 +) states show a good systematic trend. The 7~- states show a well established trend, and although we cannot quote G 7 values, we note that the cross sections corresponding to these increase slowly from 1'6Sn to 124Sn.
5. Discussion
In this section we describe briefly the different theoretical attempts which have been made in order to understand the low-lying states of even tin isotopes, and compare our experimental results with the predictions of these calculations. For nuclei with large number of particles outside assumed closed cores, exact shellmodel calculations with residual interactions are prohibitive in spite of recent developments of powerful computational techniques 59, 60). For tin nuclei even if the doubly magic core N = Z = 50 is considered completely inert, the nearly 20 valence neutrons in the five subshells 2d~, lg~, 3s~, 2d~ and lh.~ present a secular problem of the order of 101 o x 101 o. In order to make the problem manageable major simplifications about the residual nuclear interaction must be made. The first of these was the realization
360
o. BEER et al.
that a major part of configuration mixing can be taken into account by the BCS pairing which arises from the short range part of the nuclear force and which also produces the characteristic energy gap. All attempts to calculate the spectra and other properties of tin isotopes do indeed have this simplification in common. The first step is always to transform the problem from that between particles to that between "quasiparticles" in order to take account of the pairing correlations. The singleparticle energies are replaced by their counterparts, the single quasiparticle energies which are defined as Ej =
v /..
2d 5/22
3.
2,
1.
~
I
5n116
2
d
I
Sn118
~
I
Sn120
2
I
Sn122
I
Sn124
Fig. /4. Quasiparticle energies Ej according to Arvieu 6).
where 2 is the Fermi energy and Aj is the so called "energy gap". For a doubly even nucleus the excited states are made up of excitations of two, four, six, etc. quasiparticles. If the quasiparticles are considered non-interacting, the energies of quasiparticle states can be obtained simply by summing single quasiparticle energies. For example for a two quasiparticle state of spin J, the excitation energy above the 0 + ground state is given by
E(jl,j2, J)-Eg.s.
(0 +) =
EjI+Ej2.
The quasiparticles of course interact, but certain qualitative aspects of the excitation spectra follow just from the simple model of non-interacting quasiparticles.
EVEN TIN ISOTOPES
361
In fig. 14 we show the trend of quasiparticle energies according to Arvieu 6). We note that the order of the quasiparticle levels, and their A-dependence is completely different from that for the single-particle levels. We note, for example, that the d~ and gk quasiparticle energies increase rapidly with A, while the s~, h~, d~ energies change quite slowly. We expect therefore that the low-lying states will be dominated by ½, ~ and - ~ quasiparticles, with states involving -~ and ½ quasiparticles following at higher energies. Between the two kinds of states there may even develop a region of low level density, particularly for higher A. Indeed in 122Sn and 1248nsuch a region is observed between 2.6 and 3.2 MeV. Since, of the five quasiparticle levels available, only the hq has negative parity we also expect far more positive parity states. We have already remarked on this feature of our data. We expect the 8 + states to be almost pure two quasiparticle 2E~ states. Their energy should therefore decrease smoothly with increasing A. (As a matter of fact, the energies are approximately equal to 2()~-), see fig. 14.) For any other J~ more than one configuration is possible: nine 2 + states, seven 4 + states, three 6 + states, two 3 - states, f o u r 5- states and three 7 - states are predicted in this simple model. One would expect the 2 +, 4 +, 6~- 5[- and 7~- states to decrease in energy with increasing A. The 3 - states require one of the quasiparticles to be in the ~ or ~ orbit and we may therefore expect that 3[- states will rise in energy with increasing A. It is somewhat surprising that all these qualitative predictions are indeed very confirmed by the data. In any serious calculation the interaction between the quasiparticles must, of course, be taken into account. This interaction, which arises from the long-range part of the nuclear force, can be specified phenomenologically or it may be obtained from the realistic nucleon-nucleon interaction. Also the interaction can be treated in one of the numerous approximate ways. Many calculations of even tin isotopes have been reported in the literature. Each differs from the others in one or more of these ways. We summarize these calculations in table 11. The first four calculations, numbers 1,2, 3 and4in table 11, use a very simple phenomenological interaction. They all assume inert N = Z = 50 core and a configuration space of five shell-model states for the neutrons. Core excitation is not considered and in all, but ref. 8,), excitations are limited to two quasiparticles. The fits to the 2 + and 3~" excitation energies are quite good. This is not surprising since the parameters of the interaction are chosen to optimize these fits. The transition rates predicted (table 12) do not agree as well with the data, but even that agreement can be mocked up by an appropriate choice of the neutron effective charge. In view of the availability of calculations using more conventional residual interactions, we do not discuss the above calculations any further. In fig. 13 we have not shown the results ofYoshida [ref. 61)] and Veje [ref. 64)] who obtain reasonable fits to the trend of the 3[- states by introducing an octupole-octupole residual interaction whose parameters are adjusted for the best fit to the experimental data. In table 12, we tabulate the B(E3) values obtained in refs. 61,64) mainly because not many other complete predictions are available.
362
o. BErRet
al.
TABLE 11
Summary of theoretical calculations Authors
Ref.
5) 61) 62) 8a) 6) 7) sb,¢) lo) 9.,b,c) 66)
Residual interaction
pairing+quadrupole pairing + quadrupole-}- octupole pairing+octupole pairing+delta function Gaussian with exchange Gaussian with exchange Gaussian with exchange realistic-Tabakin realistic-Tabakin-Yale Gaussian with exchange
Inert core N= Z = 50 50 50 50 50 50 50 28 50 50
Number of active subshells
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Kisslinger, Sorensen Yoshida Veje Ottaviani e t a L Arvieu et al. Kuo Trieste Group Clement, Baranger Trieste Group Gillet e t aL
5 5 5 5 5 5 5 12.}.12 tt) 5 5
*) tt) *) **)
STDA denotes "second Tamm-Dancoff approximation" which permits inclusion of 4 q.p. excitations. 12 neutron and 12 proton subshells above N = Z -- 28 were included. Core excitation taken into account by means of "core polarization" renormalization. One particle-one hole excitations of the core were included.
The second g r o u p o f calculations, numbers 5, 6 and 7 in table 11, use somewhat more conventional residual interactions. Once again N = Z --- 50 inert core was assumed and only a 5 sub-shell neutron configuration space was considered. Gaussian interaction with appropriate exchange terms was first used by Arvieu 6) who also verified that there are not any major differences between T D A and R P A results. The theoretical results o f Arvieu are shown in fig. 13. K u o 7) optimized single-particle energies and force constants and obtained somewhat better fits than Arvieu using RPA. Both these calculations considered only two quasiparticle excitations. The Trieste group [ref. 8b, c)] included four quasiparticle contributions in their calculations with Gaussian interactions, but f o u n d that the predictions were extremely interaction dependent. For this reason we will only refer to their latest work using realisticinteractions 9). It supersedes their earlier work. Both Arvieu and Salusti 6d) and K u o 7) have also calculated B(E2) values for the 2+ -~ 0 transition. These are listed in table 12. In all the above calculations phenomenological forces were used and their range, depth, and other parameters, and to a certain extent the input single-particle energies (s.p.e.) were adjusted to give the best fits to the experimental data. As illustrated in fig. 13, these calculations often produced quite g o o d fits to the energies o f the lowlying states. The Z 2 search for best fit parameters leads to an "effective" interaction which masks the effect o f those aspects which are a p r i o r i neglected in the formulation o f the problem. F o r this reason these calculations do not shed any light on the importance o f four quasiparticle excitations, or effects of core excitation, etc. These effects, which may be quite large, can be meaningfully investigated only when the parameters o f the interaction are fixed, such as when one uses realistic forces (i.e., those which fit nucleon-nucleon scattering data). Calculations using realistic inter-
363
EVEN TIN ISOTOPES
of even tin isotopes 50 core excitation
4 quasiparticle excitation
N=Z=
yes
yes yes yes *) yes **)
yes
Predictions
Approximation used
Adiabatic TDA TDA STDA t TDA RPA STDA t TDA STDA t RPA
21 + energies; B(E2) 2 +, 3- energies; B(E2), B(E3) 3- energies; B(E3) 0 +, 2 +, 4 + energies 0 +, 2 +, 4 ÷, 5-, 7- energies; B(E2) 0 +, 2 +, 4 +, 5-, 7- energies; B(E2) 0 +, 2 +, 4 +, 3-, 5-, 7- energies 0 +, 2 +, 4 +, 6 +, 3-, 5-, 7- energies; B(E2) 0 ÷, 2 +, 4 +, 3-, 5-, 6-, 7- energies; B(E2) 2 +, 3-, energies; B(E2), B ( E 3 )
a c t i o n s h a v e r e c e n t l y b e e n r e p o r t e d b y b o t h the T r i e s t e g r o u p 9) a n d t h e P i t t s b u r g h g r o u p 10). T h e m a i n t h r u s t o f t h e w o r k o f t h e T r i e s t e g r o u p 9) has b e e n to e x a m i n e t h e i m p o r t a n c e o f f o u r q u a s i p a r t i c l e effects, w h e r e a s the c a l c u l a t i o n s o f C l e m e n t a n d B a r a n g e r lO) h a v e c o n c e n t r a t e d m a i n l y o n t h e i m p o r t a n c e o f c o r e e x c i t a t i o n s . W e i l l u s t r a t e s o m e o f t h e results o f t h e s e c a l c u l a t i o n s in fig. 15, a n d discuss t h e m briefly below.
TABLE 12 Transition strengths for even isotopes of tin Gl
G2 21 ÷ ~ 0
Ga 3~- --->0
A
This expt.
116 118 120 122 124
20.44-1.2 17.84-1.1 16.64-1.0 15.04-1.1 14.04-1.0
116 118 120 122 124
36.44-3.0 31.04-4.0 26.44-2.4 21.64-2.2 19.6±2.0
KS [ref. 5)]
Y [ref. 61)]
15.5 16.8 15.9 13.9 10.9
8.9 8.1 8.5 8.3 7.2 12.3
Theoretical predictions a) V AS K CB [ref. 6,)] [ref. 6a)l [ref. 7)] [ref. 16)] 5.0 6.1 7.1 7.2 6.5 32.0 34.4 36.5 38.6 40.4
9.3 9.6 8.2 8.2 6.8
2.5
1.8
GRSW [ref. 9)]
GGR [ref. 66)]
6.2 --> 7.5
0.43 0.48 0.47 0.57 0.48 20.0 21.3 19.8 15.8 9.8
~) The values of G~ quoted in this table are for effective charge = 1.0e for neutrons except for the calculation V of ref. 64) where eoff = 0.Se for neutrons and protons, and calculation CB and G G R in which no effective charge was used. B(E/) from theoretical calculations were transformed into G~ values by using R = 1.2 A4".
364
o. BEERet al.
In the new calculations o f the Trieste group, G m i t r o e t al. 9b) examine four quasiparticle c o n t r i b u t i o n s (in w h a t they call second T a m m - D a n c o f f a p p r o x i m a t i o n , S T D A ) a n d take core excitations into a c c o u n t a p p r o x i m a t e l y by using Bertsch 62),
1165n
MeV 3-!--
7-
..8* 0÷ 6+
3.1-4,+
4+
(6.71
"
-
~
,~-
!,8,*)
o+
,
62+
I
/
2.9-
) 4+
\
/ ~ ~,+ 1 i A~
4+ (5,9)
,r'~6-/-5-(2.7,, ;[ -"
2.7-
6-(3.6) "G" [4.2)
/
' I /" I 0+(16 O) I // ' 4 + ( ~ /
. 5=mmr~
/ I
2+ 7-
/
/
I
5, - -
2+
/
] 4'~
I
t
/
! \,
" ) f,,- 2+f 7'5)
2+
I
, ' ~ '
5-
5-(2.6)
, 7-
I
~,/ ,/',,
2.5
,~,-..,,
2+
~
6+
/
//
.4.+.
/
"-'a.~',
~.
4.
3"
2..~ \
2+
',
0+
~1I 2+
2.1
/~\
0+ O+ 2
0+
0+(5.7) --
0
3"
/
~L~J(4.4.)-'.
0+ 2+
-.
2+
/ I /
/
2+
0+
0+(36.2)
0+
0 4.`•
0 +
TDA
STDA
Exp
N3~PI
NI-PI
ALZETTA eL
=I. (Ref. Sd)
CLEMENT ~nd E.BARANGER(Ref.IO)
Fig. 15. The experimental spectrum of lZ6Sn and results of calculations with realistic forces. The first 3- state is predicted above 3.3 MeV in both TDA and STDA calculations ofref. 9d) and is not shown. The numbers in parentheses next to excitation energies refer to four quasiparticle components. K u o a n d Brown 63) c o r e - p o l a r i z a t i o n r e n o r m a l i z a t i o n . The m a i n p a r t o f these calculations was confined to l l6Sn for which several choices o f s.p.e, were e x a m i n e d
EVEN TIN ISOTOPES
365
with both Tabakin and Yale-Shakin interactions, with core excitations through N = Z = 28 as well as N = Z = 8 being included. Recently the Trieste group has used an improved inverse-gap-equation (I.G.E.) method to determine the input parammeters 9a). This way of specifying input parameters may be the most preferable one, but it is still not free of uncertainties, as discussed in detail in ref. 9d). When the I.G.E. method was used the results for the Tabakin and Yale-Shakin interaction were found to be generally very similar for all cases. It was found that the inclusion of four quasiparticle configurations depresses all energies including the ground state energy, so that the excitation as measured from the ground state do not differ significantly between TDA and STDA calculations. B(E2, 2 + ~ 0 +) predictions were found to be actually 30 to 40 % poorer. As illustrated in fig. 15, where results of calculations using the LG.E. method are shown, no significant improvements in fits to the higher 0 +, 2 + and 4 + states were obtained in STDA ,. At the present it is difficult to appreciate the possible improvements obtained in STDA calculations as long as there persist the larger uncertainties due to the choice of input parameters and due to the extent to which core polarization corrections are made. It may however be noted at this point that the discrepancies between experimental 4 + and 4 + states and the two quasiparticle predictions for the same (see fig. 13c) are maximum for 1 2 4 S n and minimum for 116Sn. Also in 124Sn the quasiparticle energies are higher (see fig. 14), so that two quasiparticle excitations overlap considerably with four quasiparticle excitations. It may therefore be expected the four quasiparticle admixtures will be maximum in a~4Sn, and that a STDA calculation for aZ4Sn would be far more instructive than the a a 6 S n calculation made so far. Clement and Barange~ xo) start with the assumption that for most of the states of interest four quasiparticle effects are unimportant. Having neglected four quasiparticle components, Clement and Baranger were able to take a more detailed account of core-polarization by considering the N = Z = 28 core inert, and explicitly considering the next 12 sub-shells as being active for both neutrons and protons. The importance of excitation of the N = Z = 50 core was dramatized by the result that for the 3~ states only 35 % of the wave function was found to be from the usual fivelevel valence neutron space. For the 2 + states the corresponding fraction was 72 %. This explains why 2~- energies are reasonably reproduced in calculations which do not take account of core-excitation, such as those of Arvieu 6) and Kuc 7), whereas 3~- energies are not obtained even approximately correctly. "[he results of Clement and Baranger are shown in fig. 13 and also in fig. 15. In fig. 15 we have presented their results for two different choices of single-particle energies N1-P1 (the same as in fig. 13) and N3-P1 (with neutron single-particle energies being different). From these * G m i t r o et al. 9b) have p o i n t e d out, t h a t the g r o u n d state shift is connected with //4o, which is left o u t in S T D A calculations for j~r :/~ 0 +. If the g r o u n d state shift is ignored (as p r o p o s e d in ref. 9)) the S T D A energy eigenvalues as n o t e d in fig. 15 s h o u l d be lowered by 206 keV. F o r m o s t o f the positive parity states the a g r e e m e n t with experimental excitation energies would t h e n be considerably ameliorated. T h e fit to m o s t o f the negative parity states would however become worse.
366
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al.
two results it is apparent that the choice of single-particle energies can cause quite large changes in the spectra, and that this represents the largest uncertainty in the present calculations. In the calculations of Clement and Baranger 1o) proton excitations were explicitly considered and it was hoped that B(E2) values could be obtained without use of effective charges. The calculated B(E2) values were found to be a factor eight smaller than the measured values, indicating that deeper excitations of the core play an important part in the enhancement. Similar conclusions about the importance of the lf~ subshell were also reached by Gmitro e t al. 9b). Gillet, Giraud and Rho 66) have recently taken a somewhat different approach to the question of core excitation. In a RPA calculation in the two quasiparticle subspace they have included one particle-one hole excitations of the N = Z -- 50 core. In these calculations single-particle energies and gaussian force parameters were determined by the inverse gap method. The main thrust of the work was to try to reproduce experimental transition rates without use of effective charges. As indicated in table 12, reasonable agreement with experimental values of G 3 was obtained, but G2 values obtained were factors 30 to 40 smaller than the experimental values. These results, in conjunction with those of Clement and Baranger lo), indicate that while core excitations for the 31 states are indeed mostly of the lp-lh type, for the 2~" states more complicated configurations play a dominant role. In summarizing the present status of theoretical calculations we can state safely that while the use of realistic interactions has reduced the number of adjustable parameters, the uncertainty in the input single-particle energies remains a serious problem. Core excitations have been found to have an important effect in determining the energies of not only the collective 2~- and 3~- states but also the higher lying states, and it appears that calculations must at least include shells as deep as the 1~ for both neutrons and protons in order to achieve reasonable agreements in transition strengths, without use of effective charges. In the calculations reported so far for 116Sn four quasiparticle effects have been found to be generally small (they may be larger for 124Sn, but no STDA calculations for 124Sn have so far been reported), and no 2 + or 4 + states lend themselves to a two-phonon description, although the experimental energy systematics of 4 + states is highly suggestive of such an interpretation. We wish to thank Drs. J. Thirion and J. Saudinos for their continued interest in the present investigations. We are grateful to M. Laspalles for work on the improvement of the performance of the spark chamber and to the cyclotron crew for the excellent performance of the accelerator. The authors are also thankful to Dr. E. Baranger for communicating the results of her calculations prior to their publication. One of us (K.K.S.) also wishes to thank Drs. A. Messiah and E. Cotton for their hospitality during the very profitable year of his stay at Saclay.
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