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39K(d, 3He)38Ar and the s-d shell structure of 38Ar
]
Nuclear Physics A l l 8 (1968) 347--360; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
39K(d, 3He)SSAr A N D T H E s-d S H E L L
STRUCTURE
OF
SeAr
B. H. WILDENTHAL t and E. NEWMAN Oak Ridge National Laboratory, Oak Ridge, Tennessee tt Received 1 July 1968 Abstract: The positive-parity level structure of 8aAr is investigated with the 89K(d, 3He)SaAr reaction
on a KI target at a bombarding energy of 34.5 MeV. Seven residual levels are found to be strongly excited, three with lp = 2 characteristics at 0.00, 2.17 and 7.11 MeV excitation energy and four with predominant lp = 0 characteristics at 3.94, 4.57, 5.15 and 5.55 MeV excitation energy. These experimental results are examined from the viewpoint of several recent shellmodel calculations. No extant calculations provide a quantitative interpretation of the occurrence of as many as four strong l = 0 levels. It seems likely that the extra l = 0 levels arise from 2p-4h excitations into the If shell and are populated via the large components of the (s~t-d~~) configuration mixed into these 2p-4h states. The (d, 3He) reaction on the x*vI in the target yields evidence that the [+ ground state of this nucleus contains significant 2d~ admixtures. E ]
I
NUCLEAR REACTIONS SaK, 1271 (d, aHe), E = 34.5 MeV; measured o'(EZHe, 0), Q. 3eAr, 126Te deduced levels, l, ~r. Natural target.
[
1
1. Introduction
N u c l e a r systems which can be described as either two particles o r two holes rem o v e d f r o m m a j o r shell closures are o f p a r t i c u l a r interest in the s t u d y o f the shell m o d e l because o f their c o n c e p t u a l simplicity. Such a nucleus is a SAt with 18 p r o t o n s and 20 n e u t r o n s ; its low-lying level structure can be described to a first a p p r o x i m a t i o n (ref. 1)) b y couplings o f two holes in the 2s+ a n d ld+ shells. H y p o t h e t i c a l l y , such nuclei with their small n u m b e r o f a l l o w a b l e shell-model states can p r o v i d e i m m e d i a t e a n d basic i n f o r m a t i o n a b o u t the t w o - p a r t i c l e (hole) effective interaction. In practice, h o w ever, such h o p e s are often frustrated because the effects o f configurations outside the n o m i n a l l y a d e q u a t e basis d e s t r o y the idealized simplicity o f the system. T w o n o t o r i o u s examples o f this type o f c o m p l i c a t i o n are the spectra o f 18 0 a n d +2Ca, each o f which possesses an excess o f low-lying 0 + and 2 ÷ levels; their e x p l a n a t i o n requires d e f o r m a t i o n s o r excitations o f the ~60 a n d +°Ca cores, respectively 2). Shell-model calculations have been p e r f o r m e d for both the positive- a n d negativep a r i t y levels o f the nuclei below +°Ca. T h e p o s i t i v e - p a r i t y levels have been t r e a t e d with a 2 s , - l d + basis ~,3), a n d similar calculations 4) in a ld+-lf~ basis have been p e r f o r m e d for n e g a t i v e - p a r i t y levels f o r m e d b y exciting one particle into the f~ * USAEC Postdoctoral Fellow under appointment from Oak Ridge Associated Universities; present address: Texas A and M University, College Station, Texas. t* Research sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation. 347
348
B . H . W I L D E N T H A L A N D E. N E W M A N
shell and for 0 + levels formed with two particles excited into the fk shell and coupled to J = 0. Recently, calculations for the positive-parity levels have been performed in a complete 2s- 1d basis 5). Comparisons of the predictions of some of these calculations to experimental results 6) from the 39K(d, 3He)3 BAr reaction are of multiple interest. One aspect of the investigation is to determine the extent to which the simple 2s½-1d~: model can explain the low-lying positive-parity levels of 3aAr; that is, do effects like those mentioned for 180 and ~2Ca intrude into the level structure of 3BAr? In a related sense, we want to identify exactly which levels of 3 BAr have a predominant (s,r-d~) -2 character. Another facet of interest is the search for evidence of f~ admixtures. Finally, the detailed results of energy spacings and spectroscopic factors for the (s½-d~) -2 levels, provided these can be assigned without too much ambiguity, will yield tests of the internal consistency of the calculations in the s-d shell model.
2. Experimental procedure Most of the data for the present study were obtained from bombardment of a K I target with 34.5 MeV deuterons from the ORIC. Reaction products were detected and identified in a A E - E solid state detector telescope. The resultant E + AE and AE signals were displayed and stored in a 50 x 400-channel map, and the 3He events were extracted from the array off-line. A sample 3He spectrum is shown in fig. 1. As can be seen, the energy resolution for these data was about 100 keV F W H M . A significant amount of this broadening came from the target thickness. The target was prepared by vacuum evaporation of KI (containing the naturally occurring ratio of potassium isotopes) onto thin carbon foils. Supplementary studies were carried out with targets of metallic potassium evaporated onto mylar film, the potassium being partially oxidized before bombardment. Seven of the particle groups prominent in fig. 1 are identified with transitions to levels of 3aAr and are labeled by the corresponding energies of excitation in the 3aAr system. The energy calibration of the spectra is based primarily on a value of 2.168 MeV for the spacing between the ground and first excited states 7). The angular distributions of these seven transitions are presented in sect. 3. No attempt was made to extract and analyse data for the multitude of weakly excited levels comprising the background evident in fig. 1 because of the ambiguities in their parentages resulting from the 7 ~ 41K and the 127I in the target. Some particle groups from the (d, 3He) reaction on these nuclei are identified in fig. 1, and the angular distributions to the first two levels of 126Te are presented and discussed later.
3. Angular distributions and DWBA analysis The differential cross sections of the transitions to the seven levels of 3 SAr for which data were reduced are presented in fig. 2. The data points are a superposition of results from the oxidized metal target (four points) and the K1 target (nine points).
I
.}"
100
~r~.~7o;,/ o t40
f..; " :
-
• *e a
!/
38Ar(5.55 M~
Ed = 54.5 MeV 8L= 13.0 o
KI (d, 3HeI)
t80
II
~~°V~
38Ar
220 260 CHANNEL NUMBER
:38Ar(3.94 MeV)
MeV)
]
126Te(0.661MeV)]~
~
300
/
340
380
i'~
(g .s)
38Ar(g.s.)
,'2~e~,,~.eV,|~
I
-
/
"38Ar(2.47 MeV)
30
34
%
v
Fig. 1. Spectrum of 8He particles emitted from a KI target. Levels of 3BAr which are excited with measurable strength by the (d, aHe) reaction on SgK are explicitly noted. States in 3SAr known to exist but unexcited in the present reaction are noted by arrows above the energy calibration line. States in 4°Ar [from 41K(d, 3He)] are indicated by arrows below the line. States in l~eTe[from lz~I(d, 3He)] are explicitly designated.
0
50
60
t50
© {00
CO ,I-Z
n" tad t]
-F Q)
J taJ Z Z
200
250
~D
=-
350
B.H. WILDENTHAL AND E. NEWMAN
The curves in fig. 2 are the results of local zero-range DWBA calculations s) for the (d, 3He) reaction to the various levels. The appropriateness of such an analysis for the present reaction has been fully discussedelsewhere 9). The optical-model para-
0.~ 0..~ 4C
2 2 ~
0.~ 0.2
0.~ 0.2 OJ 0.0~
0.02 0.0' 0
t0
20
30
40
OC.M(deg)
50
60
0
t0
20
30
40
50
60
ec.M.(deg)
F i g . 2. A n g u l a r distributions o f the 3*K(d, 3He)88Ar reaction. The curves are D W B A predictions for these transitions. The dashed curves represent the predictions for the alternate/-values (l = 0 in left c o l u m n and l = 2 in right column).
meters used in the present calculations are given in table 1. The deuteron parameters are obtained from the analysis 10) of the elastic scattering of 34.4 MeV deuterons on 40 Ca. The 3 He parameters are from a re-analysis of data 11) for 29 MeV 3He scattering
89K(d, 8He)
351
o n 4°Ca with the optical-model search code H U N T E R ~2). I n this re-analysis, the geometries of the real a n d i m a g i n a r y potentials were held c o n s t a n t to values suggested by G i b s o n e t al. ~ 3), a n d the e n s u i n g results for the analysis of the 29 MeV scattering resemble the results of G i b s o n e t al. for 3He scattering f r o m 4°Ca at 22, 37.7 a n d 64.3 MeV. The use in the D W B A calculations of a 3He potential of the form employed in ref. ~1) results in theoretical a n g u l a r d i s t r i b u t i o n s which deviate seriously f r o m the experimental distributions. This same general effect has been noticed for other nuclei 9, 14, 1 5). TABLE 1
Optical-model parameters used in DWBA calculations
39K (deuteron) 1'7I (deuteron) ~Ar (aHe) 126Te (SHe) Bound state
V
ro
roe
a
W
r'o
a'
WD
Vs.o.
(MeV)
(fm)
(fro)
(fro)
(MeV)
(fm)
(fm)
(MeV)
(MeV)
95.13 99.42 178.9 175.0
1.066 1.118 1.14 1.14 1.25
1.30 1.30 1.40 1.40 1.25
0.81 0.822 0.723 0.723 0.65
1.33 1.24 1.60 1.60 25
0.754 0.859 0.81 0.81
11.8 15.6 0 0
0 0 16.1 16.0 2s.o. :
7.0 7.0 0 0
As can be seen from fig. 2 a n d as listed in table 2, the g r o u n d a n d first excited states of 3BAr are p o p u l a t e d b y l = 2 transitions. The excited states at 3.94, 4.57, 5.15 a n d 5.55 MeV have a n g u l a r distributions which are d o m i n a t e d by the l = 0 shape; the d i s t r i b u t i o n of the state at 7.11 M e V excitation has a n l = 2 shape. Spectroscopic factors c a n be extracted with confidence from the l = 2 distributions o f fig. 2, since the good fit of the D W prediction to the purely l = 2 g r o u n d state t r a n s i t i o n shows that even small l = 0 admixtures would be noticeable. I n the case of TABLE 2
Experimental results for levels of SSAr from 89K(d, 3He)aSAr Ex (exp) (MeV) ~Ar
12eTe
0.00 2.17 3.94 4.57 5.15 5.55 7.11 0.00 0.66
l
2 2 0(-t-2) 0 0 0 2 2 (2)
Jn
0+ a) 2+ ~) 2+ n) (1, 2)+ (1, 2)+ (1, 2)+ (0-4) + 0+ 2+
C*S(I = O) b)
0.00-0.03 0.26-0.22 0.62-0.51 0.33-0.28 0.78-0.70
C*S(I = 2) b)
0.49 2.50-2.41 0.13-0.32 0.00-0.32 0.00-4).17 0.00--0.22 0.44(j=~) 0.39 0.18
s) Ref. le). b) Values the of C2S are normalized so that the sum of the maximum l : 0 strengths in 3SAr is equal to 2.
352
B.H.
W I L D E N T H A L A N D E. N E W M A N
predominantly l = 0 distributions, however, the amount of l = 2 spectroscopic strength is not easy to ascertain, because the I = 2 distribution has an intrinsic magnitude that is about one third that of the l = 0 distribution, and in addition is less structured. The fits of l = 0 calculations to the four "l = 0" states are such as would be considered satisfactory were the transitions known to be restricted to a single /-transfer (with perhaps some reservation about the 3.94 MeV distribution). The fact that the l = 2 mode is allowed in the present reaction, however, and that addition of some l = 2 strength, by smoothing the extreme structure of the l = 0 calculations, improves the agreement between calculation and experiment, creates uncertainty in the amount of l = 2 strength which should properly be attributed to the "l = 0" levels. Spectroscopic factors are presented which are based both on the assumptions that the transitions are purely l = 0 and, alternatively, that they proceed with sufficient l = 2 strength to fill in the first l = 0 minimum to the experimental values. In the latter cases, there is an attendant diminution of the l = 0 strength. An accurate value for the absolute cross section of the (d, 3He) reaction on 39K was not obtained. The cross-section scales in fig. 2 are, however, interconsistent. Hence, in lieu of absolute units, the spectroscopic factors extracted for the various l = 0 and l = 2 transitions with the D W B A analysis are expressed in a normalization such that the total o~served l = 0 strength satisfies the assumptions that the 2s½ shell is completely full in the ground state of 39K and that all l = 0 pickup strength has been measured, i.e., ~ C 2 S ( l = O) = 2. The experimental energies of the seven states of 3SAr under discussion and their spectroscopic properties as derived from the DW analysis are listed in table 2. We see that by using the m a x i m u m l = 0 strengths for the purpose of establishing the normalization, the summed l = 2 strength from the ground state and the 2.17 MeV and 3.94 MeV excited states equals 3.1, which is very close to the expected value of 3 for d~ pick-up. This seems to confirm the assumption that the l = 0 transitions are fairly pure, since otherwise the l = 2 strengths would amount to significantly more than the sum rule limit for pick-up from the ld~ shell. The differential cross sections for the 1271(d, 3He)126Te reaction to the ground and to the 0.663 MeV first excited states of 126Te are shown in fig. 3. The cross-section scale is the same as that for fig. 2. The curves in fig. 3 are the results of D W calculations for these transitions made under the assumption that the extracted protons come from the 2d~ shell. The deuteron and 3He optical-model parameters listed in table 1 are again taken from the formulae of refs. lO, 13), respectively. The spectroscopic factors for these two levels of 126Te are also listed in table 2. The normalization used for the potassium numbers was carried over for the 126[ values, it being assumed that the elemental and isotopic ratios of K I remained constant during the evaporation of the target material. The three protons exterior to the Z = 50 core in 127I are expected to be distributed over the 2dt and lg~ orbits. The spectroscopic strength observed for l = 2 pick-up from 127I indicates that the ~+ ground state contains at least a single 2dl particle in
agK(d, aHe)
353
its dominant configuration, thus ruling out the possibility that the 2~+ state arises primarily from the coupling of three Ig~ particles to J = {. 2
0.5
02 0.4
g ~0.05 o
0.2
0.1
0.0~
0.02 0.04
0
40
20 30 OC.M. (deg)
40
50
Fig. 3. Angular distributions ofthelaTI(d, aHe)l~6Te reaction and DWBApredictionsfor 2d¢ pick-up.
4. Discussion 4.1. I D E N T I F I C A T I O N O F P R E S E N T L Y O B S E R V E D LEVELS O F aSAr W I T H T H O S E OBSERVED VIA DIFFERENT REACTIONS
The published literature 16) on the levels of 3SAr includes reports of their population via the (t, g) [ref. 17)], (p, ~) [ref. is)], (p, 7) [ref. ~)], (g, 7) [ref. 19)] and (p, 2p) [ref. 20)] mechanisms. The establishment of identities between levels excited in the various experiments is important for both the preliminary and concluding aspects of the analysis of the present data. The (p, g) data and the recent (p, 7) data were used as the primary standards in establishing such a correspondence, the absolute energy standard being taken from the Ge-Li spectrometer 7-ray data of Engelbertink et al. 7). The excitation energies from the (p, ~) data 1 s) taken with a magnetic spectrograph are about 10 keV high at 5 MeV compared with the 7-ray
354
B. H . W I L D E N T H A L
A N D E. N E W M A N
energies, and the energies of the several levels populated in this region only by the (p, ~) reaction were accordingly modified for the purpose o f the present discussion. The levels o f 3SAr up to an excitation of 5.70 M e V are listed in table 3. We assume for the remainder o f the discussion that this represents a complete catalog o f levels for this energy range, although such m a y well n o t be the case. The identification o f the levels presently observed below an excitation energy of 4 MeV seems clear. We note that the 0 + level at 3.38 M e V excitation, which is strongly excited with the 4°Ar(p, t)3SAr reaction 21), is n o t populated at all here. Similarly the 3 - state 16) at 3.81 MeV excitation is very weakly populated, although in this case the proximity of other levels prevents a rigorous limit being set o n its m a x i m u m cross section. TABLE3 Catalog of levels of UAr with excitation energies 2.0 ~ Ex _~ 5.8 MeV and the reactions by which they are observed ~) (MeV) 2.168 3.377 3.810 3.936 4.480 4.566 4.585 4.710 4.877 5.083 5.155 5.347 5.513 5.551 5.591 5.658 5.733 5.825 5.853
(P, 7) ~)
(13,e¢) b)
(t, ~) e)
y y y y y y y y y y
y y y y y y y y y y y y y y y y y y y
y
y
y y
y
y
y
y y y y y
(d, erie) a)
j~ a, e) 2+ 0+ 32+ 4(1, 2) + a) 5-
y y y
y
(1, 2)+ a)
y
y
(1, 2)+ a) 5-
y
The symbol y signifies the observation of a particular level by the indicated reaction. s) Ref. 7). b) Ref. xs). c) Ref. x~). d) Present work. e) Ref. xe). The F W H M resolution values o f ~ 100 keV obtained for the particle groups in the present experiment together with the energy calibration o f the spectra based on the 0.00-2.17 MeV and 2.17-3.94 MeV level spacings enable energy assignments to be made to the higher-lying levels with an uncertainty o f ___45 keV. Unfortunately, as is evident f r o m inspection of table 3, this a m o u n t o f uncertainty in the measured energies is such as to allow ambiguity in the identity o f the 4.57 and 5.55 MeV levels observed in this (d, 3He) experiment. We postulate that our "4.57 M e V " level corresponds to the 4.566 MeV level o f
3*K(d,SHe)
355
Engelbertink rather than to the 4.585 MeV level. In addition to the slight bias available for such an assignment from the energy calibration, it is motivated by the defioite 5assignment made to the higher level by Engelbertink, an l = 0 transition being forbidden to such a level. The tentative 2 - assignment for the 4.566 MeV level should, on the basis of the present data, be changed to 2 +, as will be discussed further in subsect. 4.2. There is only a small uncertainty in the assignment of the "5.15 MeV" level to the 5.154 MeV level, which is seen in the (p, 0t) reaction but not observed in the 7 ray data. Three levels seen in the (p, ~t) work could possibly be identified with the "5.55 MeV" level. Of these, the 5.513 MeV level has been assigned as 3t-). The energy calibration favors the identification of the (d, 3He) level with the 5.550 MeV level. Confirmation is available for the foregoing assignments from the results of the (t, ~) reaction a7), which should tend to populate the same states as the (d, aHe) reaction. Levels observed in the (t, ct) experiment are noted in table 3. Additional confirmation comes from results of a (d, 3He) experiment on a9K at 21 MeV that have appeared 22) since the present work was first reported 6). The four strong l = 0 levels observed in the present work were there assigned energies of 3.94, 4.57, 5.16 and 5.56 MeV in agreement with the original energy values of the (p, ct) states we have chosen as corresponding to these (d, 3He) transitions and agreeing to within 10 keV with our assigned energies. This later report identified a level at 4.72 MeV excitation as l = 0, while the (t, ct) results indicate the population only of the 4.88 MeV level between the 4.57 and 5.15 MeV levels. The work of Phillips ~9) with the (ct, 7) reaction yields assignments of 2 + to a "4.01 MeV" level and 0 + to a "4.60 MeV" level. While this "4.01 MeV" level would appear to be identical with the present "3.94 MeV" level and such a shift of the energies reported from the (~, 7) work would make the "4.60 MeV" level correspond most closely in energy with the present 4.57 MeV level, a 0 + assignment cannot be reconciled with the observed I = 0 character of the 4.57 MeV level. Of course, the present reaction would not be expected 1) to populate any 0 + state other than the ground state with significant intensity. The (p, 2p) reaction should tend to populate the same two-hole configurations as does the (d, aHe) reaction. Although the energy resolution of the (p, 2p) data 2o) is such to prevent arty detailed comparison, the results of the two experiments are consistent. In particular, the (p, 2p) data suggest a state at "6.6 MeV", which is postulated to be the 4 + state formed by the configuration d~ 1 d~"1. This is a description which most aptly fits the 7.11 MeV level seen here. 4.2. RELATION OF EXPERIMENTAL RESULTS TO GENERAL FEATURES OF SHELL MODELS A shell model basis space of 2s½ and ld I orbits provides for the aSAr nucleus two 0 + states {[d~'2]]=o and [S~2]d=O}, tWO 2 + states {[d~2]i=2 and [s~1d~111=2} and a 1 + state {[s~'ld~a]s= 1}- The mixing between the states of the same J~ value
356
B . H . W I L D E N T H A L A N D E. N E W M A N
is expected to be small 1). All states except the [s~2]:= o state should be strongly populated by proton pick up of the appropriate l value from the dominant [d~-1]: = configuration of the ground state of 39K. (We express the configurations of 3SAr and 39K as holes relative to a closed 4°Ca core.) Inspection of the experimental results of the 39K(d, 3He)aSAr reaction, which are presented in fig. 2 and in table 2, shows that the ground and first excited states are excited by l = 2 transitions with about the proper ratio of spectroscopic strength for them to be identified, respectively, as the [d~ 2 ]: = o and [d~ 2 ]: = 2 states. However, instead of the two l = 0 transitions expected on the basis of the 2s~-ld~ model, four have been observed. In the following paragraphs, we explore possible explanations for these additional levels. If the various observed l = 0 levels are strongly mixed, the extra levels can originate either from configurations involving a hole in the d~ shell or from configurations of the type [d~3s~lf~]. That is, if we let ~N(I = 0) represent the wave function of one of the states of 3SAr populated by l = 0 pick up, then ~N(I-- 0) = AN[d~ls~-I]+BN[d~3 s~71f~], or
~kN(/ = 0) = A~ [ d i l s ~ l ] + B / q [di 1 s~-l], where AN and BN are of comparable magnitude. The B-components are only typical examples of the types of configurations which could mix with the [d~ls~ 1] components of these states. Because of the mixing, all such states could be populated via pick-up from 39K to just their [s~'~d~ ~] component. Thus, there would be no need, for example, to invoke significant ft admixtures in the ground state of 39K in order to explain pick-up to states which originate (before mixing) from excitations into the fp shell. If, on the other hand, the extra two I = 0 levels are not mixed strongly with the [s~ ld~ ~] levels, they again most likely have configurations of the form [d~ 3s~"1f~ ] but now cart ordy be populated via pick-up from components in the ground state of 39K of the type [d~ 3f~]j = ~" Consider the second alternative first. The strengths of the individual 1 = 0 transitions then imply considerable [f~] admixture in the 39K ground state and were this the case, an I = 2 transition to the 0 + level in 3SAr formed by the [d~-*f~] configuration should also be observed. The logical candidate for this state in the 3SAr spectrum exists at an excitation of 3.38 MeV [refs. 4,16,21)]. As noted, this state is populated with vanishing probability in the present experiment. If the [d~4f~] configuration for it is assumed, the limit on its spectroscopic factor sets an upper limit of 5 % on the [d~'3f~] admixture into 39K (g.s.). The absence of this or any other 1 = 2 transition of the required strength thus mitigates against extensive f~ admixtures in 39K (g.s.) and hence against the unmixed character of the 1 = 0 levels of 3SAr in question. It might be noted that evidence of fi admixtures in the target is difficult to obtain by directly observing l = 3 transitions. This is because their intrinsic D W cross sections are a factor of ~ 6 lower than those for l = 0 transitions
89K(d, 3He)
357
at, say, 18 ° , and also because any existing strength would be fragmented over several negative-parity levels. It thus remains to try to identify the origins of the extra l = 0 levels either as d~t hole states or as 2p-4h states. Evidence on the possibility of the existence of d~I hole states in the 4-5 M e V region of excitation is available f r o m recent shell-model calculations which have been performed in the complete sd shell basis 5). In these calculations, the k n o w n energy levels o f nuclei f r o m A = 35-39 were treated as a group with effective interactions of the "realistic" 2 3) and the "'surface delta" 24) forms. Tbe results indicate that there are no sd shell states other than those f r o m the s½-d~ subshells below 7-8 MeV (see subsect. 4.3). Thus, we are led to the probability that the extra " l = 0 " levels involve excitations of even numbers of particles into the f~ shell, that they are strongly mixed with the regular (s~ 1-d~ 1) states, and that they are populated in the pick-up reaction by way of these (s~-1-d~ 1) components. TABLE4 Predictions of s-d shell models for energies and (d, aHe) S-factors of positive-parity levels of 3aAr State .Inn
Exp. Ex
C2S(j)
2-shell (1964) Ex
C2S(j)
2-shell (1968) Ex
C2S(j)
"Realistic" Ex
C2S(j)
MSDI Ex
C2S(j)
(MeV) 1=0/1=2 (MeV) 1=0/1=2 (MeV) l=0/1=2 (MeV) l = 0 / l = 2 (MeV) 1=0/1=2 01+ 0.00 02+ ? 11+ 5.55 21+ 2 . 1 7 22+ "4.59" 41+ ( 7 . 1 1
0/0.49 ? 0.78/~0 ~0/2.5 1.2/~0.2 0/0.44)
0.00 6.67 6.94 2.55 4.39
0/0.47 0/0.03 0.75/0 0.16/2.26 1.09/0.33
0.00 5.49 5.59 2.29 4.4l
0/0.45 0/0.05 0.75/0 0.23/2.05 1.03/0.45
0.00 8.03 5.96 1.56 5.09 7.14
0/0.46 0/0.009 0.75/0 0.016/2.40 1.13/0.06 0/2.2
0.00 0/0.45 6.81 0/0.03 5.17 0.75/0 2.03 0.11/2.24 4.38 1.10/0.24 8.85 0/2.2
F r o m the preceding arguments, it follows that the distribution o f l = 0 spectroscopic strength between the 2 + and 1 + levels that is predicted by the (2s,-ld~) calculations should still hold g o o d if the fragmented strengths are properly summed. In table 4 some shell-model predictions for the spectroscopic factors are compared with the experimental values. G o o d agreement between theory and experiment can be obtained if the levels at 3.94, 4.57 and 5.15 M e V are all assumed to be 2 ÷, and the 5.55 MeV level is 1 +. This is a logical step on other grounds since 2 ÷ states f r o m the various possible configurations with f2n admixtures should come at considerably lower energies than 1 + states. As was mentioned previously, we assume that the 3SAr [0 + ] level at 3.38 MeV excitation arises f r o m a [ ( d ~ a ) 1 = o - ( f ~ ) j = 0 ] configuration. This assumption is consistent with the calculations o f Ern6 4) for such d-f levels and with all of the s-d shell-model calculations. In all o f the latter calculations, the 0 + level formed l:y [s~ 2 ]i = o is predicted to lie at a r o u n d 6 M e V excitation.
358
B.H. WILDENTHAL AND E. NEWMAN
4.3. B E A R I N G O F E X P E R I M E N T A L R E S U L T S U P O N D E T A I L S O F S H E L L - M O D E L C A L CULATIONS
As discussed in subsect. 4.2, shell-model calculations which hope to provide a quantitative explanation of the observed positive-parity level structure must include mechanisms for freely exciting at least two particles into the lf÷ shell. To our knowledge, such calculations do not exist at present. We make the assumption in the following discussion that the only significant perturbation of the s-d spectrum in 3SAr by 2p-4h excitations occurs in the aforementioned fragmentation into three main components of the 2 + state formed by the [s~ld~ 1] configuration. Summing the spectroscopic strengths of these states results irt a center of gravity for the [s~'ld~l]2 + state at 4.59 MeV excitation. We now discuss various s-d shell calculations in the light of this "simplified" level structure. The calculation of the level structure of 3SAr that is most independent of preconceptions of what is "proper" is performed with a shell-model effective interaction calculated from the Hamada-Johnston two-nucleon scattering potential with reaction matrix techniques 23). (The model space for the calculation encompasses all ld~-2s~ld~ configurations allowed by the exclusion principle.) The values of the singleparticle energies of the ld~ and 2s~ orbits relative to the ld~ orbit are o~tained from the 170 spectrum. The results for 3SAr e~ergy levels are shown in table 4 with predicted spectroscopic factors. Another calculation in this model space was made by using the modified surface delta interaction 25) as the effective Hamiltonian. The four parameters of the MSDI and the two single-particle energy splittings are fixed by fitting 30-odd energy level splittings for A = 35-39 nuclei. These results for 3SAr are also shown in table 4. One unambiguous conclusion from both tbese calculations mentioned in subsect. 4.2 is that excitations out of the ld~ sbell do not play a significant role in the low-lying energy levels of 38Ar. This result implies that a truncation of the model space to the d~ and s~r shells would yield an equally viable model below 6 MeV excitation. Glaudemans et al. 1) have calculated the properties of low-lying nuclear energy levels for 30 < A < 40 nuclei in such a model. In these calculations, the entire effective Hamiltonian was determined from a least-squares fit to energy level data. Only the ground and first excited states of 3 SAr were included in the data set used by Glaudemans, therefore his results for 3SAr are also free from any assumptions about the higher-lying level structure irt this nucleus. The results (energy levels and S-factors) are presented in table 4. Examination of the results of the three calculations mentioned thus far led to the conclusions about the experimental data given in subsect. 4.2, namely that the 2 + state formed by the [s~ ~-d~ 1] configuration is fragmented into three components, and that the 5.55 MeV level is probably 1 ÷. A recent revision 3) of the Glaudemans calculations to include a larger data set tends to confirm these assumptions. The results of this calculation for 3SAr agree quite well with our simplified experimental spectrum as cart be seen from table 4. In particular, this calculation yields an energy of the 1 ÷ level that is close to 5.5 MeV. The results of both of the three shell calculations mentioned earlier indicate that
a°K(d, 8He)
359
the lowest dt-hole state is 4 +, and that it should have an energy of 7-9 MeV. The state observed at 7.17 MeV with an l = 2 angular distribution doubtless has a [d~ 1-d~"1] character. However, both the discrepancy between the theoretical and experimental energies and between the predicted and observed S-factors argue against the 7.11 MeV level being the pure [d~"1-d~"1]s = 4 state. Most probably this is the lowest fragment of a multiplet of J = 4 levels, which is similar to the lower energy 2 ÷ levels. 5. Conclusions
Several shell-model calculations which consider only s-d configurations have yielded mutually consistent results for nuclei of A = 35-39. Major facets of the experimental positive-parity level structure in this region are in agreement with these theories. However, in the case of the mass-38, T = 1 nuclei (3BAr and 3aCa), there are lowlying 0 ÷ states which are completely extraneous to the systematics of the s-d shell calculations. These states thus constitute the first evidence for important "2p-4h"like admixtures 4) in the mass-38, T = 1 system. Proton pick-up from agK does not excite this 0 + 2p-4h state in 38Ar, thus indicating that it is fairly pure, and that 39K (g.s.) possesses small 2p-3h admixtures. The same reaction does excite more lp = 0 levels (necessarily having J~ of 2 + or 1 +) than can be constructed in the s-d space. These extra levels are also the result of 2p-4h excitations, but in this case, the 2p-4h states and the s-d states are strongly mixed. We suggest that the two extra l = 0 levels have spin and parity of 2 ÷, and that the 1 + level at 5.55 MeV excitation is the pure [s~ l_d~ 1]s = 1 state. An lp = 2 level at 7.11 MeV excitation apparently results from pick-up of a ld~ particle, but calculations indicate that it is only the lowest element of a fragmented [d~-l d ~ 1 ]j = 4 state. The s-d shell-model calculations give a good account of the experimental centroids of the l = 2 and l = 0 pick-up strength and of some of the details of its distribution. The present experimental results further emphasize, however, the desirability of expanded shell-model calculations, which take into consideration both the s-d and the f-p shells. The results of Ern6 4) for the negative-parity levels and the low excited 0 ÷ states are very encouraging in this respect, but the limitations on his configuration space are such as to exclude the type of phenomena observed and discussed here. The requirement of reproducing the detailed distribution of spectroscopic strengths observed in the 39K(d, 3He)38Ar reaction will present a stringent test for the more extensive calculations of the future. References 1) 2) 3) 4) 5) 6)
P. P. P. F. B. B.
W. M. Glaudemans, G, Wiechers and P. J. Brussaard, Nucl. Phys. 56 (1964) 529 and 548 Federman and I. Talmi, Phys. Lett. 15 (1965) 165; 22 (1966) 469 W. M. Glaudemans, E. C. Halbert, J. B. McGrory and B. H. Wildenthal, to be published C. Ern6, Nucl. Phys. 84 (1966) 91 H. Wildenthal, J. B. McGrory, E. C. Halbert and P. W. M. Glaudemans, to be published H. Wildenthal, M. R. Cates arid E. Newman, Bull. Am. Phys. Soc. 12 (1967) 1189
360
B. H. WILDENTHAL AND E. NEWMAN
7) G. A. P. Engelbertink, H. Lindman and M. J. N. Jacobs, Nucl. Phys. A107 (1968) 305 8) R. H. Bassel, R. M. Drisko and G. R. Satchler, Oak Ridge National Laboratory Report ORNL-3240 (1962) unpublished 9) J. C. Hiebert, E. Newman and R. H. Bassel, Phys. Rev. 154 (1967) 898 10) E. Newman, L. C. Becket, B. M. Preedom and J. C. Hiebert, Nucl. Phys. AI00 (1967) 11) D. J. Baugh, G. J. B. Pyle, P. M. Rolph and S. M. Scarrott, Nucl. Phys. A95 (1967) 115 12) R. M. Drisko, unpublished 13) E. F. Gibson, B. W. Ridley, J. J. Kraushaar, M. E. Rickey and R. H. Bassel, Phys. Rev. (1967) 1194 14) B. M. Preedom, E. Newman and J. C. Hiebert, Phys. Rev. 166 (1968) 1156 15) B. H. Wildenthal and E. Newman, Phys. Rev. 167 (1968) 1027 16) P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 17) I. J. Taylor, Nucl. Phys. 41 (1963) 227 18) R. G. Alias, L. Meyer-Schiitzmeister and D. von Ehrenstein, Nucl. Phys. 61 (1965) 289 19) W. R. Phillips, Nucl. Phys. 60 (1964) 544 20) D. Newton, G. L. Salmon and A. B. Clegg, Nucl. Phys. 82 (1966) 513 21) C. A. Whitten, Bull. Am. Phys. Soc. 10 (1965) 121 22) T. Wei, W. S. Gray, J. J~inecke and R. M. Polichar, Bull. Am. Phys. Soc. 12 (1967) 681 23) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71; T. T. S. Kuo and E. C. Halbert, private communication 24) P. W. M. Glaudemans, B. H. Wildenthal and J. B. McGrory, Phys. Lett. 21 (1966) 427 25) P. W. M. Glaudemans, P. J. Brussaxtrd and B. H. Wildenthal, Nucl. Phys. A102 (1967)
No.
225
155
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Nuclear Physics A l l 8 (1968) 347--360; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
39K(d, 3He)SSAr A N D T H E s-d S H E L L
STRUCTURE
OF
SeAr
B. H. WILDENTHAL t and E. NEWMAN Oak Ridge National Laboratory, Oak Ridge, Tennessee tt Received 1 July 1968 Abstract: The positive-parity level structure of 8aAr is investigated with the 89K(d, 3He)SaAr reaction
on a KI target at a bombarding energy of 34.5 MeV. Seven residual levels are found to be strongly excited, three with lp = 2 characteristics at 0.00, 2.17 and 7.11 MeV excitation energy and four with predominant lp = 0 characteristics at 3.94, 4.57, 5.15 and 5.55 MeV excitation energy. These experimental results are examined from the viewpoint of several recent shellmodel calculations. No extant calculations provide a quantitative interpretation of the occurrence of as many as four strong l = 0 levels. It seems likely that the extra l = 0 levels arise from 2p-4h excitations into the If shell and are populated via the large components of the (s~t-d~~) configuration mixed into these 2p-4h states. The (d, 3He) reaction on the x*vI in the target yields evidence that the [+ ground state of this nucleus contains significant 2d~ admixtures. E ]
I
NUCLEAR REACTIONS SaK, 1271 (d, aHe), E = 34.5 MeV; measured o'(EZHe, 0), Q. 3eAr, 126Te deduced levels, l, ~r. Natural target.
[
1
1. Introduction
N u c l e a r systems which can be described as either two particles o r two holes rem o v e d f r o m m a j o r shell closures are o f p a r t i c u l a r interest in the s t u d y o f the shell m o d e l because o f their c o n c e p t u a l simplicity. Such a nucleus is a SAt with 18 p r o t o n s and 20 n e u t r o n s ; its low-lying level structure can be described to a first a p p r o x i m a t i o n (ref. 1)) b y couplings o f two holes in the 2s+ a n d ld+ shells. H y p o t h e t i c a l l y , such nuclei with their small n u m b e r o f a l l o w a b l e shell-model states can p r o v i d e i m m e d i a t e a n d basic i n f o r m a t i o n a b o u t the t w o - p a r t i c l e (hole) effective interaction. In practice, h o w ever, such h o p e s are often frustrated because the effects o f configurations outside the n o m i n a l l y a d e q u a t e basis d e s t r o y the idealized simplicity o f the system. T w o n o t o r i o u s examples o f this type o f c o m p l i c a t i o n are the spectra o f 18 0 a n d +2Ca, each o f which possesses an excess o f low-lying 0 + and 2 ÷ levels; their e x p l a n a t i o n requires d e f o r m a t i o n s o r excitations o f the ~60 a n d +°Ca cores, respectively 2). Shell-model calculations have been p e r f o r m e d for both the positive- a n d negativep a r i t y levels o f the nuclei below +°Ca. T h e p o s i t i v e - p a r i t y levels have been t r e a t e d with a 2 s , - l d + basis ~,3), a n d similar calculations 4) in a ld+-lf~ basis have been p e r f o r m e d for n e g a t i v e - p a r i t y levels f o r m e d b y exciting one particle into the f~ * USAEC Postdoctoral Fellow under appointment from Oak Ridge Associated Universities; present address: Texas A and M University, College Station, Texas. t* Research sponsored by the U. S. Atomic Energy Commission under contract with Union Carbide Corporation. 347
348
B . H . W I L D E N T H A L A N D E. N E W M A N
shell and for 0 + levels formed with two particles excited into the fk shell and coupled to J = 0. Recently, calculations for the positive-parity levels have been performed in a complete 2s- 1d basis 5). Comparisons of the predictions of some of these calculations to experimental results 6) from the 39K(d, 3He)3 BAr reaction are of multiple interest. One aspect of the investigation is to determine the extent to which the simple 2s½-1d~: model can explain the low-lying positive-parity levels of 3aAr; that is, do effects like those mentioned for 180 and ~2Ca intrude into the level structure of 3BAr? In a related sense, we want to identify exactly which levels of 3 BAr have a predominant (s,r-d~) -2 character. Another facet of interest is the search for evidence of f~ admixtures. Finally, the detailed results of energy spacings and spectroscopic factors for the (s½-d~) -2 levels, provided these can be assigned without too much ambiguity, will yield tests of the internal consistency of the calculations in the s-d shell model.
2. Experimental procedure Most of the data for the present study were obtained from bombardment of a K I target with 34.5 MeV deuterons from the ORIC. Reaction products were detected and identified in a A E - E solid state detector telescope. The resultant E + AE and AE signals were displayed and stored in a 50 x 400-channel map, and the 3He events were extracted from the array off-line. A sample 3He spectrum is shown in fig. 1. As can be seen, the energy resolution for these data was about 100 keV F W H M . A significant amount of this broadening came from the target thickness. The target was prepared by vacuum evaporation of KI (containing the naturally occurring ratio of potassium isotopes) onto thin carbon foils. Supplementary studies were carried out with targets of metallic potassium evaporated onto mylar film, the potassium being partially oxidized before bombardment. Seven of the particle groups prominent in fig. 1 are identified with transitions to levels of 3aAr and are labeled by the corresponding energies of excitation in the 3aAr system. The energy calibration of the spectra is based primarily on a value of 2.168 MeV for the spacing between the ground and first excited states 7). The angular distributions of these seven transitions are presented in sect. 3. No attempt was made to extract and analyse data for the multitude of weakly excited levels comprising the background evident in fig. 1 because of the ambiguities in their parentages resulting from the 7 ~ 41K and the 127I in the target. Some particle groups from the (d, 3He) reaction on these nuclei are identified in fig. 1, and the angular distributions to the first two levels of 126Te are presented and discussed later.
3. Angular distributions and DWBA analysis The differential cross sections of the transitions to the seven levels of 3 SAr for which data were reduced are presented in fig. 2. The data points are a superposition of results from the oxidized metal target (four points) and the K1 target (nine points).
I
.}"
100
~r~.~7o;,/ o t40
f..; " :
-
• *e a
!/
38Ar(5.55 M~
Ed = 54.5 MeV 8L= 13.0 o
KI (d, 3HeI)
t80
II
~~°V~
38Ar
220 260 CHANNEL NUMBER
:38Ar(3.94 MeV)
MeV)
]
126Te(0.661MeV)]~
~
300
/
340
380
i'~
(g .s)
38Ar(g.s.)
,'2~e~,,~.eV,|~
I
-
/
"38Ar(2.47 MeV)
30
34
%
v
Fig. 1. Spectrum of 8He particles emitted from a KI target. Levels of 3BAr which are excited with measurable strength by the (d, aHe) reaction on SgK are explicitly noted. States in 3SAr known to exist but unexcited in the present reaction are noted by arrows above the energy calibration line. States in 4°Ar [from 41K(d, 3He)] are indicated by arrows below the line. States in l~eTe[from lz~I(d, 3He)] are explicitly designated.
0
50
60
t50
© {00
CO ,I-Z
n" tad t]
-F Q)
J taJ Z Z
200
250
~D
=-
350
B.H. WILDENTHAL AND E. NEWMAN
The curves in fig. 2 are the results of local zero-range DWBA calculations s) for the (d, 3He) reaction to the various levels. The appropriateness of such an analysis for the present reaction has been fully discussedelsewhere 9). The optical-model para-
0.~ 0..~ 4C
2 2 ~
0.~ 0.2
0.~ 0.2 OJ 0.0~
0.02 0.0' 0
t0
20
30
40
OC.M(deg)
50
60
0
t0
20
30
40
50
60
ec.M.(deg)
F i g . 2. A n g u l a r distributions o f the 3*K(d, 3He)88Ar reaction. The curves are D W B A predictions for these transitions. The dashed curves represent the predictions for the alternate/-values (l = 0 in left c o l u m n and l = 2 in right column).
meters used in the present calculations are given in table 1. The deuteron parameters are obtained from the analysis 10) of the elastic scattering of 34.4 MeV deuterons on 40 Ca. The 3 He parameters are from a re-analysis of data 11) for 29 MeV 3He scattering
89K(d, 8He)
351
o n 4°Ca with the optical-model search code H U N T E R ~2). I n this re-analysis, the geometries of the real a n d i m a g i n a r y potentials were held c o n s t a n t to values suggested by G i b s o n e t al. ~ 3), a n d the e n s u i n g results for the analysis of the 29 MeV scattering resemble the results of G i b s o n e t al. for 3He scattering f r o m 4°Ca at 22, 37.7 a n d 64.3 MeV. The use in the D W B A calculations of a 3He potential of the form employed in ref. ~1) results in theoretical a n g u l a r d i s t r i b u t i o n s which deviate seriously f r o m the experimental distributions. This same general effect has been noticed for other nuclei 9, 14, 1 5). TABLE 1
Optical-model parameters used in DWBA calculations
39K (deuteron) 1'7I (deuteron) ~Ar (aHe) 126Te (SHe) Bound state
V
ro
roe
a
W
r'o
a'
WD
Vs.o.
(MeV)
(fm)
(fro)
(fro)
(MeV)
(fm)
(fm)
(MeV)
(MeV)
95.13 99.42 178.9 175.0
1.066 1.118 1.14 1.14 1.25
1.30 1.30 1.40 1.40 1.25
0.81 0.822 0.723 0.723 0.65
1.33 1.24 1.60 1.60 25
0.754 0.859 0.81 0.81
11.8 15.6 0 0
0 0 16.1 16.0 2s.o. :
7.0 7.0 0 0
As can be seen from fig. 2 a n d as listed in table 2, the g r o u n d a n d first excited states of 3BAr are p o p u l a t e d b y l = 2 transitions. The excited states at 3.94, 4.57, 5.15 a n d 5.55 MeV have a n g u l a r distributions which are d o m i n a t e d by the l = 0 shape; the d i s t r i b u t i o n of the state at 7.11 M e V excitation has a n l = 2 shape. Spectroscopic factors c a n be extracted with confidence from the l = 2 distributions o f fig. 2, since the good fit of the D W prediction to the purely l = 2 g r o u n d state t r a n s i t i o n shows that even small l = 0 admixtures would be noticeable. I n the case of TABLE 2
Experimental results for levels of SSAr from 89K(d, 3He)aSAr Ex (exp) (MeV) ~Ar
12eTe
0.00 2.17 3.94 4.57 5.15 5.55 7.11 0.00 0.66
l
2 2 0(-t-2) 0 0 0 2 2 (2)
Jn
0+ a) 2+ ~) 2+ n) (1, 2)+ (1, 2)+ (1, 2)+ (0-4) + 0+ 2+
C*S(I = O) b)
0.00-0.03 0.26-0.22 0.62-0.51 0.33-0.28 0.78-0.70
C*S(I = 2) b)
0.49 2.50-2.41 0.13-0.32 0.00-0.32 0.00-4).17 0.00--0.22 0.44(j=~) 0.39 0.18
s) Ref. le). b) Values the of C2S are normalized so that the sum of the maximum l : 0 strengths in 3SAr is equal to 2.
352
B.H.
W I L D E N T H A L A N D E. N E W M A N
predominantly l = 0 distributions, however, the amount of l = 2 spectroscopic strength is not easy to ascertain, because the I = 2 distribution has an intrinsic magnitude that is about one third that of the l = 0 distribution, and in addition is less structured. The fits of l = 0 calculations to the four "l = 0" states are such as would be considered satisfactory were the transitions known to be restricted to a single /-transfer (with perhaps some reservation about the 3.94 MeV distribution). The fact that the l = 2 mode is allowed in the present reaction, however, and that addition of some l = 2 strength, by smoothing the extreme structure of the l = 0 calculations, improves the agreement between calculation and experiment, creates uncertainty in the amount of l = 2 strength which should properly be attributed to the "l = 0" levels. Spectroscopic factors are presented which are based both on the assumptions that the transitions are purely l = 0 and, alternatively, that they proceed with sufficient l = 2 strength to fill in the first l = 0 minimum to the experimental values. In the latter cases, there is an attendant diminution of the l = 0 strength. An accurate value for the absolute cross section of the (d, 3He) reaction on 39K was not obtained. The cross-section scales in fig. 2 are, however, interconsistent. Hence, in lieu of absolute units, the spectroscopic factors extracted for the various l = 0 and l = 2 transitions with the D W B A analysis are expressed in a normalization such that the total o~served l = 0 strength satisfies the assumptions that the 2s½ shell is completely full in the ground state of 39K and that all l = 0 pickup strength has been measured, i.e., ~ C 2 S ( l = O) = 2. The experimental energies of the seven states of 3SAr under discussion and their spectroscopic properties as derived from the DW analysis are listed in table 2. We see that by using the m a x i m u m l = 0 strengths for the purpose of establishing the normalization, the summed l = 2 strength from the ground state and the 2.17 MeV and 3.94 MeV excited states equals 3.1, which is very close to the expected value of 3 for d~ pick-up. This seems to confirm the assumption that the l = 0 transitions are fairly pure, since otherwise the l = 2 strengths would amount to significantly more than the sum rule limit for pick-up from the ld~ shell. The differential cross sections for the 1271(d, 3He)126Te reaction to the ground and to the 0.663 MeV first excited states of 126Te are shown in fig. 3. The cross-section scale is the same as that for fig. 2. The curves in fig. 3 are the results of D W calculations for these transitions made under the assumption that the extracted protons come from the 2d~ shell. The deuteron and 3He optical-model parameters listed in table 1 are again taken from the formulae of refs. lO, 13), respectively. The spectroscopic factors for these two levels of 126Te are also listed in table 2. The normalization used for the potassium numbers was carried over for the 126[ values, it being assumed that the elemental and isotopic ratios of K I remained constant during the evaporation of the target material. The three protons exterior to the Z = 50 core in 127I are expected to be distributed over the 2dt and lg~ orbits. The spectroscopic strength observed for l = 2 pick-up from 127I indicates that the ~+ ground state contains at least a single 2dl particle in
agK(d, aHe)
353
its dominant configuration, thus ruling out the possibility that the 2~+ state arises primarily from the coupling of three Ig~ particles to J = {. 2
0.5
02 0.4
g ~0.05 o
0.2
0.1
0.0~
0.02 0.04
0
40
20 30 OC.M. (deg)
40
50
Fig. 3. Angular distributions ofthelaTI(d, aHe)l~6Te reaction and DWBApredictionsfor 2d¢ pick-up.
4. Discussion 4.1. I D E N T I F I C A T I O N O F P R E S E N T L Y O B S E R V E D LEVELS O F aSAr W I T H T H O S E OBSERVED VIA DIFFERENT REACTIONS
The published literature 16) on the levels of 3SAr includes reports of their population via the (t, g) [ref. 17)], (p, ~) [ref. is)], (p, 7) [ref. ~)], (g, 7) [ref. 19)] and (p, 2p) [ref. 20)] mechanisms. The establishment of identities between levels excited in the various experiments is important for both the preliminary and concluding aspects of the analysis of the present data. The (p, g) data and the recent (p, 7) data were used as the primary standards in establishing such a correspondence, the absolute energy standard being taken from the Ge-Li spectrometer 7-ray data of Engelbertink et al. 7). The excitation energies from the (p, ~) data 1 s) taken with a magnetic spectrograph are about 10 keV high at 5 MeV compared with the 7-ray
354
B. H . W I L D E N T H A L
A N D E. N E W M A N
energies, and the energies of the several levels populated in this region only by the (p, ~) reaction were accordingly modified for the purpose o f the present discussion. The levels o f 3SAr up to an excitation of 5.70 M e V are listed in table 3. We assume for the remainder o f the discussion that this represents a complete catalog o f levels for this energy range, although such m a y well n o t be the case. The identification o f the levels presently observed below an excitation energy of 4 MeV seems clear. We note that the 0 + level at 3.38 M e V excitation, which is strongly excited with the 4°Ar(p, t)3SAr reaction 21), is n o t populated at all here. Similarly the 3 - state 16) at 3.81 MeV excitation is very weakly populated, although in this case the proximity of other levels prevents a rigorous limit being set o n its m a x i m u m cross section. TABLE3 Catalog of levels of UAr with excitation energies 2.0 ~ Ex _~ 5.8 MeV and the reactions by which they are observed ~) (MeV) 2.168 3.377 3.810 3.936 4.480 4.566 4.585 4.710 4.877 5.083 5.155 5.347 5.513 5.551 5.591 5.658 5.733 5.825 5.853
(P, 7) ~)
(13,e¢) b)
(t, ~) e)
y y y y y y y y y y
y y y y y y y y y y y y y y y y y y y
y
y
y y
y
y
y
y y y y y
(d, erie) a)
j~ a, e) 2+ 0+ 32+ 4(1, 2) + a) 5-
y y y
y
(1, 2)+ a)
y
y
(1, 2)+ a) 5-
y
The symbol y signifies the observation of a particular level by the indicated reaction. s) Ref. 7). b) Ref. xs). c) Ref. x~). d) Present work. e) Ref. xe). The F W H M resolution values o f ~ 100 keV obtained for the particle groups in the present experiment together with the energy calibration o f the spectra based on the 0.00-2.17 MeV and 2.17-3.94 MeV level spacings enable energy assignments to be made to the higher-lying levels with an uncertainty o f ___45 keV. Unfortunately, as is evident f r o m inspection of table 3, this a m o u n t o f uncertainty in the measured energies is such as to allow ambiguity in the identity o f the 4.57 and 5.55 MeV levels observed in this (d, 3He) experiment. We postulate that our "4.57 M e V " level corresponds to the 4.566 MeV level o f
3*K(d,SHe)
355
Engelbertink rather than to the 4.585 MeV level. In addition to the slight bias available for such an assignment from the energy calibration, it is motivated by the defioite 5assignment made to the higher level by Engelbertink, an l = 0 transition being forbidden to such a level. The tentative 2 - assignment for the 4.566 MeV level should, on the basis of the present data, be changed to 2 +, as will be discussed further in subsect. 4.2. There is only a small uncertainty in the assignment of the "5.15 MeV" level to the 5.154 MeV level, which is seen in the (p, 0t) reaction but not observed in the 7 ray data. Three levels seen in the (p, ~t) work could possibly be identified with the "5.55 MeV" level. Of these, the 5.513 MeV level has been assigned as 3t-). The energy calibration favors the identification of the (d, 3He) level with the 5.550 MeV level. Confirmation is available for the foregoing assignments from the results of the (t, ~) reaction a7), which should tend to populate the same states as the (d, aHe) reaction. Levels observed in the (t, ct) experiment are noted in table 3. Additional confirmation comes from results of a (d, 3He) experiment on a9K at 21 MeV that have appeared 22) since the present work was first reported 6). The four strong l = 0 levels observed in the present work were there assigned energies of 3.94, 4.57, 5.16 and 5.56 MeV in agreement with the original energy values of the (p, ct) states we have chosen as corresponding to these (d, 3He) transitions and agreeing to within 10 keV with our assigned energies. This later report identified a level at 4.72 MeV excitation as l = 0, while the (t, ct) results indicate the population only of the 4.88 MeV level between the 4.57 and 5.15 MeV levels. The work of Phillips ~9) with the (ct, 7) reaction yields assignments of 2 + to a "4.01 MeV" level and 0 + to a "4.60 MeV" level. While this "4.01 MeV" level would appear to be identical with the present "3.94 MeV" level and such a shift of the energies reported from the (~, 7) work would make the "4.60 MeV" level correspond most closely in energy with the present 4.57 MeV level, a 0 + assignment cannot be reconciled with the observed I = 0 character of the 4.57 MeV level. Of course, the present reaction would not be expected 1) to populate any 0 + state other than the ground state with significant intensity. The (p, 2p) reaction should tend to populate the same two-hole configurations as does the (d, aHe) reaction. Although the energy resolution of the (p, 2p) data 2o) is such to prevent arty detailed comparison, the results of the two experiments are consistent. In particular, the (p, 2p) data suggest a state at "6.6 MeV", which is postulated to be the 4 + state formed by the configuration d~ 1 d~"1. This is a description which most aptly fits the 7.11 MeV level seen here. 4.2. RELATION OF EXPERIMENTAL RESULTS TO GENERAL FEATURES OF SHELL MODELS A shell model basis space of 2s½ and ld I orbits provides for the aSAr nucleus two 0 + states {[d~'2]]=o and [S~2]d=O}, tWO 2 + states {[d~2]i=2 and [s~1d~111=2} and a 1 + state {[s~'ld~a]s= 1}- The mixing between the states of the same J~ value
356
B . H . W I L D E N T H A L A N D E. N E W M A N
is expected to be small 1). All states except the [s~2]:= o state should be strongly populated by proton pick up of the appropriate l value from the dominant [d~-1]: = configuration of the ground state of 39K. (We express the configurations of 3SAr and 39K as holes relative to a closed 4°Ca core.) Inspection of the experimental results of the 39K(d, 3He)aSAr reaction, which are presented in fig. 2 and in table 2, shows that the ground and first excited states are excited by l = 2 transitions with about the proper ratio of spectroscopic strength for them to be identified, respectively, as the [d~ 2 ]: = o and [d~ 2 ]: = 2 states. However, instead of the two l = 0 transitions expected on the basis of the 2s~-ld~ model, four have been observed. In the following paragraphs, we explore possible explanations for these additional levels. If the various observed l = 0 levels are strongly mixed, the extra levels can originate either from configurations involving a hole in the d~ shell or from configurations of the type [d~3s~lf~]. That is, if we let ~N(I = 0) represent the wave function of one of the states of 3SAr populated by l = 0 pick up, then ~N(I-- 0) = AN[d~ls~-I]+BN[d~3 s~71f~], or
~kN(/ = 0) = A~ [ d i l s ~ l ] + B / q [di 1 s~-l], where AN and BN are of comparable magnitude. The B-components are only typical examples of the types of configurations which could mix with the [d~ls~ 1] components of these states. Because of the mixing, all such states could be populated via pick-up from 39K to just their [s~'~d~ ~] component. Thus, there would be no need, for example, to invoke significant ft admixtures in the ground state of 39K in order to explain pick-up to states which originate (before mixing) from excitations into the fp shell. If, on the other hand, the extra two I = 0 levels are not mixed strongly with the [s~ ld~ ~] levels, they again most likely have configurations of the form [d~ 3s~"1f~ ] but now cart ordy be populated via pick-up from components in the ground state of 39K of the type [d~ 3f~]j = ~" Consider the second alternative first. The strengths of the individual 1 = 0 transitions then imply considerable [f~] admixture in the 39K ground state and were this the case, an I = 2 transition to the 0 + level in 3SAr formed by the [d~-*f~] configuration should also be observed. The logical candidate for this state in the 3SAr spectrum exists at an excitation of 3.38 MeV [refs. 4,16,21)]. As noted, this state is populated with vanishing probability in the present experiment. If the [d~4f~] configuration for it is assumed, the limit on its spectroscopic factor sets an upper limit of 5 % on the [d~'3f~] admixture into 39K (g.s.). The absence of this or any other 1 = 2 transition of the required strength thus mitigates against extensive f~ admixtures in 39K (g.s.) and hence against the unmixed character of the 1 = 0 levels of 3SAr in question. It might be noted that evidence of fi admixtures in the target is difficult to obtain by directly observing l = 3 transitions. This is because their intrinsic D W cross sections are a factor of ~ 6 lower than those for l = 0 transitions
89K(d, 3He)
357
at, say, 18 ° , and also because any existing strength would be fragmented over several negative-parity levels. It thus remains to try to identify the origins of the extra l = 0 levels either as d~t hole states or as 2p-4h states. Evidence on the possibility of the existence of d~I hole states in the 4-5 M e V region of excitation is available f r o m recent shell-model calculations which have been performed in the complete sd shell basis 5). In these calculations, the k n o w n energy levels o f nuclei f r o m A = 35-39 were treated as a group with effective interactions of the "realistic" 2 3) and the "'surface delta" 24) forms. Tbe results indicate that there are no sd shell states other than those f r o m the s½-d~ subshells below 7-8 MeV (see subsect. 4.3). Thus, we are led to the probability that the extra " l = 0 " levels involve excitations of even numbers of particles into the f~ shell, that they are strongly mixed with the regular (s~ 1-d~ 1) states, and that they are populated in the pick-up reaction by way of these (s~-1-d~ 1) components. TABLE4 Predictions of s-d shell models for energies and (d, aHe) S-factors of positive-parity levels of 3aAr State .Inn
Exp. Ex
C2S(j)
2-shell (1964) Ex
C2S(j)
2-shell (1968) Ex
C2S(j)
"Realistic" Ex
C2S(j)
MSDI Ex
C2S(j)
(MeV) 1=0/1=2 (MeV) 1=0/1=2 (MeV) l=0/1=2 (MeV) l = 0 / l = 2 (MeV) 1=0/1=2 01+ 0.00 02+ ? 11+ 5.55 21+ 2 . 1 7 22+ "4.59" 41+ ( 7 . 1 1
0/0.49 ? 0.78/~0 ~0/2.5 1.2/~0.2 0/0.44)
0.00 6.67 6.94 2.55 4.39
0/0.47 0/0.03 0.75/0 0.16/2.26 1.09/0.33
0.00 5.49 5.59 2.29 4.4l
0/0.45 0/0.05 0.75/0 0.23/2.05 1.03/0.45
0.00 8.03 5.96 1.56 5.09 7.14
0/0.46 0/0.009 0.75/0 0.016/2.40 1.13/0.06 0/2.2
0.00 0/0.45 6.81 0/0.03 5.17 0.75/0 2.03 0.11/2.24 4.38 1.10/0.24 8.85 0/2.2
F r o m the preceding arguments, it follows that the distribution o f l = 0 spectroscopic strength between the 2 + and 1 + levels that is predicted by the (2s,-ld~) calculations should still hold g o o d if the fragmented strengths are properly summed. In table 4 some shell-model predictions for the spectroscopic factors are compared with the experimental values. G o o d agreement between theory and experiment can be obtained if the levels at 3.94, 4.57 and 5.15 M e V are all assumed to be 2 ÷, and the 5.55 MeV level is 1 +. This is a logical step on other grounds since 2 ÷ states f r o m the various possible configurations with f2n admixtures should come at considerably lower energies than 1 + states. As was mentioned previously, we assume that the 3SAr [0 + ] level at 3.38 MeV excitation arises f r o m a [ ( d ~ a ) 1 = o - ( f ~ ) j = 0 ] configuration. This assumption is consistent with the calculations o f Ern6 4) for such d-f levels and with all of the s-d shell-model calculations. In all o f the latter calculations, the 0 + level formed l:y [s~ 2 ]i = o is predicted to lie at a r o u n d 6 M e V excitation.
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B.H. WILDENTHAL AND E. NEWMAN
4.3. B E A R I N G O F E X P E R I M E N T A L R E S U L T S U P O N D E T A I L S O F S H E L L - M O D E L C A L CULATIONS
As discussed in subsect. 4.2, shell-model calculations which hope to provide a quantitative explanation of the observed positive-parity level structure must include mechanisms for freely exciting at least two particles into the lf÷ shell. To our knowledge, such calculations do not exist at present. We make the assumption in the following discussion that the only significant perturbation of the s-d spectrum in 3SAr by 2p-4h excitations occurs in the aforementioned fragmentation into three main components of the 2 + state formed by the [s~ld~ 1] configuration. Summing the spectroscopic strengths of these states results irt a center of gravity for the [s~'ld~l]2 + state at 4.59 MeV excitation. We now discuss various s-d shell calculations in the light of this "simplified" level structure. The calculation of the level structure of 3SAr that is most independent of preconceptions of what is "proper" is performed with a shell-model effective interaction calculated from the Hamada-Johnston two-nucleon scattering potential with reaction matrix techniques 23). (The model space for the calculation encompasses all ld~-2s~ld~ configurations allowed by the exclusion principle.) The values of the singleparticle energies of the ld~ and 2s~ orbits relative to the ld~ orbit are o~tained from the 170 spectrum. The results for 3SAr e~ergy levels are shown in table 4 with predicted spectroscopic factors. Another calculation in this model space was made by using the modified surface delta interaction 25) as the effective Hamiltonian. The four parameters of the MSDI and the two single-particle energy splittings are fixed by fitting 30-odd energy level splittings for A = 35-39 nuclei. These results for 3SAr are also shown in table 4. One unambiguous conclusion from both tbese calculations mentioned in subsect. 4.2 is that excitations out of the ld~ sbell do not play a significant role in the low-lying energy levels of 38Ar. This result implies that a truncation of the model space to the d~ and s~r shells would yield an equally viable model below 6 MeV excitation. Glaudemans et al. 1) have calculated the properties of low-lying nuclear energy levels for 30 < A < 40 nuclei in such a model. In these calculations, the entire effective Hamiltonian was determined from a least-squares fit to energy level data. Only the ground and first excited states of 3 SAr were included in the data set used by Glaudemans, therefore his results for 3SAr are also free from any assumptions about the higher-lying level structure irt this nucleus. The results (energy levels and S-factors) are presented in table 4. Examination of the results of the three calculations mentioned thus far led to the conclusions about the experimental data given in subsect. 4.2, namely that the 2 + state formed by the [s~ ~-d~ 1] configuration is fragmented into three components, and that the 5.55 MeV level is probably 1 ÷. A recent revision 3) of the Glaudemans calculations to include a larger data set tends to confirm these assumptions. The results of this calculation for 3SAr agree quite well with our simplified experimental spectrum as cart be seen from table 4. In particular, this calculation yields an energy of the 1 ÷ level that is close to 5.5 MeV. The results of both of the three shell calculations mentioned earlier indicate that
a°K(d, 8He)
359
the lowest dt-hole state is 4 +, and that it should have an energy of 7-9 MeV. The state observed at 7.17 MeV with an l = 2 angular distribution doubtless has a [d~ 1-d~"1] character. However, both the discrepancy between the theoretical and experimental energies and between the predicted and observed S-factors argue against the 7.11 MeV level being the pure [d~"1-d~"1]s = 4 state. Most probably this is the lowest fragment of a multiplet of J = 4 levels, which is similar to the lower energy 2 ÷ levels. 5. Conclusions
Several shell-model calculations which consider only s-d configurations have yielded mutually consistent results for nuclei of A = 35-39. Major facets of the experimental positive-parity level structure in this region are in agreement with these theories. However, in the case of the mass-38, T = 1 nuclei (3BAr and 3aCa), there are lowlying 0 ÷ states which are completely extraneous to the systematics of the s-d shell calculations. These states thus constitute the first evidence for important "2p-4h"like admixtures 4) in the mass-38, T = 1 system. Proton pick-up from agK does not excite this 0 + 2p-4h state in 38Ar, thus indicating that it is fairly pure, and that 39K (g.s.) possesses small 2p-3h admixtures. The same reaction does excite more lp = 0 levels (necessarily having J~ of 2 + or 1 +) than can be constructed in the s-d space. These extra levels are also the result of 2p-4h excitations, but in this case, the 2p-4h states and the s-d states are strongly mixed. We suggest that the two extra l = 0 levels have spin and parity of 2 ÷, and that the 1 + level at 5.55 MeV excitation is the pure [s~ l_d~ 1]s = 1 state. An lp = 2 level at 7.11 MeV excitation apparently results from pick-up of a ld~ particle, but calculations indicate that it is only the lowest element of a fragmented [d~-l d ~ 1 ]j = 4 state. The s-d shell-model calculations give a good account of the experimental centroids of the l = 2 and l = 0 pick-up strength and of some of the details of its distribution. The present experimental results further emphasize, however, the desirability of expanded shell-model calculations, which take into consideration both the s-d and the f-p shells. The results of Ern6 4) for the negative-parity levels and the low excited 0 ÷ states are very encouraging in this respect, but the limitations on his configuration space are such as to exclude the type of phenomena observed and discussed here. The requirement of reproducing the detailed distribution of spectroscopic strengths observed in the 39K(d, 3He)38Ar reaction will present a stringent test for the more extensive calculations of the future. References 1) 2) 3) 4) 5) 6)
P. P. P. F. B. B.
W. M. Glaudemans, G, Wiechers and P. J. Brussaard, Nucl. Phys. 56 (1964) 529 and 548 Federman and I. Talmi, Phys. Lett. 15 (1965) 165; 22 (1966) 469 W. M. Glaudemans, E. C. Halbert, J. B. McGrory and B. H. Wildenthal, to be published C. Ern6, Nucl. Phys. 84 (1966) 91 H. Wildenthal, J. B. McGrory, E. C. Halbert and P. W. M. Glaudemans, to be published H. Wildenthal, M. R. Cates arid E. Newman, Bull. Am. Phys. Soc. 12 (1967) 1189
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7) G. A. P. Engelbertink, H. Lindman and M. J. N. Jacobs, Nucl. Phys. A107 (1968) 305 8) R. H. Bassel, R. M. Drisko and G. R. Satchler, Oak Ridge National Laboratory Report ORNL-3240 (1962) unpublished 9) J. C. Hiebert, E. Newman and R. H. Bassel, Phys. Rev. 154 (1967) 898 10) E. Newman, L. C. Becket, B. M. Preedom and J. C. Hiebert, Nucl. Phys. AI00 (1967) 11) D. J. Baugh, G. J. B. Pyle, P. M. Rolph and S. M. Scarrott, Nucl. Phys. A95 (1967) 115 12) R. M. Drisko, unpublished 13) E. F. Gibson, B. W. Ridley, J. J. Kraushaar, M. E. Rickey and R. H. Bassel, Phys. Rev. (1967) 1194 14) B. M. Preedom, E. Newman and J. C. Hiebert, Phys. Rev. 166 (1968) 1156 15) B. H. Wildenthal and E. Newman, Phys. Rev. 167 (1968) 1027 16) P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 17) I. J. Taylor, Nucl. Phys. 41 (1963) 227 18) R. G. Alias, L. Meyer-Schiitzmeister and D. von Ehrenstein, Nucl. Phys. 61 (1965) 289 19) W. R. Phillips, Nucl. Phys. 60 (1964) 544 20) D. Newton, G. L. Salmon and A. B. Clegg, Nucl. Phys. 82 (1966) 513 21) C. A. Whitten, Bull. Am. Phys. Soc. 10 (1965) 121 22) T. Wei, W. S. Gray, J. J~inecke and R. M. Polichar, Bull. Am. Phys. Soc. 12 (1967) 681 23) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71; T. T. S. Kuo and E. C. Halbert, private communication 24) P. W. M. Glaudemans, B. H. Wildenthal and J. B. McGrory, Phys. Lett. 21 (1966) 427 25) P. W. M. Glaudemans, P. J. Brussaxtrd and B. H. Wildenthal, Nucl. Phys. A102 (1967)
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