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Energy levels of 34S from the 33S(d, p)34S reaction
1
Nuclear Physics A198 (1972) 209--227; (~) North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilmwithout written permission from the publisher
ENERGY LEVELS OF 34S FROM THE 33S(d, p)34S REACTION D. J. CROZIER t Argonne National Laboratory, Argonne, Illinois 60439 and University of Chicago, Chicago, Illinois 60637 tt Received 4 February 1972 (Revised 14 August 1972)
Abstract: The energy levels of 34S have been studied with the 33S(d, p)34S reaction at Ea = 12 MeV. A split-pole magnetic spectrograph was used to record the deuteron spectra at 10 angles from 9 ° to 59°. An overall energy resolution of 16 keV was achieved. The results make possible the identification of the major components of the ld~ lf_~, T = 1 configuration. The two-body matrix elements of the d~r f't-interaction are compared with other known d~r f~ multiplets and with theoretical calculations. E
NUCLEAR REACTIONS 3aS(d, p), E = 12 MeV; measured cr(Ep, 0). a4S deduced levels, In, J, zt, spectroscopic strengths.
1. Introduction The nucleus 32S m a y be considered as a fairly g o o d closed-sheU nucleus. A study 1) o f the 32S(z, d)33C1 reaction revealed that both the ~+ g r o u n d state and a ~ - excited state o f 33C1 were g o o d single-particle states. The observation o f a ~imilar result in the 325(d, p)338 reaction at 18 MeV [ref. 2)] indicates that the ~+ g r o u n d state and the ½- state at 2937 keV are g o o d single-particle states. The mass-34 nuclei 34C1 and 34S m a y therefore be t h o u g h t o f as two particles coupled to a 32S core. Thus the spectra o f these nuclei should be characterized primarily by the residual t w o - b o d y interaction between these two particles. Their spectra should therefore be fairly simple and easily interpreted. In a recent study 3) o f 34C1 by means o f the 33S(z, d)34C1 reaction, the m a j o r components o f the (d~) 2 multiplet and the T = 0 members o f the d~f~ multiplet were identified. The d~ f~ interaction has been rather thoroughly studied also in the particleparticle spectrum o f 38C1 and in the particle-hole spectrum o f 4 ° K and 4°Ca. The latter nucleus provides a complete set o f b o t h T = 0 and T = 1 matrix elements. It would be interesting to obtain the T = 1 members o f the d~f~ multiplet in mass-34 + Submitted in partial fulfillment of the requirements of the University of Chicago for the degree of Doctor of Philosophy. Present address: University of Pennsylvania, Philadelphia, Pennsylvania 19104. t* Work performed under the auspices of the US Atomic Energy Commission and supported in part by an AUA Fellowship. 209
210
D.J. CROZIER
nuclei as well. Although it is theoretically possible to observe these states in 34C1, the high level density in the excitation region in which they are expected to occur makes it very difficult in practice to do so by means of the 335('[, d)a4c1 reaction. The most reasonable method for studying the d~f~, T = 1 interaction in states of mass-34 nuclei therefore is by means of the 33S(d, p)34 S reaction. A previous study 4) of this reaction revealed the excitation energies of the states that are populated by the 338(d, p)345 reaction, but had not determined the angular momentum transfers. In addition to the work with the 33S(d, p)345 reaction, an attempt was made to study the mass-34 system by means of the 32S(ct, d)34C1 and 34S(z, t)34C1 reactions. Both of these proved unfeasible: the first because of the difficulty of choosing differential absorbers, the second because of the contamination of the spectra with deuterons produced in (r, d) reactions. These two experiments might be possible with position-sensitive detectors or, in the latter one, with a higher energy 3He beam or a gas target.
2. Experimental procedure In order to study the 33S(d, p)34S reaction, it was first necessary to prepare a suitable sulfur target. The difficulty in using sulfur as a target is that elemental sulfur sublimes under vacuum. The heating by bombardment with a charged-particle beam, of course, increases the rate of sublimation and in addition may induce sputtering o f the sulfur. Some method must therefore be found to retain the sulfur on the target. In the 33S(z, d)34C1 experiment, a cadmium sulfide target was used. The beam energy E~ = 14 MeV was low enough so that the cross section for the (z, d) reaction on cadmium was negligible. A bombarding energy at which the cross seetion for the (d, p) reaction on cadmium is expected to be negligible is also too low for the satisfactory performance of the 33S(d, p)34S experiment. Therefore it was necessary to employ a different technique for the preparation of the 33S target in the present experiment. Natural sulfur was used in trying out several methods of preparing sulfur targets. The one that was found to satisfy the requirements of the present experiment is as follows. A sample of sulfur enriched to 83 ~ in 33S was obtained from the Isotopes Division of Oak Ridge National Laboratory, and 1.2 mg of sulfur was loaded into a closed tantalum boat into which a hole 1 mm in diameter had previously been drilled. The sulfur was then evaporated, under vacuum, onto a 30/tg/cm 2 carbon foil which had not been removed from a glass slide. The carbon foil was then floated off the slide and mounted on a formvar backing. A second carbon foil was then picked-up upon a thin film of formvar and placed over the sulfur. Thus the sulfur was sandwiched between two layers of carbon. Targets prepared in this manner were found to withstand a beam o f 60 nA of 12 MeV deuterons incident through a 1.5 x 3 mm collimator without noticeable attribution of the sulfur. During the present experiment, the beam current was maintained at less than 30 nA.
a4S LEVELS
211
The 12 MeV deuteron beam used in this experiment was supplied by the Argonne F N tandem Van de Graaff accelerator. The protons resulting from the (d, p) reaction were analysed in an Enge split-pole magnetic spectrograph 5). Nuclear track emulsions were placed in the focal plane of the spectrograph to record the protons resulting from the (d, p) reaction, and enough cellulose triacetate absorbers were employed to stop all other particles. The emulsions were scanned with the Argonne automatic plate s c a n n e r 6). Ten exposures were taken at angles ranging from 9 ° to 59 °. The peaks resulting from the 335(d, p)345 reaction exhibited a F W H M of 16 keV. The proton groups in the resulting spectra were fitted with the spectrum-fitting program A U T O F I T 7). A solid-state detector was used to monitor the target during the exposures, and a multichannel analyser was used to record deuterons elastically scattered by the sulfur. Cross sections were determined by assuming the ratio to Rutherford scattering for the elastically scattered deuterons at 30 ° to be 1.26, as predicted by the optical-model calculations. Because of the volatility and lack of uniformity of the sulfur target this was the most reliable procedure. Since the ground state transition was weak in most of the spectra, excitation energies were determined with reference to the 2127.52_ 0.2 keV [ref. 8)] first excited state. The spectrograph had been previously calibrated with a Zl°Po a-source with an assumed energy of 5304.5 keV. Exposures at nine angles on a natural sulfur target were taken to aid in identifying impurities. In another recent study of the 33S(d, p)34S reaction 9), position-sensitive detectors and an aluminum-backed target were employed. This technique allowed spectroscopic information on several of the weaker states to be extracted, but made the study o f the states at higher excitation more difficult. For the states seen in both experiments the /-value assignments are essentially the same, and the l = 2 and l = 3 spectroscopic factors obtained in the two experiments are comparable. The 1 = 0 and l = 1 spectroscopic strengths are larger in the present experiment: this is due partially to a slightly different procedure in deducing spectroscopic factors. The spectroscopic strengths, G, were defined as G-
2 J e + l C2 S _ ae,,p 2 Ji + 1 1.53anw
which is a meaningful quantity regardless of the final-state spin. Since the absolute cross sections are somewhat uncertain, the absolute spectroscopic factors obtained in this manner are uncertain as well. In addition, they depend, as always, on the details of the distorted-wave analysis. It seems desirable therefore to attempt to eliminate these uncertainties in a non-arbitrary manner. In a recent investigation of the 3xS(d, p)335 reaction 2) the transitions to the ground state and the 2.94 MeV excited state were observed to comprise nearly all the observed l = 2(d¢) and l = 3(f~_) strengths. In addition, the transition to the 3.22 MeV state represented most of the l = l(p~) strengths. The spectroscopic strengths were
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34S LEVELS
213
normalized in such a way that the aZS(d, p)338 spectroscopic factors had the full single-particle value. The normalization factors defined in this manner were 2.40 for l = 3 and 1.41 for I = 1 and 2. The 1 = 0 transition is weak in 3zS(d, p)a3S because this orbit is very nearly full; therefore the average of the above normalizations 1.74 was used. '
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This procedure removes the major uncertainties in the distorted-wave analysis, since the distortions for both targets are likely to be very similar, and in the absolute cross section, since both isotopes were present in the target with accurately known abundance. The distorted-wave calculations were still used to reproduce the dependence on the Q-value when applied to 33S. The procedure used here is slightly different from the standard one of relating the measured cross section to a distorted-wave calculation; the symbol G' is used instead of G above to denote this different normalization procedure. Relating the cross section to that seen in a nearby "simple" nucleus has ample precedent in the literature, for instance ref. 3), and reasonable justification. As a test the values of G' given in table 2 were summed for each 1; they are seen, in table 3, to obey the sum rules quite well (the one slight exception is l = 1 where some p t admixture may be contributing). The implicit assumption is that the ground state, the 2 - and the 3 - states of 3~S are "single-particle states" in some sense and that in 34S the focus is on understanding
214
D.J. CROZIER
the coupling of the 33S ground state to these single-particle modes. To the extent that these are not "pure" single-particle states, one is studying the coupling of the more complex configurations, which are the best candidates for single-particle states, and arguments on whether these are the " t r u e " single-particle states in some idealized shell-model representation are largely irrelevant to the present analysis. i~
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3. Experimental results The spectrum of protons seen in the spectrograph at a scattering angle of 24 ° is shown in fig. l. In addition to the proton groups resulting from the 335(d, p)348 reaction and several groups resulting from the carbon and oxygen impurities in the target, one also sees several peaks resulting from the 32S(d, p)335 reaction occasioned by the 16 % concentration of 32S in the target. The latter are useful for normalizing the spectroscopic strengths from the 335(d, p)345 reaction. Figs. 2-6 show the angular distributions obtained in this study. As can be seen, it is possible to distinguish between transitions corresponding to different values of orbital
s4S L E V E L S
215
angular momentum transfer I. The solid lines in the figures are the distorted-wave calculations which were performed with the program D W U C K lo). The opticalmodel parameters 1, 11) used are given in table 1. The excitation energies and spectroscopic strengths obtained in the present experiment are listed in table 2. The spectroscopic strengths given here have been obtained by normalizing to the strengths of the transitions of the same/-value in the spectrum obtained from the 16 ~o concentration of 32S in the target. :~
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4. Spin assignments Of the early spin information on 34S which has been compiled 12), the most important for the present discussion are the 32S(t, p)Z4S L-values 13). These are given incolumn 6 of table 2. Pertinent information from recent studies of the 35CI(d, r)3*S reaction 1,) and the 7-decay 1s - t 7) of 3,S are also given in columns 7 and 8 of table 2, respectively. The best J~ values, in the author's opinion, based upon these results as well as the present experiment are given in column 9 of table 2. The energy-level diagram for 34S, as determined by the present experiment, is shown in fig. 7. 4.1. T H E G R O U N D
S T A T E O F a4S
It is now appropriate to ascertain what information can be extracted from our spectroscopic data. The 1 = 2 transitions will be examined first• The ground state in
216
D.J. CROZIER
34S appears to be reached by a pure l = 2 transition, as is expected because o f its 0 ÷ assignment. In table 4, which compares the observed spectroscopic strengths G' with the expected strengths Gth = ( 2 J r + l ) / ( 2 J i + 1), it is seen that the g r o u n d state approximately exhausts the expected 0 + strength.
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4.2. THE STATES AT 2.13, 3.31 AND 4.12 MeV In addition to the pure l = 2 transition to the g r o u n d state, three states in 34S at 2.13, 3.31 and 4.12 MeV excitation are reached by a mixture o f / = 0 and l = 2 transitions. This is consistent with a 2 + assignment for these states. The l = 2 strengths o f the two states at 2.13 and 3.31 M e V are in very g o o d agreement with those o f their analog states in a4C1, as seen in the 33S(~, d)34C1 reaction. The 4.12 M e V state in 34S is m o s t likely the analog o f the 4.14 M e V state in 34C1, the angular distribution o f which was noted by Erskine et aL 3) to be consistent with that o f a mixed I = 0 and l = 2 transition. It is seen in table 4 that these three 2 ÷ states approximately exhaust the expected 1 = 2, J~ = 2 + strength.
a4S LEVELS I
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TABLE 1 Optical-model parameters Projectile
d a) pc)
V (MeV)
W a) (MeV)
179•7 26.3 V(p) b) W(p) c)
Ro (fm)
A (fro)
R'o (fm)
A" (fm)
Rc (fm)
Vs.o. (MeV)
R .... (MeV)
A .... (fro)
0.656 1.15
1.086 0.57
1.488 Ro
0.535 0.50
1.30 Ro
5.5
Ro
0.57
a) Surface absorption was used for both d- and p-potentials. b) V(p) = 60--0•3 E + 0 . 0 4 Z/A{+27 (N--Z)/A (E in MeV). t0.64 E E < 13.8 MeV. ¢) W(p) = | 9 . 6 - - 0 . 0 6 E E _ 13.8 MeV. a) Ref. 2). =) Ref. 11).
4.3• T H E STATES A T 5.38 A N D 6.83 MeV
In addition to the mixed I -- 0 and I -- 2 transitions, there appear to be pure l = 0 transitions to two states at 5.38 and 6.83 MeV. These states may therefore be assigned
D. J. C R O Z I E R
218
TABLE 2 Summary o f spectroscopic strengths G ' and spin assignments Excitation (keV) 0.0 2127.5-- 0.2 ~) 3305 4- 2 4118 4627
5758 6128
± 4 ± 5 -- 6 4- 6 7 4- 7 4- 7 h)
6174 6256 6346 6422
4444-
5326 5384 5683 5694
l = 0
G ' ") l : 1 l-- 2
(0.13)
0.46 1.32
0 2
2 0+2
0.16 0.41
0.36 0.80
(2) 2 3
0
0.21
l-- 3
0.46 1.15
(t, p) b) (d, r) c) L lp
0.23
2
y-decay jrr
0+ 2 e,f) 2 e'f) 2 r) ~ 4 ¢), 3 - f) 2(+) r) 1 r)
0.65 3.12 0.42
8 8 8 8
0.25
6483 4- 8 6644 4- 9 6690 4- 9 6832 4- 9 6959 4-10 7115 4-10 7388 I)
0.91
5 - ~) 1 2
0.71 0.57
0.20
(2)
2+
2+ 321+ (1 --3)" 512+ (3)(3, 4)" 1(3, 4)"
1 0.22
(1-3) (4)(0-3)-
(1.64) 0.32 2
0.14 0.21 0.13
jz, d)
0.44
2+ (0--3)(3)-
2or3
7547 h) 7633 7659
4-11 4-11
0.98
7732 7755 7785 8142
4-11 4-11 4-11 4-12
(0.o3)
3
3-
(3, 4 ) (2, 3 )
(0.57) (0.14)
(0-3)-
0.08 0.27 0.26
(0--3) (0--3)
a) Spectroscopic strength, as explained in the text. b) Ref. 12). c) Ref. 14). a) The j~r value, based upon the present experiment and other available data. o) Ref. 14) f) Ref. xs). ~) Ref. 16). h) Weak. i) Obscured by an impurity. J) Ref. 7) TABLE 3 Summed strengths
I
EG'
EGth
I
4.939
4.0
2 3
2.949 9.027
3.0 8.0
a*S LEVELS
219
(1, 2) +. The 5.38 MeV state has been determined to have J = 1 [ref. 16)] and the positive-parity assignment can also be inferred from the (d, T) results. The 6.83 MeV state has been assigned 2 + on the basis of the (t, p) results. EXC(keV)
jr
(d,p)
EXC(keV)
d 7r
8142
4627
3-
4118
2+
2,(0)
7755. 7732~7785 . 7659~ 7633 7547~ 7388 /
2,(0)
6959
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7115 3305
2+
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__2
+
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3 I WEAK WEAK 3,(I)
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I 0
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0 3
Fig. 7. Energy-level diagram o f a4s. TABLE 4 l = 2 strengths j~r
Excitation (MeV)
G"
Gth
0+ 2+
0 2.13 3.30 4.12
0.46 1.32~ 0.3612.48 0.80
0.50 2.50
4.4. T H E STATES A T 4.63 A N D 6.17 MeV
Let us now discuss the states reached by l = 3 transitions. The spectroscopic strengths for the l = 3 transitions are shown in histogram form in fig. 8. The l = 1 admixture in the transition to the 4.63 MeV state is consistent with the previous 3 assignment for this state. The 6.17 MeV state also has an I = 1 admixture. Moreover,
220
D.J. CROZIER
in a study o f the 335(p, y)34C1 reaction by H y d e r 18), the E x = 6.17 M e V resonance in 34C1 was assigned J~ --- 3 - , T = 1 and was f o u n d to have a large d-f component. The most likely candidate for the analog o f this resonance appears to be the 6.17 MeV state in 34S, which therefore has been assigned ( 3 - ) . 4.5. THE STATES AT 5.32, 5.69 AND 6.64 MeV It is immediately apparent that the 5.69 MeV state is too strong to be anything other than the 5 - m e m b e r o f the d~f~ multiplet. In fact, this state has recently been assigned 5 - on the basis o f y-ray polarization direction correlation measurements 17).
33S(d,p)34S f,=3 3.0 2.0,.o-
-
4-
51-
3- ----
UNASSIGNE II
-..,2 d
Fig. 8. The spectroscopic strengths G' for l = 3 transitions. The values expected for various ,/-values are indicated by horizontal dashed lines. O f the remaining states reached by l = 3 transitions, the 5.32 and 6.64 M e V states appear to be too strong to be the ones associated with the remaining 3 - strength. They are also a m o n g the weaker states seen in 32S(t, p)345 and therefore g o o d candidates for unnatural-parity states. The 5.32 MeV state has been assigned J = 2 by Mulhern e t al. 16). The 1 = 3 (d, p) angular distribution indicates a negative-parity assignment. The 6.64 MeV state is then most likely the 4 - member o f the multiplet. 4.6. THE STATES AT 6.26, 6.42, 7.11, 7.66 AND 7.73 MeV A 2 - or 3 - assignment for the 7.11 and 7.73 M e V states seems to be implied by the l = 1 admixtures in their angular distributions *. The remaining three states at 6.26, 6.42 and 7.66 M e V presumably account for the missing 3 - and 4 - strengths and therefore m a y be assigned (3, 4 ) - . t The assignment 1- is a remote possibility if the l = 3 transition were A j ~ ~, which would seem rather unlikely.
345 LEVELS
221
4.7. THE STATE AT 7.63 MeV Finally let us consider the l = 1 transitions. The l = 1 strength is considerably fragmented, and the possibility of P~r mixing cannot be ignored. Therefore it is rather difficult to make assignments on the basis of the spectroscopic strengths. The states at 4.63 and 6.17 MeV have been considered in subsect. 4.4 and are designated 3-. The rather strong state at 7.63 MeV may correspond to the 3- state observed in this region in the (t, p) reaction and has therefore been designated (3)- in table 3. 4.8. THE STATES AT 56.8, 5.76, 6.34 AND 6.48 MeV The states at 6.34 and 5.76 MeV have previously been assigned 1- on the basis of the 328(t, p)345 data. The states at 5.68 and 6.48 MeV are both too strong to be 0 and therefore are assigned ( 1 - 3)-. 4.9. THE STATES AT 6.69, 6.96, 7.76, 7.79 AND 8.14 MeV The five remaining l = 1 states at 6.69, 6.96, 7.76, 7.78 and 8.14 MeV have been assigned ( 0 - 3 ) - since the possibilities could not be restricted further.
5. Two-body matrix elements 5.1. THE d~rf~, T = 1 INTERACTION Having identified the major components of the d~f~, T = 1 multiplet, it is interesting to try to compare these results with other known d~f~ multiplets. In order to do so, we must define the matrix elements of the residual two-body interaction. The matrix element for a given value of J is the difference between the excitation energy E Xof the state with that value of J and a reference energy Er for the multiplet. The reference energy Er, the excitation energy at which the state would be expected to occur if there were no residual two-body interaction, is given by Er = M(32S)+2M(n)-2Q[3ZS(n, y)33S]-M(34S)-I-Ex(335)~ - = 5.717 MeV, for the d~f~ interaction, where E r is the excitation energy in 345, M(325), M ( 3 4 5 ) and M ( n ) the masses of 325, 345 and the neutron, respectively, Q the ground state Q-value of the 325(n, ~)335 reaction and Ex the excitation energy of the ~ - level in 335. When the strength for a given J is fragmented, the centroid of the spectroscopic strength is used for Ex in calculating the matrix element. The configurations of the known d~f~ particle-particle interactions are shown in schematic form in fig. 9. The nucleus 34C1 can be thought of as a neutron and a proton coupled to a 325 core. This gives rise to two possible configurations corresponding to T = 0 and T = 1, the two possible values of isospin. In the nucleus 345, the circumstances are similar except that now both of the particles outside the core are neutrons. In this case, therefore, the isospin of the pair is T = 1. Finally, in the 365 core of the nucleus 38C1, the d~ shell is completely filled for neutrons. It is therefore possible to
222
D.J. CROZIER
say which of the particles occupies which orbit. Therefore, as far as the two particles are concerned, the configuration is the same as the right-hand diagram for 34C1. Hence the isospin of the pair is mixed. The complete set of d~f~ matrix elements obtained in 34C1 and 34S is given in table 5, along with values determined by other authors 19-22) and theoretical calculations 23. z4). It is apparent that the mixed-isospin matrix elements Ej of the residual two-body interaction in 38C1 should be related to the pure-isospin matrix elements E j r observed in 34C1 and 34S. In fact, one has
Ej
=
½(Ejo+Ej,).
In fig. I0 the predicted mixed-isospin matrix elements deduced from the mass-34 spectra are compared with the matrix elements actually observed in 38CI. The uncertainties in the predicted 3- and 4- matrixelementsare due to the unassignedstates TABLE 5 d~f~ Source
T= J=2
matrix elements (MeV)
0
T=
3
4
5
J=2
1
3
4
5
1.12 0.82 +1.45 + 1.35
--0.02
34C1 and 34S
--2.98
--1.71
--1.61
--2.06
--0.39
4°Ca [ref. 19)] Moinester and Alford 22) Erne 2o) Dieperink and
--4.14 --4.23
--1.92 -- 1.89
--2.03 --2.26
--2.35 --2.11
+0.91 +0.64
0.35 0.72 +0.21 --0.25
--3.65 --4.01
--1.85 --2.01
--1.77 -- 1.87
--2.90 --2.37
+0.38 +0.62
--0.08 --0.07
+0.92 +0.98
-!-0.43 +0.20
--4.75
-- 1 . 4 1
--0.96
-- 1.73
+0.27
--0.04
+0.81
--0.88
--3.11
--1.63
--0.87
--1.29
+0.28
0.07
0.34
--0.59
+0.52 --0.08
Brussaard 21) Sartoris and
Zamick 23) K u o and Brown 24)
in 34S. It is immediately apparent that the observed 2 - and 5- matrix elements are very close to the predicted values. The observed 4- matrix element is slightly lower than its predicted value. The most important discrepancy occurs for the 3- matrix element, whose observed value is seen to be considerably lower than the predicted value. One possible explanation for this discrepancy can be found upon examining the spectrum of asC1. The relative l = 3 spectroscopic strengths obtained from the 37C1 (d, p)38Cl experiment of Rapaport and Buechner 25) are displayed in histogram form in fig. 11. The expected strength for each value of J is indicated by a dashed line. It can be seen that the strength of the 3- state at 0.76 MeV falls about 30 ~ short of the expected 3- strength. It should be noted also that this state exhibits an l = 1 admixture. Since there are several observed 1 = 1 transitions to states of higher excitation
a4S LEVELS
223
PARTICLE-PARTICLE d~,2-f7/2 SPECTRA r---T---
f 7121 ',
~e I
1 I
d5 / 2 ~
34 S
p
F--r--1 l_l~__ I ----I
T= I n
r--l---lq I .... , • l / T=O
"c,
~--,%-I
~
i____1
I
½[(T.O).CT-,)]
Fig. 9. Schematic representation of the different nuclei in which d~f~ particle-particle spectra have been observed.
4-
-0.5
\\
-hO
5-
b.I
-I.5 2-
m
2-
PREDICTED FROM -2.0
MASS 34
SPECTRA
~8CI
OBSERVED
E,- ½ ( E.,o~E,.,I Fig. 10. Comparison of the mixed-isospin d~.f~, particle-particle matrix elements with those actually observed in 38C1. The uncertainties in the 3- and 4 - predictions are due to the unassigned levels in 3~S.
224
D.J. CROZIER
in 3Sc1, it seems reasonable to assume that some l = 3 admixture in these transitions m a y have been overlooked. I f this is indeed the case, it would conceivably give a somewhat higher value to the 3 - matrix element in 38C1 and thus would lead to better agreement with the prediction. However, this explanation raises another problem.
37CI(d,p)3eCI
STRENGTHS
5-
4-
ZUNASSIGNED
Fig. 11. Relative spectroscopic strengths G" for l = 3 transitions observed in the a~Cl(d, p)38C1 reaction. Horizontal dashed lines indicate the expected strengths for each value of./.
PARTICLE-HOLE
40K
d3/2--fT/2 SPECTRA
d31
I/2 [(T=O}+(T,I )]
~-l,--I 40(::o ~ -I
,---r--~
+-
and T=O
Fig. 12. Schematic representation of the different nuclei in which d~rf~ particle-hole spectra have been observed.
The residual t w o - b o d y interaction is observed not only in particle-particle spectra but in particle-hole spectra as well. As one would expect, the particle-particle spectra and the particle-hole spectra are not unrelated. In fact, they are related by a simple
34S LEVELS
225
Racah transformation known as the Pandya transformation 19), which may be written as E~ "h(p'p) =
- -
~ ( 2 J ' + 1) W(jlj2J2Jl ;JJ')E~-p(p'h).
(l)
j,
There are two nuclei in which d~f~ particle-hole multiplets have been observed. The configurations of these nuclei are shown schematically in fig. 12. In 4°Ca there are two
1.5
m
5_z... 2-
-- 7 / / / / 5 -
>
1.0
0.5
0
4-
P (mass-34)
40K
P(38CI)
Fig. 13. Comparison of dgrf~ matrix elements in 4°K (center) with the Pandya transforms of the 3sc1 spectrum (right) and the mass-34 predictions (left).
+1
- -
3"/I----
3" ~-~_
4 - ~
4- ~x.x~x.~
A
>
/
=E
2
- -
2-/5-{'L~"~'~
-2
-3 3-
4°Ca
P( mass -34 )
Fig. 14. Comparison of the pure-isospin d~.f~_ particle-hole spectra from 4°Ca and mass-34. The mass-34 spectra have been transformed by means of the Pandya transformation.
226
D.J. CROZIER
possible configurations, again corresponding to the two possible values of isospin T = 0 and T = 1. In 4°K, since there are two more neutrons than protons, it is possible to distinguish the nature of the particle and of the hole. Therefore the isospin of the pair is mixed. It has been well known for some time 26.27) that the Pandya transformation works extremely well between the d~fz~ spectra of 38C1 and 4°K. Therefore if the 3- matrix element in 38C1 should indeed be somewhat higher as has been proposed above, one should consider the effect upon the Pandya transformation between 38C1 and 4°K. TABLE 6 T h e (dg_) z m a t r i x elements (MeV) j~r, T mass-34 G l a u d e m a n s e t al. c) Ern6 a) ~) F r o m a4C1 [ref. ~)].
0 +, 1
1 +, 0
2 +, 1
--2.87 ~) --2.27 -- 1.71
--2.23 ~) --0.92 --2.11
+ 0 . 1 6 b) +0.16 +0.26
b) F r o m 34S.
c) Ref. 28).
3 +, 0 --2.72 a) --2.64 --2.51
d) Ref. 20).
A cursory examination of eq. (I) will reveal that the predictions for all of the particlehole states will be affected. This effect is shown in fig. 13. In the center of the figure are the observed particle-hole matrix elements in 4°K, while on the right are those predicted from the 3aC1 particle-particle spectra. It can be seen, as was mentioned before, that the agreement is very good. On the left-hand side of the figure are the predicted particle-hole matrix elements from the mass-34 spectra. The nucleus 40Ca provides another source of a complete set of d~ f~ matrix elements. These may be related to the mass-34 elements by means of the Pandya transformation with isospin, namely EPTp(p'h) :
--~-~ ( 2 J ' + 1 ) ( 2 T ' +
1)W(jlj2J2J
1 ; JJ')
l , l / [ 1 1 l l . T"/"t~ K'P'h(p-P) × , , ~7z~r:r, i x :~s'r" •
J'T'
In fig. 14 the Pandya transformation of the mass-34 spectrum is compared with the observed particle-hole spectrum in 4°Ca. In all of these comparisons some rather sizable discrepancies are noticed. Therefore some attempts at an explanation should be made. In the case of 4°K, one possible explanation is that the reference energy is too low. This might be caused by the true closed-shell state of 4°Ca not being just the ground state but containing some admixture of the 3.35 MeV (0 +) state as well. Such an admixture in the 4°Ca ground state would also be consistent with the 4°Ca spectrum, but considerable collective admixtures would still be required in the 3-, T = 0 and 5-, T = 0 states. Since the lowest 3- and 5- states in 4°Ca have been taken as the members of a multiplet, it should be pointed out that the states are known to have strong collective enhancement with B(E3), and B(E5) for transitions to the ground state much larger than single-particle values. It is therefore very likely that the collective admixtures significantly depress these states in energy.
a4S LEVELS
227
5.2. THE (d~.) 2 I N T E R A C T I O N
It should be pointed out that the identification of the mixed l = 0 and 1 = 2 transitions with the 4118 keV 2 + state gives a higher (more negative) value to the 2 + matrix element of the (d~_)2 interaction than was reported for 34C1 by Erskine et al. 3). The complete set of (d~) 2 matrix elements with this correction is given in table 6, along with theoretical calculations by other authors 20.28). The author wishes to thank his advisor, Professor J. P. Schiffer, for making available the facilities of Argonne National Laboratory and for much very valuable advice and encouragement. The author would also like to thank A. K. Hyder, Jr., for making his results available prior to publication and Dr. J. R. Erskine, Dr. H. T. Fortune and Dr. A. Richter for many helpful discussions. The author appreciates the assistance of W. Horath in target preparation, C. Bolduc in the operation of the spectrograph and the cooperation of the operational group at the tandem accelerator. Referenees 1) 2) 3) 4) 5) 6) 7)
R. A. Morrison, Nucl. Phys. A140 (1970) 97 M. C. Mermaz et aL, Phys. Rev. C4 (1971) 1778 J. R. Erskine, D. J. Crozier, J. P. Schiffer and W. P. Alford, Phys. Rev. C3 (1971) 1976 M. W. Brenner, Phys. Rev. 129 (1963) 765 J. E. Spencer and H. A. Enge, Nucl. Instr. 49 (1967) 181 J. R. Erskine and R. H. Vonderohe, Nucl. Instr. 81 (1970) 221 P. Spink and J. R. Erskine, Argonne National Laboratory Physics Division informal report PHY-1965B, unpublished; J. R. Comfort, Argonne National Laboratory Physics Division informal report PHY-1970B, unpublished 8) G. J. Bock, E. A. Samworth, J. W. Olness a n d E . K. Warburton, Phys. Rev. C5 (1972) 284 9) J. G. van der Baan and B. R. Sikora, Nucl. Phys. A173 (1971) 456 10) P. D. Kunz, the program D W U C K , private communication 11) B. A. Watson, thesis, Indiana University, July 1968 12) P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 13) S. Hinds, private communication described by P. M. Endt and C. van der Leun, Nucl. Phys. A105 (1967) 1 14) N. G. Puttaswamy and J. L. Yntema, Phys. Rev. 177 (1969) 1624 15) C. E. Moss, R. V. Poore and N. R. Roberson, Nucl. Phys. A144 (1970) 577 16) P. J. Mulhern, et aL, Nucl. Phys. A162 (1971) 259 17) M. W. Greene, J. A. Kuehner and A. A. Pilt, Phys. Lett. 35B (1971) 560; Bull. Am. Phys. Soc. 16 (1971) 554 18) A. K. Hyder, private communication; A. K. Hyder and G. I. Harris, Phys. Rev. C4 (1971) 2046 19) J. R. Erskine, Phys. Rev. 149 (1966) 854; K. K. Seth, J. A. Biggerstaff, P. D. Miller and G. R. Satchler, Phys. Rev. 164 (1967) 1450; J. S. Forster, K. Bearpark, J. L. Hutton and J. F. Sharpey-Schafer, Nucl. Phys. A150 (1970) 30 20) F. C. Ern6, Nucl. Phys. 84 (1966) 91 21) A. E. L. Dieperink and P. J Brussaard, N u c l Phys. A106 (1968) 177 22) M. Moinester and W. P. Alford, Nucl. Phys. A144 (1970) 305 23) G. Sartoris and L. Zanaick, Phys. Lett. 25B (1967) 5 24) T. T. S. Kuo and G. E. Brown, Nucl. Phys. A l l 4 (1968) 241 25) J. Rapaport and W. W. Buechner, Nucl. Phys. 80 (1966) 83 26) S. P. Pandya, Phys. Rev. 103 (1956) 956 27) S. Goldstein and 1. Talmi, Phys. Rev. 102 (1956) 589 28) P. W. M. Glaudemans, G. Wiechers and P. J. Brussaard, Nucl. Phys. 56 (1964) 548
Nuclear Physics A198 (1972) 209--227; (~) North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilmwithout written permission from the publisher
ENERGY LEVELS OF 34S FROM THE 33S(d, p)34S REACTION D. J. CROZIER t Argonne National Laboratory, Argonne, Illinois 60439 and University of Chicago, Chicago, Illinois 60637 tt Received 4 February 1972 (Revised 14 August 1972)
Abstract: The energy levels of 34S have been studied with the 33S(d, p)34S reaction at Ea = 12 MeV. A split-pole magnetic spectrograph was used to record the deuteron spectra at 10 angles from 9 ° to 59°. An overall energy resolution of 16 keV was achieved. The results make possible the identification of the major components of the ld~ lf_~, T = 1 configuration. The two-body matrix elements of the d~r f't-interaction are compared with other known d~r f~ multiplets and with theoretical calculations. E
NUCLEAR REACTIONS 3aS(d, p), E = 12 MeV; measured cr(Ep, 0). a4S deduced levels, In, J, zt, spectroscopic strengths.
1. Introduction The nucleus 32S m a y be considered as a fairly g o o d closed-sheU nucleus. A study 1) o f the 32S(z, d)33C1 reaction revealed that both the ~+ g r o u n d state and a ~ - excited state o f 33C1 were g o o d single-particle states. The observation o f a ~imilar result in the 325(d, p)338 reaction at 18 MeV [ref. 2)] indicates that the ~+ g r o u n d state and the ½- state at 2937 keV are g o o d single-particle states. The mass-34 nuclei 34C1 and 34S m a y therefore be t h o u g h t o f as two particles coupled to a 32S core. Thus the spectra o f these nuclei should be characterized primarily by the residual t w o - b o d y interaction between these two particles. Their spectra should therefore be fairly simple and easily interpreted. In a recent study 3) o f 34C1 by means o f the 33S(z, d)34C1 reaction, the m a j o r components o f the (d~) 2 multiplet and the T = 0 members o f the d~f~ multiplet were identified. The d~ f~ interaction has been rather thoroughly studied also in the particleparticle spectrum o f 38C1 and in the particle-hole spectrum o f 4 ° K and 4°Ca. The latter nucleus provides a complete set o f b o t h T = 0 and T = 1 matrix elements. It would be interesting to obtain the T = 1 members o f the d~f~ multiplet in mass-34 + Submitted in partial fulfillment of the requirements of the University of Chicago for the degree of Doctor of Philosophy. Present address: University of Pennsylvania, Philadelphia, Pennsylvania 19104. t* Work performed under the auspices of the US Atomic Energy Commission and supported in part by an AUA Fellowship. 209
210
D.J. CROZIER
nuclei as well. Although it is theoretically possible to observe these states in 34C1, the high level density in the excitation region in which they are expected to occur makes it very difficult in practice to do so by means of the 335('[, d)a4c1 reaction. The most reasonable method for studying the d~f~, T = 1 interaction in states of mass-34 nuclei therefore is by means of the 33S(d, p)34 S reaction. A previous study 4) of this reaction revealed the excitation energies of the states that are populated by the 338(d, p)345 reaction, but had not determined the angular momentum transfers. In addition to the work with the 33S(d, p)345 reaction, an attempt was made to study the mass-34 system by means of the 32S(ct, d)34C1 and 34S(z, t)34C1 reactions. Both of these proved unfeasible: the first because of the difficulty of choosing differential absorbers, the second because of the contamination of the spectra with deuterons produced in (r, d) reactions. These two experiments might be possible with position-sensitive detectors or, in the latter one, with a higher energy 3He beam or a gas target.
2. Experimental procedure In order to study the 33S(d, p)34S reaction, it was first necessary to prepare a suitable sulfur target. The difficulty in using sulfur as a target is that elemental sulfur sublimes under vacuum. The heating by bombardment with a charged-particle beam, of course, increases the rate of sublimation and in addition may induce sputtering o f the sulfur. Some method must therefore be found to retain the sulfur on the target. In the 33S(z, d)34C1 experiment, a cadmium sulfide target was used. The beam energy E~ = 14 MeV was low enough so that the cross section for the (z, d) reaction on cadmium was negligible. A bombarding energy at which the cross seetion for the (d, p) reaction on cadmium is expected to be negligible is also too low for the satisfactory performance of the 33S(d, p)34S experiment. Therefore it was necessary to employ a different technique for the preparation of the 33S target in the present experiment. Natural sulfur was used in trying out several methods of preparing sulfur targets. The one that was found to satisfy the requirements of the present experiment is as follows. A sample of sulfur enriched to 83 ~ in 33S was obtained from the Isotopes Division of Oak Ridge National Laboratory, and 1.2 mg of sulfur was loaded into a closed tantalum boat into which a hole 1 mm in diameter had previously been drilled. The sulfur was then evaporated, under vacuum, onto a 30/tg/cm 2 carbon foil which had not been removed from a glass slide. The carbon foil was then floated off the slide and mounted on a formvar backing. A second carbon foil was then picked-up upon a thin film of formvar and placed over the sulfur. Thus the sulfur was sandwiched between two layers of carbon. Targets prepared in this manner were found to withstand a beam o f 60 nA of 12 MeV deuterons incident through a 1.5 x 3 mm collimator without noticeable attribution of the sulfur. During the present experiment, the beam current was maintained at less than 30 nA.
a4S LEVELS
211
The 12 MeV deuteron beam used in this experiment was supplied by the Argonne F N tandem Van de Graaff accelerator. The protons resulting from the (d, p) reaction were analysed in an Enge split-pole magnetic spectrograph 5). Nuclear track emulsions were placed in the focal plane of the spectrograph to record the protons resulting from the (d, p) reaction, and enough cellulose triacetate absorbers were employed to stop all other particles. The emulsions were scanned with the Argonne automatic plate s c a n n e r 6). Ten exposures were taken at angles ranging from 9 ° to 59 °. The peaks resulting from the 335(d, p)345 reaction exhibited a F W H M of 16 keV. The proton groups in the resulting spectra were fitted with the spectrum-fitting program A U T O F I T 7). A solid-state detector was used to monitor the target during the exposures, and a multichannel analyser was used to record deuterons elastically scattered by the sulfur. Cross sections were determined by assuming the ratio to Rutherford scattering for the elastically scattered deuterons at 30 ° to be 1.26, as predicted by the optical-model calculations. Because of the volatility and lack of uniformity of the sulfur target this was the most reliable procedure. Since the ground state transition was weak in most of the spectra, excitation energies were determined with reference to the 2127.52_ 0.2 keV [ref. 8)] first excited state. The spectrograph had been previously calibrated with a Zl°Po a-source with an assumed energy of 5304.5 keV. Exposures at nine angles on a natural sulfur target were taken to aid in identifying impurities. In another recent study of the 33S(d, p)34S reaction 9), position-sensitive detectors and an aluminum-backed target were employed. This technique allowed spectroscopic information on several of the weaker states to be extracted, but made the study o f the states at higher excitation more difficult. For the states seen in both experiments the /-value assignments are essentially the same, and the l = 2 and l = 3 spectroscopic factors obtained in the two experiments are comparable. The 1 = 0 and l = 1 spectroscopic strengths are larger in the present experiment: this is due partially to a slightly different procedure in deducing spectroscopic factors. The spectroscopic strengths, G, were defined as G-
2 J e + l C2 S _ ae,,p 2 Ji + 1 1.53anw
which is a meaningful quantity regardless of the final-state spin. Since the absolute cross sections are somewhat uncertain, the absolute spectroscopic factors obtained in this manner are uncertain as well. In addition, they depend, as always, on the details of the distorted-wave analysis. It seems desirable therefore to attempt to eliminate these uncertainties in a non-arbitrary manner. In a recent investigation of the 3xS(d, p)335 reaction 2) the transitions to the ground state and the 2.94 MeV excited state were observed to comprise nearly all the observed l = 2(d¢) and l = 3(f~_) strengths. In addition, the transition to the 3.22 MeV state represented most of the l = l(p~) strengths. The spectroscopic strengths were
212
D . J . CROZIER
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normalized in such a way that the aZS(d, p)338 spectroscopic factors had the full single-particle value. The normalization factors defined in this manner were 2.40 for l = 3 and 1.41 for I = 1 and 2. The 1 = 0 transition is weak in 3zS(d, p)a3S because this orbit is very nearly full; therefore the average of the above normalizations 1.74 was used. '
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This procedure removes the major uncertainties in the distorted-wave analysis, since the distortions for both targets are likely to be very similar, and in the absolute cross section, since both isotopes were present in the target with accurately known abundance. The distorted-wave calculations were still used to reproduce the dependence on the Q-value when applied to 33S. The procedure used here is slightly different from the standard one of relating the measured cross section to a distorted-wave calculation; the symbol G' is used instead of G above to denote this different normalization procedure. Relating the cross section to that seen in a nearby "simple" nucleus has ample precedent in the literature, for instance ref. 3), and reasonable justification. As a test the values of G' given in table 2 were summed for each 1; they are seen, in table 3, to obey the sum rules quite well (the one slight exception is l = 1 where some p t admixture may be contributing). The implicit assumption is that the ground state, the 2 - and the 3 - states of 3~S are "single-particle states" in some sense and that in 34S the focus is on understanding
214
D.J. CROZIER
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o*
,
I
,
,
3o*
l
60*
8¢ m Fig. 3. Angular distributions identified as l = 1. The lines are the DWBA calculations.
3. Experimental results The spectrum of protons seen in the spectrograph at a scattering angle of 24 ° is shown in fig. l. In addition to the proton groups resulting from the 335(d, p)348 reaction and several groups resulting from the carbon and oxygen impurities in the target, one also sees several peaks resulting from the 32S(d, p)335 reaction occasioned by the 16 % concentration of 32S in the target. The latter are useful for normalizing the spectroscopic strengths from the 335(d, p)345 reaction. Figs. 2-6 show the angular distributions obtained in this study. As can be seen, it is possible to distinguish between transitions corresponding to different values of orbital
s4S L E V E L S
215
angular momentum transfer I. The solid lines in the figures are the distorted-wave calculations which were performed with the program D W U C K lo). The opticalmodel parameters 1, 11) used are given in table 1. The excitation energies and spectroscopic strengths obtained in the present experiment are listed in table 2. The spectroscopic strengths given here have been obtained by normalizing to the strengths of the transitions of the same/-value in the spectrum obtained from the 16 ~o concentration of 32S in the target. :~
1.0- ' 0.5
'
i
,
I
I.c ~-,~ ¢
I__, ~E2XC g.s•
0.. 7
','~
'
I----I
-4_
EXC - ~
1~..e"...~e
--
"
t
---~=o
.y>i
keY) ~ 3305 -~
(
\'~
0•0~
,
•
l o-V
~,.
--~-l-o
"~ o 5~-\ ".. . . . . . • ",/
%,
Exc
\
....... ~=0 ---~-2
• :-
2
•.\
I•O~- ~ , ~ o o.sV ~
>"
]
EXC (keY) 4,,8 -_-
_-
( keV )
-I
"~S,
_--
÷
0.05 \
I
\
o.,L_
\
0.010.
I
I
310.
x t X I
L-.,, 60*
o*
30°
6o*
8crn Fig. 4. A n g u l a r d i s t r i b u t i o n s identified as l = 2. The lines are the D W B A calculations. Some of the t r a n s i t i o n s a p p e a r to have an 1 = 0 c o m p o n e n t . The d a s h e d lines represents the c o m p o n e n t s w i t h d i f f e r e n t / - v a l u e s . The solid lines are the s ums of the d a s h e d curves.
4. Spin assignments Of the early spin information on 34S which has been compiled 12), the most important for the present discussion are the 32S(t, p)Z4S L-values 13). These are given incolumn 6 of table 2. Pertinent information from recent studies of the 35CI(d, r)3*S reaction 1,) and the 7-decay 1s - t 7) of 3,S are also given in columns 7 and 8 of table 2, respectively. The best J~ values, in the author's opinion, based upon these results as well as the present experiment are given in column 9 of table 2. The energy-level diagram for 34S, as determined by the present experiment, is shown in fig. 7. 4.1. T H E G R O U N D
S T A T E O F a4S
It is now appropriate to ascertain what information can be extracted from our spectroscopic data. The 1 = 2 transitions will be examined first• The ground state in
216
D.J. CROZIER
34S appears to be reached by a pure l = 2 transition, as is expected because o f its 0 ÷ assignment. In table 4, which compares the observed spectroscopic strengths G' with the expected strengths Gth = ( 2 J r + l ) / ( 2 J i + 1), it is seen that the g r o u n d state approximately exhausts the expected 0 + strength.
L J.oE-
e •
--
EXC (keY)
EXC (keY) 5326 e
6422
•
0.1 .--
-
t~
~:~
5694
0.5~
6256
0,0~ I .')"~--
0
E
•
.
~'-
6644
~
~
b
-r
• ~
÷ *
Z O.
o,
.0
0,05 0*
I 30"
II 1 60"
0era 0*
I
I
30*
60*
Fig. 5. Angular distributions identified as l = 3. The lines are the DWBA calculations.
4.2. THE STATES AT 2.13, 3.31 AND 4.12 MeV In addition to the pure l = 2 transition to the g r o u n d state, three states in 34S at 2.13, 3.31 and 4.12 MeV excitation are reached by a mixture o f / = 0 and l = 2 transitions. This is consistent with a 2 + assignment for these states. The l = 2 strengths o f the two states at 2.13 and 3.31 M e V are in very g o o d agreement with those o f their analog states in a4C1, as seen in the 33S(~, d)34C1 reaction. The 4.12 M e V state in 34S is m o s t likely the analog o f the 4.14 M e V state in 34C1, the angular distribution o f which was noted by Erskine et aL 3) to be consistent with that o f a mixed I = 0 and l = 2 transition. It is seen in table 4 that these three 2 ÷ states approximately exhaust the expected 1 = 2, J~ = 2 + strength.
a4S LEVELS I
I
,
'
I
'
l-
---I=l . . . . . . . 1=5
'
I
'
'
'
I
_]
I.C
EXC (keV)
hO
217
0•~
~.~_ \~
--
(keY) -/115
_1
0.5
,..~" .... liD'... \ ..... •
...•.,'\ • • .."
•......
\
•
0.1
...•
\\
...~'~.~
~ - ~ - ~
~0.1
0.0~
0.~=
1.0
~'~4I:~I~N~.
_
6'74
\\.
-
0.5
0.1
--
.,\ ..... .,. ),0.
,,''"
~
0.1 "-
o*
I
i
I
i
30*
l
I 6o*
3.01 i
I
I
)*
I 30*
I
I
I 60*
8¢ m Fig. 6. Angular distributions that were identified as mixed l = 1 and l = 3 transitions• The dashed lines represent the c o m p o n e n t s with different angular m o m e n t u m transfer. The solid lines are the sums o f the dashed curves.
TABLE 1 Optical-model parameters Projectile
d a) pc)
V (MeV)
W a) (MeV)
179•7 26.3 V(p) b) W(p) c)
Ro (fm)
A (fro)
R'o (fm)
A" (fm)
Rc (fm)
Vs.o. (MeV)
R .... (MeV)
A .... (fro)
0.656 1.15
1.086 0.57
1.488 Ro
0.535 0.50
1.30 Ro
5.5
Ro
0.57
a) Surface absorption was used for both d- and p-potentials. b) V(p) = 60--0•3 E + 0 . 0 4 Z/A{+27 (N--Z)/A (E in MeV). t0.64 E E < 13.8 MeV. ¢) W(p) = | 9 . 6 - - 0 . 0 6 E E _ 13.8 MeV. a) Ref. 2). =) Ref. 11).
4.3• T H E STATES A T 5.38 A N D 6.83 MeV
In addition to the mixed I -- 0 and I -- 2 transitions, there appear to be pure l = 0 transitions to two states at 5.38 and 6.83 MeV. These states may therefore be assigned
D. J. C R O Z I E R
218
TABLE 2 Summary o f spectroscopic strengths G ' and spin assignments Excitation (keV) 0.0 2127.5-- 0.2 ~) 3305 4- 2 4118 4627
5758 6128
± 4 ± 5 -- 6 4- 6 7 4- 7 4- 7 h)
6174 6256 6346 6422
4444-
5326 5384 5683 5694
l = 0
G ' ") l : 1 l-- 2
(0.13)
0.46 1.32
0 2
2 0+2
0.16 0.41
0.36 0.80
(2) 2 3
0
0.21
l-- 3
0.46 1.15
(t, p) b) (d, r) c) L lp
0.23
2
y-decay jrr
0+ 2 e,f) 2 e'f) 2 r) ~ 4 ¢), 3 - f) 2(+) r) 1 r)
0.65 3.12 0.42
8 8 8 8
0.25
6483 4- 8 6644 4- 9 6690 4- 9 6832 4- 9 6959 4-10 7115 4-10 7388 I)
0.91
5 - ~) 1 2
0.71 0.57
0.20
(2)
2+
2+ 321+ (1 --3)" 512+ (3)(3, 4)" 1(3, 4)"
1 0.22
(1-3) (4)(0-3)-
(1.64) 0.32 2
0.14 0.21 0.13
jz, d)
0.44
2+ (0--3)(3)-
2or3
7547 h) 7633 7659
4-11 4-11
0.98
7732 7755 7785 8142
4-11 4-11 4-11 4-12
(0.o3)
3
3-
(3, 4 ) (2, 3 )
(0.57) (0.14)
(0-3)-
0.08 0.27 0.26
(0--3) (0--3)
a) Spectroscopic strength, as explained in the text. b) Ref. 12). c) Ref. 14). a) The j~r value, based upon the present experiment and other available data. o) Ref. 14) f) Ref. xs). ~) Ref. 16). h) Weak. i) Obscured by an impurity. J) Ref. 7) TABLE 3 Summed strengths
I
EG'
EGth
I
4.939
4.0
2 3
2.949 9.027
3.0 8.0
a*S LEVELS
219
(1, 2) +. The 5.38 MeV state has been determined to have J = 1 [ref. 16)] and the positive-parity assignment can also be inferred from the (d, T) results. The 6.83 MeV state has been assigned 2 + on the basis of the (t, p) results. EXC(keV)
jr
(d,p)
EXC(keV)
d 7r
8142
4627
3-
4118
2+
2,(0)
7755. 7732~7785 . 7659~ 7633 7547~ 7388 /
2,(0)
6959
I,(3)
7115 3305
2+
2127
__2
+
6690 6644 6422~ 6346
2,(0)
(0-3)-
_/(o-3)~(0-3):-f__-------(2,3):~(3,4)~
3~ /
3-
~(I 6832
6256-6174--
- -
0+
2
3 I WEAK WEAK 3,(I)
( I, 3)42-
5
-(3,4)I-
5
~
2+
/ I_-------52-
5384 34 S
I I (1,3)
I 0
-(3,4)3-
5694 5.758 ~, 5683
I
,3)2"
~
6483
6128 ~ '
(d,p)
( I, 2) +
5326
2-
I
I I 3 3,(I)
WEAK I 3 I
0 3
Fig. 7. Energy-level diagram o f a4s. TABLE 4 l = 2 strengths j~r
Excitation (MeV)
G"
Gth
0+ 2+
0 2.13 3.30 4.12
0.46 1.32~ 0.3612.48 0.80
0.50 2.50
4.4. T H E STATES A T 4.63 A N D 6.17 MeV
Let us now discuss the states reached by l = 3 transitions. The spectroscopic strengths for the l = 3 transitions are shown in histogram form in fig. 8. The l = 1 admixture in the transition to the 4.63 MeV state is consistent with the previous 3 assignment for this state. The 6.17 MeV state also has an I = 1 admixture. Moreover,
220
D.J. CROZIER
in a study o f the 335(p, y)34C1 reaction by H y d e r 18), the E x = 6.17 M e V resonance in 34C1 was assigned J~ --- 3 - , T = 1 and was f o u n d to have a large d-f component. The most likely candidate for the analog o f this resonance appears to be the 6.17 MeV state in 34S, which therefore has been assigned ( 3 - ) . 4.5. THE STATES AT 5.32, 5.69 AND 6.64 MeV It is immediately apparent that the 5.69 MeV state is too strong to be anything other than the 5 - m e m b e r o f the d~f~ multiplet. In fact, this state has recently been assigned 5 - on the basis o f y-ray polarization direction correlation measurements 17).
33S(d,p)34S f,=3 3.0 2.0,.o-
-
4-
51-
3- ----
UNASSIGNE II
-..,2 d
Fig. 8. The spectroscopic strengths G' for l = 3 transitions. The values expected for various ,/-values are indicated by horizontal dashed lines. O f the remaining states reached by l = 3 transitions, the 5.32 and 6.64 M e V states appear to be too strong to be the ones associated with the remaining 3 - strength. They are also a m o n g the weaker states seen in 32S(t, p)345 and therefore g o o d candidates for unnatural-parity states. The 5.32 MeV state has been assigned J = 2 by Mulhern e t al. 16). The 1 = 3 (d, p) angular distribution indicates a negative-parity assignment. The 6.64 MeV state is then most likely the 4 - member o f the multiplet. 4.6. THE STATES AT 6.26, 6.42, 7.11, 7.66 AND 7.73 MeV A 2 - or 3 - assignment for the 7.11 and 7.73 M e V states seems to be implied by the l = 1 admixtures in their angular distributions *. The remaining three states at 6.26, 6.42 and 7.66 M e V presumably account for the missing 3 - and 4 - strengths and therefore m a y be assigned (3, 4 ) - . t The assignment 1- is a remote possibility if the l = 3 transition were A j ~ ~, which would seem rather unlikely.
345 LEVELS
221
4.7. THE STATE AT 7.63 MeV Finally let us consider the l = 1 transitions. The l = 1 strength is considerably fragmented, and the possibility of P~r mixing cannot be ignored. Therefore it is rather difficult to make assignments on the basis of the spectroscopic strengths. The states at 4.63 and 6.17 MeV have been considered in subsect. 4.4 and are designated 3-. The rather strong state at 7.63 MeV may correspond to the 3- state observed in this region in the (t, p) reaction and has therefore been designated (3)- in table 3. 4.8. THE STATES AT 56.8, 5.76, 6.34 AND 6.48 MeV The states at 6.34 and 5.76 MeV have previously been assigned 1- on the basis of the 328(t, p)345 data. The states at 5.68 and 6.48 MeV are both too strong to be 0 and therefore are assigned ( 1 - 3)-. 4.9. THE STATES AT 6.69, 6.96, 7.76, 7.79 AND 8.14 MeV The five remaining l = 1 states at 6.69, 6.96, 7.76, 7.78 and 8.14 MeV have been assigned ( 0 - 3 ) - since the possibilities could not be restricted further.
5. Two-body matrix elements 5.1. THE d~rf~, T = 1 INTERACTION Having identified the major components of the d~f~, T = 1 multiplet, it is interesting to try to compare these results with other known d~f~ multiplets. In order to do so, we must define the matrix elements of the residual two-body interaction. The matrix element for a given value of J is the difference between the excitation energy E Xof the state with that value of J and a reference energy Er for the multiplet. The reference energy Er, the excitation energy at which the state would be expected to occur if there were no residual two-body interaction, is given by Er = M(32S)+2M(n)-2Q[3ZS(n, y)33S]-M(34S)-I-Ex(335)~ - = 5.717 MeV, for the d~f~ interaction, where E r is the excitation energy in 345, M(325), M ( 3 4 5 ) and M ( n ) the masses of 325, 345 and the neutron, respectively, Q the ground state Q-value of the 325(n, ~)335 reaction and Ex the excitation energy of the ~ - level in 335. When the strength for a given J is fragmented, the centroid of the spectroscopic strength is used for Ex in calculating the matrix element. The configurations of the known d~f~ particle-particle interactions are shown in schematic form in fig. 9. The nucleus 34C1 can be thought of as a neutron and a proton coupled to a 325 core. This gives rise to two possible configurations corresponding to T = 0 and T = 1, the two possible values of isospin. In the nucleus 345, the circumstances are similar except that now both of the particles outside the core are neutrons. In this case, therefore, the isospin of the pair is T = 1. Finally, in the 365 core of the nucleus 38C1, the d~ shell is completely filled for neutrons. It is therefore possible to
222
D.J. CROZIER
say which of the particles occupies which orbit. Therefore, as far as the two particles are concerned, the configuration is the same as the right-hand diagram for 34C1. Hence the isospin of the pair is mixed. The complete set of d~f~ matrix elements obtained in 34C1 and 34S is given in table 5, along with values determined by other authors 19-22) and theoretical calculations 23. z4). It is apparent that the mixed-isospin matrix elements Ej of the residual two-body interaction in 38C1 should be related to the pure-isospin matrix elements E j r observed in 34C1 and 34S. In fact, one has
Ej
=
½(Ejo+Ej,).
In fig. I0 the predicted mixed-isospin matrix elements deduced from the mass-34 spectra are compared with the matrix elements actually observed in 38CI. The uncertainties in the predicted 3- and 4- matrixelementsare due to the unassignedstates TABLE 5 d~f~ Source
T= J=2
matrix elements (MeV)
0
T=
3
4
5
J=2
1
3
4
5
1.12 0.82 +1.45 + 1.35
--0.02
34C1 and 34S
--2.98
--1.71
--1.61
--2.06
--0.39
4°Ca [ref. 19)] Moinester and Alford 22) Erne 2o) Dieperink and
--4.14 --4.23
--1.92 -- 1.89
--2.03 --2.26
--2.35 --2.11
+0.91 +0.64
0.35 0.72 +0.21 --0.25
--3.65 --4.01
--1.85 --2.01
--1.77 -- 1.87
--2.90 --2.37
+0.38 +0.62
--0.08 --0.07
+0.92 +0.98
-!-0.43 +0.20
--4.75
-- 1 . 4 1
--0.96
-- 1.73
+0.27
--0.04
+0.81
--0.88
--3.11
--1.63
--0.87
--1.29
+0.28
0.07
0.34
--0.59
+0.52 --0.08
Brussaard 21) Sartoris and
Zamick 23) K u o and Brown 24)
in 34S. It is immediately apparent that the observed 2 - and 5- matrix elements are very close to the predicted values. The observed 4- matrix element is slightly lower than its predicted value. The most important discrepancy occurs for the 3- matrix element, whose observed value is seen to be considerably lower than the predicted value. One possible explanation for this discrepancy can be found upon examining the spectrum of asC1. The relative l = 3 spectroscopic strengths obtained from the 37C1 (d, p)38Cl experiment of Rapaport and Buechner 25) are displayed in histogram form in fig. 11. The expected strength for each value of J is indicated by a dashed line. It can be seen that the strength of the 3- state at 0.76 MeV falls about 30 ~ short of the expected 3- strength. It should be noted also that this state exhibits an l = 1 admixture. Since there are several observed 1 = 1 transitions to states of higher excitation
a4S LEVELS
223
PARTICLE-PARTICLE d~,2-f7/2 SPECTRA r---T---
f 7121 ',
~e I
1 I
d5 / 2 ~
34 S
p
F--r--1 l_l~__ I ----I
T= I n
r--l---lq I .... , • l / T=O
"c,
~--,%-I
~
i____1
I
½[(T.O).CT-,)]
Fig. 9. Schematic representation of the different nuclei in which d~f~ particle-particle spectra have been observed.
4-
-0.5
\\
-hO
5-
b.I
-I.5 2-
m
2-
PREDICTED FROM -2.0
MASS 34
SPECTRA
~8CI
OBSERVED
E,- ½ ( E.,o~E,.,I Fig. 10. Comparison of the mixed-isospin d~.f~, particle-particle matrix elements with those actually observed in 38C1. The uncertainties in the 3- and 4 - predictions are due to the unassigned levels in 3~S.
224
D.J. CROZIER
in 3Sc1, it seems reasonable to assume that some l = 3 admixture in these transitions m a y have been overlooked. I f this is indeed the case, it would conceivably give a somewhat higher value to the 3 - matrix element in 38C1 and thus would lead to better agreement with the prediction. However, this explanation raises another problem.
37CI(d,p)3eCI
STRENGTHS
5-
4-
ZUNASSIGNED
Fig. 11. Relative spectroscopic strengths G" for l = 3 transitions observed in the a~Cl(d, p)38C1 reaction. Horizontal dashed lines indicate the expected strengths for each value of./.
PARTICLE-HOLE
40K
d3/2--fT/2 SPECTRA
d31
I/2 [(T=O}+(T,I )]
~-l,--I 40(::o ~ -I
,---r--~
+-
and T=O
Fig. 12. Schematic representation of the different nuclei in which d~rf~ particle-hole spectra have been observed.
The residual t w o - b o d y interaction is observed not only in particle-particle spectra but in particle-hole spectra as well. As one would expect, the particle-particle spectra and the particle-hole spectra are not unrelated. In fact, they are related by a simple
34S LEVELS
225
Racah transformation known as the Pandya transformation 19), which may be written as E~ "h(p'p) =
- -
~ ( 2 J ' + 1) W(jlj2J2Jl ;JJ')E~-p(p'h).
(l)
j,
There are two nuclei in which d~f~ particle-hole multiplets have been observed. The configurations of these nuclei are shown schematically in fig. 12. In 4°Ca there are two
1.5
m
5_z... 2-
-- 7 / / / / 5 -
>
1.0
0.5
0
4-
P (mass-34)
40K
P(38CI)
Fig. 13. Comparison of dgrf~ matrix elements in 4°K (center) with the Pandya transforms of the 3sc1 spectrum (right) and the mass-34 predictions (left).
+1
- -
3"/I----
3" ~-~_
4 - ~
4- ~x.x~x.~
A
>
/
=E
2
- -
2-/5-{'L~"~'~
-2
-3 3-
4°Ca
P( mass -34 )
Fig. 14. Comparison of the pure-isospin d~.f~_ particle-hole spectra from 4°Ca and mass-34. The mass-34 spectra have been transformed by means of the Pandya transformation.
226
D.J. CROZIER
possible configurations, again corresponding to the two possible values of isospin T = 0 and T = 1. In 4°K, since there are two more neutrons than protons, it is possible to distinguish the nature of the particle and of the hole. Therefore the isospin of the pair is mixed. It has been well known for some time 26.27) that the Pandya transformation works extremely well between the d~fz~ spectra of 38C1 and 4°K. Therefore if the 3- matrix element in 38C1 should indeed be somewhat higher as has been proposed above, one should consider the effect upon the Pandya transformation between 38C1 and 4°K. TABLE 6 T h e (dg_) z m a t r i x elements (MeV) j~r, T mass-34 G l a u d e m a n s e t al. c) Ern6 a) ~) F r o m a4C1 [ref. ~)].
0 +, 1
1 +, 0
2 +, 1
--2.87 ~) --2.27 -- 1.71
--2.23 ~) --0.92 --2.11
+ 0 . 1 6 b) +0.16 +0.26
b) F r o m 34S.
c) Ref. 28).
3 +, 0 --2.72 a) --2.64 --2.51
d) Ref. 20).
A cursory examination of eq. (I) will reveal that the predictions for all of the particlehole states will be affected. This effect is shown in fig. 13. In the center of the figure are the observed particle-hole matrix elements in 4°K, while on the right are those predicted from the 3aC1 particle-particle spectra. It can be seen, as was mentioned before, that the agreement is very good. On the left-hand side of the figure are the predicted particle-hole matrix elements from the mass-34 spectra. The nucleus 40Ca provides another source of a complete set of d~ f~ matrix elements. These may be related to the mass-34 elements by means of the Pandya transformation with isospin, namely EPTp(p'h) :
--~-~ ( 2 J ' + 1 ) ( 2 T ' +
1)W(jlj2J2J
1 ; JJ')
l , l / [ 1 1 l l . T"/"t~ K'P'h(p-P) × , , ~7z~r:r, i x :~s'r" •
J'T'
In fig. 14 the Pandya transformation of the mass-34 spectrum is compared with the observed particle-hole spectrum in 4°Ca. In all of these comparisons some rather sizable discrepancies are noticed. Therefore some attempts at an explanation should be made. In the case of 4°K, one possible explanation is that the reference energy is too low. This might be caused by the true closed-shell state of 4°Ca not being just the ground state but containing some admixture of the 3.35 MeV (0 +) state as well. Such an admixture in the 4°Ca ground state would also be consistent with the 4°Ca spectrum, but considerable collective admixtures would still be required in the 3-, T = 0 and 5-, T = 0 states. Since the lowest 3- and 5- states in 4°Ca have been taken as the members of a multiplet, it should be pointed out that the states are known to have strong collective enhancement with B(E3), and B(E5) for transitions to the ground state much larger than single-particle values. It is therefore very likely that the collective admixtures significantly depress these states in energy.
a4S LEVELS
227
5.2. THE (d~.) 2 I N T E R A C T I O N
It should be pointed out that the identification of the mixed l = 0 and 1 = 2 transitions with the 4118 keV 2 + state gives a higher (more negative) value to the 2 + matrix element of the (d~_)2 interaction than was reported for 34C1 by Erskine et al. 3). The complete set of (d~) 2 matrix elements with this correction is given in table 6, along with theoretical calculations by other authors 20.28). The author wishes to thank his advisor, Professor J. P. Schiffer, for making available the facilities of Argonne National Laboratory and for much very valuable advice and encouragement. The author would also like to thank A. K. Hyder, Jr., for making his results available prior to publication and Dr. J. R. Erskine, Dr. H. T. Fortune and Dr. A. Richter for many helpful discussions. The author appreciates the assistance of W. Horath in target preparation, C. Bolduc in the operation of the spectrograph and the cooperation of the operational group at the tandem accelerator. Referenees 1) 2) 3) 4) 5) 6) 7)
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