Proton Capture Reactions and Nuclear Structure Proton Capture Reactions and Nuclear Structure

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Proton capture reactions and nuclear structure Proton capture reactions and nuclear structure

Protonvangstreacties en kernstructuur

(met een samenvat ting in het Nederlands)

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Rijksuniversiteit te Utrecht, op gezag van de rector magnificus prof. dr. J.A. van Ginkel, volgens besluit van het College van Decanen in het openbaar te verdedigen op woensdag 18 oktober 1989 des namiddags te 12.45 uur

door

Sybren Wouter Kikstra

geboren op 14 augustus 1962 te Almelo PROMOTOR: PROF. DR. C. VAN DER LEUN

Dit werk is een onderdeel van het onderzoeksprogramma van de "Stichting voor Fundamenteel Onderzoek der Materie" (FOM), die financieel gesteund wordt door de "Nederlandse Organisatie voor Wetenschappelijk Onderzoek" (NWO). Adagio cuUbilr

— Ludwig van Beethoven Contents

1 Introduction 9 1.1 Historical notes 9 1.2 Proton capture reactions 10 1.3 Resonant absorption 11 1.4 Summary 11

2 Resonant absorption by the lowest two T = 3/2 levels of 9Be 15 2.1 Introduction 15 2.2 Experimental 16 2.3 Measurements and data analysis 18 2.4 Results 20 2.5 Theory and discussion 22 2.5.1 The Ex = 14.393 MeV, J*;T = 3/2"; 3/2 level 24 2.5.2 The Ex = 16.975 MeV, J*;T = 1/2"; 3/2 level 25

3 Superallowed 42Sc(/?+)42Ca decay 31 3.1 Introduction 31 3.2 The 41Ca(n,7)42Ca measurements 32 3.3 The 41Ca(p,7)42Sc measurements 32 3.3.1 Experimental 32 42 3.3.2 Determination of 5p( Sc) 35 3.3.3 Levels of 42Sc 38 3.4 The 42Sc->42Ca superallowed ft value 41

4 Investigation of the 40Ca level scheme 51 4.1 Introduction 51 4.2 Experimental 52 39 4.3 The K+p resonances for Ep = 0.3-2.9 MeV 53 4.3.1 Measurements 53 4.3.2 Analysis 53 4.3.3 Results and discussion 60 Contents

4.4 Gamma-ray spectra 65 4.4.1 Measurements 65 4.4.2 Analysis 65 4.4.3 Results and discussion 70 4.5 Shell model calculations 81 4.5.1 Introduction 81 4.5.2 The WBMB interaction 83 4.5.3 Calculations 83 4.5.4 Results and discussion 85 4.6 Summary and conclusions 91

Statistical nuclear spin assignments 95 5.1 Introduction 95 5.2 Average spins 95 5.3 Weighted averages 99 5.4 Chi-squared tests 102 5.4.1 Resonances 102 5.4.2 Bound states 108 5.5 New assignments 109 5.6 Summary and conclusion 109 Appendix 113 Samenvatting 115 Nawoord 117 Curriculum vitae 119 Chapter 1

Introduction

In this thesis experimental studies are described of the structure of some atomic nuclei. Where possible, an attempt has been made to interpret the results of the measurements in terms of existing models. The 40Ca and 42Sc nuclides were excited by bombarding 39K and 41Ca targets, respectively, with low energy protons (Ep = 0.3-3.0 MeV), that were produced by the Utrecht 3 MV Van de GraafF accelerator. From the measured energy and intensity of the 7-rays created in the subsequent decay of the nuclei, accurate information was obtained on the existence and properties of their excited states. The nuclear reactions of this type are referred to as proton-capture or (p,7) reactions. In addition, properties of two levels at high excitation energy of the 9Be nucleus were investigated. These levels were excited by the resonant absorption of 7-rays from the nB(p,7)12C reaction. The above, essentially independent investigations are described in chapters 2-5 of this thesis in the style presently required by the international physics journals, i.e. rather concisely. The purpose of the rest of this introduction is to provide the reader with some background information which may be considered 'common knowledge' in publications. We will start with some historical notes, proceed with a short discussion of the merits of capture reactions and resonant absorption in nuclear spectroscopy, and finally give some details specific to the above mentioned subjects of this thesis.

1.1 Historical notes The discovery by Pose [1] in 1929 of a resonant effect in the bombardment of a thin aluminum foil with a-particles from a radioactive source, marks the start of the exploitation of resonances in capture reactions for obtaining data on properties of nuclear energy levels. In 1935, resonances in the yield of a proton induced reaction were observed by Hafstadt and Tuve [2] who used a 10 Introduction

Van de Graaff accelerator, and narrow (few keV) separated (p,7) resonances were first measured with a Geiger Miiller counter by Tangen [3] during world war II. Giant steps forward in the detection techniques were made around 1955, when the high efficiency Nal detectors became generally available, and again in 1966 with the introduction of high-resolution semiconductor detectors of the Ge(Li) type. Presently, Ge and Ge(Li) detectors are used, often surrounded by a scintillator which is effective in the reduction of the counting of unwanted background radiation [4,5]. Nal detectors are still in use when energy resolution is less important than efficiency [6]. The importance of the developments in computer facilities and data acquisition techniques hardly needs mentioning.

1.2 Proton capture reactions Proton capture resonance reactions are an excellent tool for nuclear structure investigations. Firstly, the reaction process is very well understood. Provided that the reaction proceeds through separated resonances, which is the case when the bombarding energy is not too high, the compound system has a well-defined angular momentum and the unquestioned single-level Breit-Wigner theory may be applied. Secondly, the electromagnetic interaction is the best understood of all four fundamental interactions; the results of electromagnetic theory and experiment agree on the 1 ppm level. The strength of the electromagnetic interaction is ideal; it is sufficiently large to cause easily observable effects of the charge and current distribution in the nucleus, and small enough, compared to the strong interaction, to validate the application of perturbation theory for the analysis of the observed effects. The above considerations imply that the results of the measurements are model-independent, and consequently the data may be treated statistically. The availability, in most cases, of a variety of resonances with different spins and parities favourably distinguishes the (p,7) reaction from e.g. thermal neutron capture. Specific bound states may thus be excited with different intensities at different (p,7) resonances, which is especially advantageous in the identification and disentanglement of close doublets [4]. It should be mentioned, however, that the (p,7) and (n,7) reactions are complementary in that there are no p- and sd-shell nuclei which can be studied in both reactions. Another advantage of (p,7) reactions is the possibility to measure the lifetimes of nuclear excited states from the observed Doppler shifts of the energies of 7-rays emitted by the recoiling final nucleus [4,5]. The power of (p,7) reactions in combination with a good detector system was beautifully demonstrated in the study by Endt et al. [7] of 26A1, which is now by far the best known nucleus in the sd-shell. The data are ideally suited Resonant absorption 11 for tests of chaos theory (see e.g. [8]), for which complete (no missing levels) and pure (correct spin assignments) data are of utmost importance. The availability of complete and pure information has prompted the study of a new method for the assignment of nuclear level spins, which is discussed in chapter 5 of this thesis. For further reading the reader is referred to refs. [9-11].

1.3 Resonant absorption As was first shown in 1957 [12,13], it is feasible to use proton capture resonances as a source of tuneable monoenergetic 7-rays to excite nuclear levels. A 7- ray energy slightly higher than the excitation energy of the level is needed to compensate for the recoil energy losses at emission and absorption. The angle dependence of the Doppler shift provides the energy tuneability for obtaining a resonant dip in the transmission cross section, which contains the nuclear structure information. In the first experiments a level was 'self-excited' by the Doppler-shifted decay 7-rays of that same level. The feasibility of 'cross excitation' of a level in one nucleus by a 7-ray from another nucleus was first demonstrated by Sparks et al. [14]. The resonant absorption method is reviewed in ref. [15].

1.4 Summary Chapter 2 describes an investigation of the lowest two T = 3/2 levels of 9Be. As shown schematically in fig. 1.1, these levels were excited by 7-rays arising from broad resonances in the nB(p,7)12C reaction. The results of the measurements are interpreted by a comparison to the analogue /?-decay of 9Li and to shell model calculations. The basic idea behind the experiments presented in chapter 3 is contained in the following equations

+ Trip = m42Sc + Sp, and thus Tl = ~ ' «ca Sn — Sp + mp — mn,

where m denotes the mass in energy units, and 5n and Sp are the neutron separation energy in 42Ca and the proton separation energy in 42Sc, respectively. Fig. 1.2 illustrates the situation in the form of simplified level schemes. The total decay energy of the superallowed 0+ —• 0+ /?+ transition between the ground states of 42Sc and 42Ca was thus determined accurately by measure- 42 42 ments in Utrecht of Sp of Sc and in Oak Ridge of Sa of Ca. The results were 12 Introduction

f,x(MeV) f,T £x (MeV) (MeV)

18.83 . ; 3.06

16.98 l/2";3/2 16.98. [ 1.06 •• • A ' 15.96

14.39 3/2 ;3/2 B + p

4.44

[ 43/2";l/2 Be C

Figure 1.1: In the experiment described in chapter 2, the lowest two T = 3/2 levels of 9Be were excited with 7-radiation from broad resonances the uB(p,7)12C reaction.

used for verification of the conserved vector current hypothesis, which implies that the // values of all superallowed 0+ —* 0+ /J-decays are the same. Chapter 4 descibes a spectroscopic study of the 40Ca level scheme via the 39K(p,7) reaction. An attempt was made to describe properties of odd-parity states of A = 37-41 nuclei with a variant of the WBMB interaction [16]. Finally, a new method for the assignment of nuclear spins by a simple statistical analysis of spectroscopic information is proposed in chapter 5. Chapters 2-5 have been written in the style required by Nuclear Physics. Chapter 3 has already been published in this journal, chapters 2 and 5 have been submitted for publication, and chapter 4 will be submitted after incorporation of the results of a few additional measurements. 1

References 13

Ex(MeV) f;T £„ (MeV) f;T Ep (MeV)

6.OB : f |- 1.85 11.48 I 4.27 • ! Ca + n j -V L 41Ca+p

.A. Sc

A- /

Figure 1.2: The total energy of the superallowed 0+ —• 0+ 42Sc(/?+ )42Ca decay was inferred from the proton separation energy of 42Sc and the neutron separation energy of 42Ca, determined in experiments described in chapter 3.

References

[1] H. Pose, Z. Phys. 30 (1929) 780 ; 64 (1930) 1 [2] L.R. Hafstadt and M.A. Tuve, Phys. Rev. 47 (1935) 506 (L) [3] R. Tangen, Det Kgl. Norske Videnskabers Selskabs Skrifter (1946) No. 1 [4] Chapter 4 of this thesis [5] Chapter 3 of this thesis [6] Chapter 2 of this thesis [7] P.M. Endt, P. de Wit and C. Alderliesten, Nucl. Phys. A459 (1986) 61; P.M. Endt, P. de Wit and C. Alderliesten, Nucl. Phys. A476 (1988) 333; P.M. Endt, P. de Wit, C. Alderliesten and B.H. Wildenthal, Nucl. Phys. A487 (1989) 221 [8] G.E. Mitchell, E.G. Bilpuch, P.M. Endt and J.F. Shriner, Phys. Rev. Lett. 61 (1988) 1473 [9] W.A. Fowler, C.C. Lauritsen and T. Lauritsen, Rev. Mod. Phys. 20 (1948) 236 14 References

[10] H.E. Gove, in Nuclear Reactions I, eds. P.M. Endt and M. Demeur, North-Holland, Amsterdam (1959) [11] W.D. Hamilton, The electromagnetic interaction in nuclear spectroscopy, North- Holland, Amsterdam (1975) [12] S.S. Hanna and L. Meyer-Schtitzmeister, Phys. Rev. 108 (1957) 1644; 115 (1959) 986 [13] P.B. Smith and P.M. Endt, Phys. Rev. 110 (1958) 379; 110 (1958) 1442 [14] R.J. Sparks, H. Lancman and C. van der Leun, Nucl. Phys. A259 (1976) 13 [15] P.B. Smith, Am. Inst. Phys. Conf. Proc. 125 (1985) 192 [16] E.K. Warburton, J.A. Becker, D.J. Millener and B.A. Brown, BNL Report 40980 (1987) Chapter 2

Resonant absorption by the lowest two T — 3/2 levels of 9Be

Abstract

The ground-state radiative widths of the Ex = 14.393 and 16.975 MeV, T = 3/2 levels of 9Be were measured with a new 7-ray resonant absorption technique. Thin targets and broad resonances in the nB(p,7o7i)12C reaction were used to produce the required monochromatic 7-rays of variable energy. The present results are compared to results of (e,e') scattering and, along with the particle widths of the 16.98 MeV level previously determined, to shell-model calculations. The isospin mixing and giant Ml character of the levels are also discussed.

2.1 Introduction The lowest two T = 3/2 levels of9 Be at 14.393 and 16.975 MeV are of remarkably small width, since particle decay is either forbidden or inhibited by isospin or energy conservation [1-3]. The radiative transitions between these states and the ground state of 9Be are therefore well-defined in energy and, being of Ml character, are suitable for probing the spin, orbital and isospin characteristics of the transitions. The isospin forbidden particle decay widths of the upper state are a measure of the degree of isospin mixing occurring in this state. The two levels are part of the giant isovector Ml resonance and are expected to exhaust a large fraction of the sum rule strength. Comparison with the /3-decay analogues of the Ml transitions gives information on the relative importance of orbital and spin components of the isovector Ml operator [4]. Various shell-model calculations give large variations in Ml strengths of the decays from the two T = 3/2 levels to the ground state. These variations have been ascribed [5] to the difference in orbital symmetry of the main components of the T = 3/2 states and the ground state. The Ml matrix elements then

15 16 Resonant absorption by the lowest two T = 3/2 levels of 9Be

depend on small components of the wave functions that might not be determined accurately enough when the two-body matrix elements of the interaction are obtained from a fit to excitation energies alone. Recent p-shell calculations [6] show that improvement in this respect can be obtained by simultaneously fitting other observables. In a previous experiment [7] the radiative width F70 of the upper T = 3/2 state at Ex = 16.975 MeV, was determined in a resonant absorption measurement in which 7-rays from the same resonance in the inverse reaction 7Li(d,7)9Be were resonantly absorbed by the 9Be(7,X)9Be(16.98) process. The angle dependence of the Doppler shift in the capture reaction provides the energy-tuneability for obtaining a resonant dip in the absorption cross section [8,9]. The total width F and some (limits on) particle decay widths of this state were also deduced from 7Li(d,7)9Be yield measurements [7]. Since the lower state at 14.393 MeV is particle bound, the above resonant self- absorption method can not be used. However, it has been demonstrated [10] that 7-rays from very broad or overlapping resonances in suitable capture reactions can be used in such cases. The required monochromacity is obtained by using targets of the appropriate thinness and the energy tuneability is produced by changing the energy of the captured particle. We have applied this technique to the level at 14.393 MeV, as well as to the one at 16.975 MeV in order to demonstrate consistency between the two methods.

2.2 Experimental In the second method the source of 7-rays used for the resonant absorption must come from a capture reaction in which the yield remains practically constant over the energy of the absorbing resonance. The thinness of the target must be comparable to the resonance width. The 7-energy E-, changes with proton energy Ep according to

where Q is the Q-value of the capture reaction, E( the final state energy and irip and m-i are the projectile and target mass, respectively. Corrections to (2.1) for the angle-dependent Doppler shift of the in-flight emitted 7-rays and for the recoil losses at emission and absorption have to be applied to determine the resonant beam-energy, EVT. The 7-rays required for the photo-excitation of the 14.39 MeV level of 9Be were produced in the capture reaction nB(p,7i)12C(4.43) at proton energies Ep varying from 3.050 to 3.070 MeV. The yield arises here mainly from broad overlapping resonances centered at Ep =1.4, 2.6 and 3.5 MeV [11] which produce Experimental 17

PLASTIC Pb.

Li,CO:,

Pf Nal(Tl)

Pf PLASTIC

DUMMY

10 cm

Figure 2.1: Schematic drawing of the resonant absorption setup. The aluminum dummy absorber was positioned in place of the Be absorber to obtain data for normalization at each value of the proton energy Ep. The paraffin (Pf) and Li2CO3 are effective in shielding neutrons coming from the target.

a continuous 7-yield. At these proton energies the primary 7-rays to the first 12 excited state of C at 4.43 MeV scan the energy range around Ey = 14.39 MeV. The maximum Doppler shift and total recoil-energy in this case amount to +96 and —22 keV, respectively. Since the neutron channel opens at Ep =3.022 MeV, the radiative decay process competes with neutron decay which has cross- sections larger by three to four orders of magnitude. A large Nal detector was used to increase the (p,7) counting rate as much as possible. We note that at 0° a wide-angle 7-detector can be used since the Doppler shift varies slowly near 0° and produces little Doppler broadening of the 7-rays. A large amount of paraffin was used between the (p,7) source and the 7-detector the thermalize 6 the fast neutrons. A jacket of Li (Li2CO3) surrounds the Nal detector to absorb the slow neutrons. In fig. 2.1 a schematic drawing of the set-up is shown. Two 18 Resonant absorptica by the lowest two T = 3/2 levels of 9Be cylinders* of 99.9 % pure beryllium metal (9Be), the first 5 cm in diameter and 15.5 cm long and the second 12 cm in diameter and 30 cm long, were used as absorbers. The absorbers were placed, as shown, in separate boxes filled with paraffin which were 50 cm high, 81 cm wide, and 15.5 and 45 cm long, respectively. The boxes were moveable perpendicular to the beam direction and contained a second absorber of aluminum, the length of which was adjusted to give the same electronic absorption as the Be absorber. This second (dummy) absorber allowed normalization measurements in an identical geometry. The Nal crystal*, 25 cm in diameter and 30 cm long, is viewed by six EMI 9758L photomultiplier tubes and is surrounded by a plastic anticoincidence shield that vetoes any event that makes a pulse in both the Nal and the shield, chiefly the Compton events and cosmic ray background [12]. The energy-resolution was 4-5% at £7 = 15 MeV. Conventional electronic circuits were used including antipileup logic which reduced the effects caused by the neutron-induced 7-rays considerably. The Utrecht 3 MV Van de Graaff accelerator provided proton beam-currents of up to 150 /iA with an energy spread of about 200 eV at Ep — 1.0 MeV. The targets consisting of boron enriched to 99.1 % in nB were deposited onto 0.3 mm tantalum backings with thickness ranging from 17 to 25 /ig/cm2. By directly cooling the backings and continuously wobbling the beam up and down to spread the heating over an area of 2 x 5 mm2 most targets showed no deterioration during the 24-hour runs of the experiment. Spectra not vetoed by pulses in the anticoincidence shield (accepted) and those vetoed by the shield (rejected) were sorted into 2048 channels, accumulated in the memory of a PDP 11/34 computer and dumped onto magnetic tape for off-line analysis. The experiment control and on-line analysis was automatized on the basis of the ANALYS system [13].

2.3 Measurements and data analysis A typical scan consisted in measurement of the 7-yield through the 9Be absorber sandwiched between two measurements through the dummy absorber at one 7- energy, changing to the next energy and repeating the measurements through the dummy and the 9Be absorbers, and so on until all the desired energies had been scanned. The 7-energies were changed in approximately 1.0 keV steps by changing the proton energy appropriately. A region of about 20 keV was scanned over the resonant dip. Twelve scans over the energy range Ey = 14.38- 14.40 MeV were completed. Three scans were used to locate the exact resonant •Kindly loaned by ECN, Petten. 'Kindly loaned by CRN, Strasbourg. Measurements and data analysis 19

•100(1 14.39 MeV

ll I2 B(p,7o7,) C

Ep = 3.06 MeV

1B.83 MeV

1000 • . H L 1

0 • 15 20 E, (MeV) u l2 Figure 2.2: High energy part of the B(p,7) C 7-ray spectrum measured at Ep = 3.060 MeV, with the 25 x 30 cm Nal detector, showing the 70 and 71 peaks.

proton energy and three other scans had to be discarded because of (partial) target breakdown. Of the remaining six scans one was measured with a 30 cm 9Be absorber; the other five with a 45.5 cm 9Be absorber. Long term stability of target composition and thickness was checked by consecutively measuring two scans with the same target, one with increasing beam-energies and the second with decreasing beam-energies. To test the consistency of the results obtained with the present and the previous method, two runs were performed around 12 Ep = 1060 keV were the C ground state 7-transition excites the upper T = 3/2 state investigated in the previous experiment. Fig. 2.2 shows the high-energy portion of a typical pulse-height spectrum, containing the two primary 7-rays 70 and 7! which lead to the ground state and first excited state of 12C, respectively. The background is due to cosmic radiation. In order to determine the area of the 71 peak in a consistent way from low-yield spectra a superposition of two empirical lineshapes [14] was fitted to the 70 and 71 peaks. The detector- 20 Resonant absorption by the lowest two T = 3/2 levels of 9Be dependent parameters of this function were obtained by fitting a number of high-yield spectra. All but four parameters (position of upper peak, height of both peaks and background level) were kept constant during further analysis. A peakarea was obtained by integrating the background corrected lineshape over an energy region around the peak centroid of fixed width. The area was dead-time corrected (< 2%) and normalized by dividing the main-absorber yield by the sum of the two dummy absorber yields. After the complete scan was constructed, the experimental absorption integrals Ae were obtained from numerical integration of the data-points below the baseline as described in ref. [7]. The absorption integral determines the value of the ground-state radiative width I\o when the total level width F is known [7]. In the present analysis of the data on the upper level, the value F = 490±50 eV was used, as determined in the previous experiment [7]. Recently a strongly deviating value of F = 303 ± 30 eV has been reported [15] for this level. The total width F = 381 ± 33 of the lower level was taken from the literature [3]. The value of F7o of the lower level is only moderately dependent on F; a 5% change in F causes a 1% change in F^o- This is also true for the upper level since the pertinent quantity for this dependence, ncro [see ref. [16]], is close to unity for both levels. The theoretical absorption integrals for the lower state were calculated for a range of ground state widths according to equation (1) of ref. [7] for the two absorber thicknesses used in this experiment.

2.4 Results

Typical transmission curves obtained via the analysis decribed above are shown in fig. 2.3. Table 2.1 presents the results in detail. There are no indications for changes in target composition or thickness during a run; the two runs measured consecutively with the same target (see sect.2.3) lead to consistent absorption integrals which also justifies the assumption of a constant 7-ray energy spread within the period of one run. The result for the upper level, F7o = 17.6 ± 2.3 eV, is consistent with the 16.6 ± 1.2 eV obtained with the older technique [7]. Figure 2.4 gives an illustration of the graphical inversion method used in the calculation of F-yo of the lower level. The final result for the 14.39 MeV level is F^o = 5.9 ± 0.8 eV, which is in good agreement with the value of 6.2 ± 0.6 eV reported by Bergstrom et al. [17] but deviates at the 2a level from the value of 8.1 ±0.8 eV obtained by Clerc et al. [18]. Table 2.4 gives a summary of these results. Results 21

= 14.39 MeV

c I I o " t 2 1 g en C CO

01 > 9 9 Be(7,7) Be

L = 30.0 cm

1 1 1 1 1 1 1 1-

1 1 1 1 I C 1 1 . o 1 i in ( i en 6 en C £ 0.9 01 9 9 Be(7,7) Be

OS 0.8 L — 45.5 cm

3050 3055 3060 3065 ED (keV)

Figure 2.3: Transmission dips for E*, « 14.39 MeV 7-rays measured with Be absorbers of 30.0 and 45.5 cm. The lines drawn serve to guide the eye. 22 Resonant absorption by the lowest two T = 3/2 levels of 9Be

Table 2.1: Absorption integrals (At) and radiative widths (I\o) of the lowest two T = 3/2 states of 9Be Ex(MeV) : 14.393 16.975 Absorber length (cm) : 30.0 45.5 45.5 >U(eV) 300 ± 70 510 ± 90 570 ± 70 520 ±100° 510±110 520 ± 120"

3?(eV) 300 ±70 520 ±60 550 ±60

I\o (eV) 5.0±1.4c 6.3±1.0e 17.6 ± 2.3d T-,06 (eV) 5.9 ±0.8 17.6 ±2.3

"Scans performed with the same target and increasing and decreasing beam energy (see text). 'Weighted average. ^Calculated with T = 381 ± 33 eV [3] and a Debye temperature of 1160 K. ^Calculated with T = 490 ± 50 eV [7] and a Debye temperature of 1160 K.

2.5 Theory and discussion Most p-shell calculations performed so far [5,19-21] show no consistency in the predicted values of the isovector Ml strengths which are measured in the present experiment. As noted by Woods and Barker [5] the large spread in the calculated values most probably is due to the sensitivity of these observables to small components of the wave functions. In all calculations, the spatial symmetry of the main components of the lowest two T = 3/2 states is [/ = 32]*, whereas the ground state has mainly [/ = 41]. Although the spin component of the isovector Ml operator has vanishing matrix elements between states of different spatial symmetries, it can become important because the spin ^-factor is about ten times as large as the orbital g-factor. Therefore small (as 1%) admixtures of [/ = 32] components into the 9Be ground state can cause the spin and orbital parts of the two transitions to be of the same order of magnitude. In the present work the calculations of Cohen and Kurath [19] and Kumar [21] were repeated, in order to analyze separately the spin and orbital contributions to the isovector Ml transitions. (Due to the limited precision of the /./-coupling matrix elements given in refs. [19,21] some of the widths presented in table 2.4 differ slightly from the original values.)

*The standard notation is [/] = [32]. Theory and discussion 23

, I = 45.5 cm Tr0 - 5.9 r 0.8 eV /

600 •

L = 30.0 cm •

1 -400

10 (eV)

Figure 2.4: Illustration of the graphical inversion method used for the determination of F7o from the measured dip areas. The curves were calculated according to eq.(l) of ref. [7].

It appears that the interactions used in the various calculations all give roughly the same values for the orbital parts of the Ml transitions, but indeed lead to strongly differing values for the spin parts. Small components of the wave functions can be sensitive to matrix elements of the interaction which may not be well determined by a procedure of fitting energies only. In the present calculation the interaction matrix elements are determined by simultaneously fitting energies as well as static moments [6]. The effective interaction was parameterized in terms of the 15 two-body matrix elements of the p-shell. Two parameters specifying an effective three-body interaction were also included [22]. The r.m.s. deviations obtained in a fit of 73 normal parity (for the Op shell) energy levels and all available magnetic dipole and electric quadrupole moments of A = 4 — 16 nuclei is 0.38 MeV for the energy levels, and 0.11 /IN and 0.48 efm2 for magnetic dipole and electric quadrupole moments, respectively. The Ml transition strengths, separated into the orbital and spin components, as well as 24 Resonant absorption by the lowest two T = 3/2 levels of 9Be

Table 2.2: The Ml strength SMI, separated into its spin and orbital parts, of the 7-decay of the lowest T = 3/2 state of 9Be and the analog /3-decay of 9Li

log/i° 5M1 (W.u.;1 total spin orbit Experiment 5.31» 0.094 0.021 0.026 ±0.05 ± 0.012 ± 0.002 ± 0.007 Calculation: (6-16)2BME<: 5.08 0.112 0.036 0.021 (8-16)2BME<: 4.74 0.170 0.078 0018 (8-16)POTc 4.68 0.183 0.090 0.016 Kumar** 5.48 0.069 0.014 0.020 Present work 523 0.089 0.026 0.019

"Transition between the9 Li and 9Be ground states. »Ref.[31. cRef.[19]. •*Ref.[21].

the /3-decay strengths are then calculated with this interaction.

2.5.1 The Ex = 14.393 MeV, J*;T = 3/2"; 3/2 level In table 2.2 the experimental and theoretical Ml strengths of the transition from the J*;T = 3/2";3/2 state to the9 Be ground state are presented together with the analog /9-decay strength between the 9Li and9 Be ground states. The Ml operator is split into its orbital and spin parts. The 'experimental' value for the spin Ml strength was deduced from the ft value for /?-decay. This can be done since the same operator, i.e. ar, is involved in the Ml- as well as the /3-decay, [see e.g. ref. [23]]. It should be noted that the bare-nucleon isovector fir-factor g'v - g'n = 9.421, and the values K/G\ = 6163 sec and G\/G\ = 1.4 (all from ref. [23]) are used in order to relate the ft value to the B(Ml) value. The orbital Ml strength can subsequently be deduced from the total and the spin Ml transition strengths. Since there are two solutions, the additional assumption was made of constructive interference of the orbital and spin matrix elements in the Ml operator. As shown in table 2.2, all the calculations listed are consistent with the 'experimental' value obtained with this assumption. The orbital Ml matrix element can also be estimated without performing a shell- model calculation. Assuming that the ground state and excited state have spatial symmetry [/ = 41] and [/ = 32] respectively, with L = 1 and S = 1/2 (which indeed are the dominant components in all calculations), an orbital Ml strength of 0.025 W.u. is obtained, which actually gives the best agreement with the Theory and discussion 25

Table 2.3: The Ml strength SMi, split into its spin and orbital parts, of the 7-decay of the second T = 3/2 state of9 Be Sui (W.u.) total spin orbit Experiment 0.162 ± 0.009 Calculation (6-16)2BMEa 0.063 0.005 0.032 (8-16)2BME° 0.023 0.000 0.021 (8-16)POTa 0.021 0.001 0.029 Kumar* 0.232 0.066 0.050 Present work 0.078 0.056 0.043

aRef.[19]. *Ref.[21].

'experimental' value of 0.026 ± 0.007 W.u. The various calculations in table 2.2 lead to disagreeing values for the spin part of the Ml matrix elements. It is apparent that the spin component is overestimated by most calculations, whereas the present calculation gives the best agreement. Here it should be re-emphasized that the transition between the dominant components in the wavefunctions, i.e. [/ = 41] for the ground state and [/ = 32] for the excited state, does not contribute at all to the spin Ml matrix element. Nevertheless, the fact that the strength for the spin part of this Ml transition in most calculations even exceeds the 'experimental' strength, indicates its large sensitivity to small components v 1 ihe wave functions.

2.5.2 The Ex = 16.975 MeV, J*;T = 1/2"; 3/2 Jsvel In table 2.3 the Ml strength of the transition from the J*;T = l/2~;3/2 state to the 9Be ground state and the calculated strengths ar« compared. Again the strength is separated in orbital and spin components, although for this level an 'experimental' separation is not possible, sin-e the analog /9-decay from 9Li has not been observed. The various calculation3 in this case show a larger spread in the orbital Ml strength than was found ibr the lower T = 3/2 level. In analogy to the previous section, a simple calculation can be performed to show the contributions from various wave function components in the initial and final states. The orbital Ml strength of a transition xotn a pure [32]42P state (which again is the main component of the actual l/2~;3/2 state) to 28 Resonant absorption by the lowest two T - 3/2 levels of 9Be a pure [41]22P state is only 0.010 W.u., whereas a transition from a [311]425 initial state gives an orbital strength of 0.148 W.u.. Hence admixtures of the latter component into the l/2~; 3/2 state may have a considerable effect on the orbital Ml strength. Since in all calculations the [311]42S component enters into the wave function with an intensity of at most 5%, it is expected that in this transition orbital and spin matrix elements are of comparable magnitude and interfere constructively, because otherwise the experimental strength is too large to be explained with the present wavefunctions. The sensitivity to small wave function components is again demonstrated by the large value of the spin component in the Kumar calculation relative to all other calculations. The [32]24P and [32]22P components (which have intensities of less than 1.6% and 0.9%, respectively, in all calculations) have equal signs only in the Kumar case, which causes an enhancement of the Ml strength beyond the experimental value. The present calculation gives an improvement over the Cohen and Kurath values and deviates on the low side about as much as does the Kumar value on the high side. Barker has performed calculations [20,24] of isospin mixing based on the state-mixing model similar to the earlier calculations concerning 8Be [25]. W Coulomb interaction matrix elements between the Ex = 16.98 MeV, J ;T = l/2-;3/2 state and background J*;T = 1/2"; 1/2 states at 2.02, 11.88, 15.64 and 19.40 MeV were calculated to be —29, 14, 38 and 15 keV respectively (for b = 1.7 fm). Because of its proximity in energy and its larger matrix element, the 15.64 MeV state is expected to provide the main admixture. On this basis some qualitative observations can be made regarding the various particle decay widths of the 1/2"; 3/2 state. The 15.64 MeV, J*;T = l/2~; 1/2 state is 74% [32]22P and 10% [32]24P. The orbital symmetry does not allow neutron decay to low-lying 8Be states which have mainly symmetry (4), or a-decay to low-lying 5He states, whereas decay to higher states of 8Be and 5He is symmetry allowed but energetically forbidden. Thus, small neutron and alpha widths are expected. The neutron width Fno is found to be small but at present the prediction is not tested for the a channels since only an experimental upper limit of 350 eV is found. The T-forbidden deuteron decay to the 7Li ground state is allowed by symmetry, which (as in the case of 8Be) accounts for the relatively large deuteron width of 86 eV (or 0.05% of the Wigner limit for deuteron decay of 9Be). The isospin allowed proton decay to the 8Li ground state is also allowed by symmetry since the 1/2"; 3/2 state is 87% [32]42P. In the previous experiment the proton width Tp was deduced to be 12 eV (within a factor of two). Woods and Barker [5] estimated Fp = 30 eV, using formulae and values from ref. [20] with a channel radius a = 4.35 fm, a dimensionless single particle reduced width 0^ = 0.4 and Theory and discussion 27

Table 2.4: Comparison of experimental and calculated values of partial widths (in eV) of the lowest two T = 3/2 states of 9Be Experiment Theory" Present Ref. [17] Ref. [18] Ref. [27]" Present Ref. [19] Ref. [20] Ref. [2lf

Ex= 14.393 MeV, J*;T = 3/2~;3/2 c T7o 5.9 ±0.8 6.2 ±0.6 8.1 ±0.8 7.2 ±0.3 '5.5* 7.0 5.3 4.3* 10.7d 8.3e 11.4'

Ex= 16.975 MeV, J*;T = l/2~;3/2 h C 6 r70 16.7±1.0» 11.5 ±1.4 17.2±1.8 19.0 ±1.0 8.1* 6.4 15.2 23.9 2.3"1 18.2'

490 ± 50 r s rpo nt? 14.4* 20.0 25.9' 14.6* 20.2' 26.3J «35 Fdo 86 ±18 ra <350 rn <380 "The calculations of ref. [19] and ref. [21] have been repeated in this work (see text). Some of the present widths differ slightly from the original values due to the limited precision of the J J-coupIing matrix elements given in these refs. 'With bare-nucleon g-factors. C(6-16)2BME. d(8-16)2BME. 'Ref. [5]. /(8-16)POT. i Average of the present value with 16.6 ± 1.2 eV (ref. [7]) and 16 ± 2 eV (ref. [10]). ftRef. [18] as corrected in ref. [261 , then multiplied by a statistical factor 2, ref. [27]. 'a = 3.6 fm, 6gp = 0.408 [24] (see text). >a = 4.0 fm, 6jjp = 0.351 [24] (see text).

a spectroscopic factor 5 = 0.100 (and neglecting the 6% Thomas-Ehrman shift). Calculations by Cohen and Kurath [19] and Kumar [21] give 5 = 0.072 and 0.129 respectively. When Q^ is calculated as an overlap integral at the channel radius a, 4^ (2-2) 6 /u2(r)r2dr o where u(r) is the radial wavefunction determined in an appropriate Woods- Saxon potential, the results of the above mentioned calculations change to the values given in table 2.4. 28 References

In the previous paper [7] we compared the T-forbidden nucleon decay channels in 9Be (second T = 3/2) and 9B (first T = 3/2) using different recipes for the channel radius a. If a = 3.6 fm is used in both cases the (more) closely agreeing values 7^0 = 6.7 eV and 7p0 = 6.3 eV are obtained.

Acknowledgement We have profitted greatly from communications from Professor Fred Barker.

References [1] J.B. Woods and D.H. Wilkinson, Nucl. Phys. 66 (1965) 661 [2] B. Lynch, G.M. Griffiths and T. Lauritsen, Nucl. Phys. 65 (1965) 641 [3] F. Ajzenberg-Selove, Nucl. Phys. A413 (1984) 1 [4] S.S. Hanna, in Isospin in nuclear physics, ed. D.H. Wilkinson (North Holland, Am- sterdam, 1969) p. 593 [5] C.L. Woods and F.C. Barker, Nucl. Phys. A427 (1984) 73 [6] A.G.M. van Hees, A.A. Wolters and P.W.M. Glaudemans, Nucl. Phys. A476 (1988) 61 [7] F. Zijderhand, S.W. Kikstra, S.S. Hanna and C. van der Leun, Nucl. Phys. A462 (1987) 205 [8] S.S. Hanna and L. Meyer-Schutzmeister, Phys. Rev. 108 (1957) 1644 [9] P.B. Smith and P.M. Endt, Phys. Rev. 110 (1958) 397 [10] F. Zijderhand and C. van der Leun, Phys. Lett. B166 (1986) 4 [11] R.S. Segel, S.S. Hanna and R.G. Alias, Phys. Rev. 139 (1965) 818 [12] M. Suffert, W. Feldman, J. Mahieux, and S.S. Hanna, Nucl. Instr. Meth. 63 (1968) 1 [13] E.L. Bakkum and R.J. Elsenaar, Nucl. Instr. Meth. 227 (1984) 515 [14] S. Mellema and T.R. Wang, Internal Report, University of Wisconsin-Madison, unpublished [15] B. Bellenberg, H. Hemmert and E. Kuhlmann, Phys.Rev. C34 (1986) 1991 [16] W. Biesiot, thesis, Groningen University (1980) [17] F.C. Bergstrom, I.P. Auer, M. Ahmad, F.J. Kline, J.H. Hough, H.S. Caplan and J.L. Groh, Phys.Rev. C7 (1973) 2228 [18] H.G. Clerc, K.J. Wetzel and E. Spamer, Phys.Lett. 20 (1966) 667 [19] S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 [20] F.C. Barker, Nucl. Phys. 83 (1966) 418 [21] N. Kumar, Nucl. Phys. A225 (1974) 221 [22] A.G.M. van Hees, J.G.L. Booten and P.W.M. Glaudemans, to be published [23] P.J. Brussaard and P.W.M. Glaudemans, Shell model applications in nuclear spectroscopy (North Holland, Amsterdam, 1977) p. 1 [24] F.C. Barker, Australian National University, private communication References 29

[25] F.C. Barker and N. Kumar, Phys.Lett. B30 (1969) 103 [26] H. Theissen, in Springer tracts in modern physics, ed. G. Hohler, vol. 65 (Springer, Berlin, 1972) p. 1 [27] D. Schull, thesis, Technische Hochschule Darmstadt (1975) Chapter 3 Superallowed 42Sc(/?+)42Ca decay

Abstract The neutron separation energy of 42Ca and the proton separation energy of 42Sc have been measured accurately via the 41Ca(n,7)42Ca and 41Ca(p,7)42Sc reaction, respectively. The measured values imply a Q-value of 6425.92±0.19 keV for the total /?-decay energy of 42Sc. If the conserved vector current hypothesis is valid and the electromagnetic corrections are made properly, the ft values of superallowed 0+ —v 0+ /3-decays should all be the same. The ft values for the superallowed 42Sc and 140 decays were found to be equal to better than 13 ± 18 parts in 104, in agreement with the expectations.

3.1 Introduction 5uperallowed 0+ —* 0+ /?+ transitions between members of an isospin multiplet have been studied for more than three decades [1]. If isospin is a good quantum number, all the ft values are expected to be the same for these transitions [1-4]. The ft values, in turn, lead to a value for the effective weak interaction vector coupling constant G'v. The known superallowed ft values are based on accurate measurements of decay energies, half lives, and branchings (where applicable). An uncertainty of less than 0.1% in ft values is currently the goal of researchers in this field. This goal has been pursued, usually in a seesaw fashion, with significant improvements in the determination of decay energies followed by similar improvements in half-lives or vice versa. Of the 18 known superallowed 0+ -» 0+ transitions [1], the 26Alm(/?+)26Mg and 34Cl(/0+)34S cases are special in that their decay energies can be deduced accurately in a straightforward manner from combined 25Mg(n,7) and 25Mg(p,7) experiments and 33S(n,7) and 33S(p,7) measurements, respectively [5,6]. The

31 32 Superallowed 42Sc(/3+)42Ca decay corresponding targets in all remaining cases are unstable, but in the case of 42 42 41 Sc(/?+) Ca, the requisite Ca target is sufficiently long lived (t1/2 = 1.03 x 105 y) that a successful experiment can be done as is reported here. There are no known previous measurements of either the 41Ca(n,7)42Ca or the 41Ca(p,7)42Sc reaction.

3.2 The 41Ca(n,7)42Ca measurements 41 42 The Ca(nth,7) Ca reaction was studied at the Los Alamos Omega West Reac- 41 tor [6] with a 12 mg CaCO3 target enriched to 81.7% in Ca. The target material was produced by irradiation of enriched 40Ca at the Oak Ridge High-Flux Iso- tope Reactor followed by a mass separation. The 7-rays were detected with a 26 cm3 Ge(Li) detector with a Nal(Tl) annulus, which was operated in either a Compton-suppressed or pair-spectrometer mode. Even though the target was small, the thermal-neutron capture cross section (w 3 b) was sufficiently high that over 250 7-rays were identified in the E^ =0.1 to 10.0 MeV region. Fig. 3.1 shows a selected high-energy portion of the data. The energy calibration lines were mainly those observed in the 12C(n,7) and 14N(n,7) reactions. A set of calibration lines was developed, as described in ref. [6], on the basis of the neutron 2 13 separation energies Sn( H) = 2224.57 ±0.01 keV, 5n( C) = 4946.34 ±0.03 keV, 15 and Sn( N) = 10833.30 ± 0.03 keV quoted in the 1983 Atomic Mass Table [7]. This standard is known as the 'mass doublet standard'. 42 Table 3.1 shows ten strong cascades leading to Sn( Ca). Based on the entire 42 level scheme , a neutron separation energy of 5n( Ca) = 11480.66 ± 0.06 keV was deduced. The quoted uncertainty does not include the uncertainty in the primary calibration lines. Adding 30 eV algebraically to account for this, a final 42 value of Sn( Ca) = 11480.66 ± 0.09 keV for the neutron separation energy of 42Ca was obtained.

3.3 The 41Ca(p,7)42Sc measurements 3.3.1 Experimental The proton energy Ep and excitation energy Ex of a (p,7) resonance are related by

mT 2 EX = Q+ Ep , (3.1) 2 mT + mp 1 + ^1 + 2£pmT/(mT + mp) c* where mj and mp denote the masses of the target nuclide and proton, respectively. An accurate determination of Q (or equivalently the proton separation energy Sp) thus requires precision measurements of Ep and Ex. In the current The 41Ca(p,7)42Sc measurements 33

600 41Ca(n,7)42Ca --- thermal

400

200

6000 7000 8000 9000 (keV)

Figure 3.1: High-energy part of the Compton-suppressed 7-ray spectrum from the 41Ca(n,7)42Ca reaction.

42 Table 3.1: Selected cascades leading to the neutron separation energy (Sn) of Ca"

+ E, + £(73) £, Et = Sn 8727.42 H + 0.974 + 1227.65 3 + 0.019 + 1524.68 3 + 0.030 11480.77 15 8225.86 13 + 0.866 1729.19 5 + 0.038 + 1524.68 3 + 0.030 : 11480.66 14 8225.86 13 + 0.866 501.46 3 + 0.003 + 1227.65 3 + 0.019 + 1524.68 3 + 0.030 : 11480.57 14 8033.03 18 + 0.826 1922.20 7 + 0.047 + 1524.68 3 + 0.030 11480.81 20 8033.03 18 + 0.826 1022.77 3 + 0.013 + 899.42 3 + 0.010 + 1524.68 3 + 0.030 : 11480.78 19 7525.47 10+ 0.724 507.45 3 + 0.003 + 1922.20 7+0.047 + 1524.68 3 + 0.030 : 11480.60 IS 7480.07 18+ 0.716 + 2474.80 10 + 0.078 + 1524.68 3 + 0.030 11480.37 21 6790.05 17+0.590 + 3165.24 11 + 0.128 + 1524.68 3 + 0.030 11480.72 20 6720.46 18+ 0.578 + 2335.70 SO + 0.070 + 2424.17 6+ 0.075 11481.05 35 6462.79 17+ 0.534 + 1763.12 12 + 0.040 + 1729.19 5 + 0.038 + 1524.68 3 + 0.030 = 11480.42 22

Sn based on these 10 cascades 11480.65 7 S, based on all cascades [£(bonnd levels) < 5500 keV] 11480.66 6 Sn including uncertainty in the calibration energies 11480.66 9

"All energies are given in keV. The notation 8727.42 stands for 8727.42 ± 0.14 keV, etc. E, denotes the recoil energy. 34 Superallowed 42Sc(;3+)42Ca decay

experiment Ep was measured relative to a neighbouring and precisely known 40 Ca(p,7) resonance, and E% was deduced by summing the energies of cascade 7-rays. The (p,7) measurements were made with the Utrecht 3 MV Van de Graaff accelerator [8] with currents of 20-40 fiA and an energy spread of 200 eV at 1 MeV. Two closed-end hyperpure Ge ('7-X') detectors were used with active volumes of 90 and 95 cm3 and a resolution of 1.72 and 1.81 keV, respectively, at E-, =1.332 MeV. For selected measurements the 90 cm3 detector was placed in a Compton suppression shield (CSS) [9] reducing the background by a factor of 10-20 in the 1-10 MeV energy range. The distance between the target spot and the detectors was 50 mm. A 2.5 mm Pb shield was placed between the target and the detectors. The S6Co, n0Agm, and 108Agm calibration sources were positioned close (< 3 mm) to the target spot to enable simultaneous and unidirectional detection of the reaction and calibration lines. The recoil motion (with initial velocity VQ) of the nucleus formed in a (p,7) reaction causes a mean kinematic shift of the center of mass energy Ey of primary 7-rays detected at angle 0, which is in first order given by

(3.2) where F(Tm) is the Doppler-shift attenuation factor, which depends on the mean life rm of the resonance level and on the slowing-down process of the nucleus in the target. The mean kinematic shift is zero at 6 = 90°. To average out the effects of deviations of the actual detection angle from 9 = 90°, two detectors were used (see fig. 3.2). They were positioned (by optical and mechanical means) at 6+ « +90° and 0- ss -90° with a precision of a = 0+ - 0_ = 180.00 ± 0.10°. To eliminate the effect of the beam not hitting the target centre defined by the two detectors, results obtained at target angles = +45° and = +135° were averaged [10]. Only a limited amount of target material (see sect. 3.2) was available, therefore an electroplating procedure was used to prepare the targets. Five targets with thicknesses in the 10-100 /zg/cm2 range were made on 0.3 mm tantalum backings. In the plating procedure, a small volume (as 0.1 ml) of the metal nitrate solution was added to « 2 ml of isopropanol, and the metal was elec- troplated as oxide. Subsequent (p,7) yield measurements showed that all five targets had a thickness of 5-7 keV, which is thicker than planned probably due to the formation of tantalum hydride during electroplating. Besides the un- usual contaminants 17O (from 24Mg170 with mass 41) and 2SMg (from 25Mg160), troublesome amounts of 10B, 19F, and 23Na were also present in the target. The targets were therefore less than ideal for extensive (p,7) spectroscopy, but they 42 were adequate for the determination of Sp( Sc). The 41Ca(p,7)42Sc measurements 35

Target 7 X

Proton beam

Figure 3.2: Setup for the measurement of the excitation energy Ex of the Ep = 1848 keV 41Ca(p,7)42Sc resonance.

42 3.3.2 Determination of Sp( Sc) A yield curve was measured in the Ep = 0.9-2.3 MeV region in steps of 0.5- 2 keV but only the 1.7-2.1 MeV region produced useful results. Even here the data were of poor quality, but a resonance curve could be produced by setting narrow windows on known 42Sc 7-rays and subtracting the contents of equally wide background windows. Selected were the 611 keV 7-ray representing the 611-»0 keV transition in 42Sc, the 1227 and 1525 keV 7-rays following the j3- decay of the 616 keV, 7+ isomeric level to the 3189 keV level of 42Ca, and the 975 keV 7-ray representing the 1586—»611 keV transition in 42Sc. Spectra in the regions of interest were summed and studied in some detail to ascertain their identification as 41Ca(p,7)42Sc resonances. The resonances at Ep =1766, 1848, 1949, and 2030 keV were studied in greater detail (see sect. 3.3.3) and the 1848 keV resonance was selected for an accurate Q-value determination for the following reasons :

40 41 (i) the nearby Ep = 1843 keV Ca(p,7) Sc resonance serves as an internal calibration because the target also contains 18.2% 40Ca, (ii) the (p,7) strength is adequate, (iii) four-step cascades are present with 7-ray energies that can be bracketed by known standards, and 36 Superaltowed 42Sc(/3+)42Ca decay

2000 4 4! 42 a) Ca(p,7) 'Sc b) Ca(p,7) Sc H

1 (-10) f 1000 - I 5.73 :c 0.05 keV

( - \ i 0 u 1 , [ 1640 1845 1850

Figure 3.3: Gamma-ray yield curve showing reaction rates in (a) 40Ca(p,7)41Sc and (b) 41Ca(p,7)42Sc. The resonance energy is defined as the energy at which the yield, after sub- traction of the background, is at half the maximum value.

(iv) the 19F + p and 23Na + p resonances cause less interference at this energy than at the other resonances. The proton energy was determined by measuring the energy difference between the 1843 keV, 40Ca(p,7) and the 1848 keV, 41Ca(p,7) resonances. Gamma-ray yield curves were measured in the Ev = 1840-1850 keV range in steps of 0.1- 0.5 keV. A window set at a 7-ray energy of 2882 keV effectively represents the 40 Ca(p,7) reaction because the Ex = 2882 keV resonance is known to decay almost 100% to the 41Sc ground state [11]. The sum of the contents of windows set at 611,1227, and 1525 keV (see above) represents the 41Ca(p,7) reaction. The result of a typical scan is shown in figure 3.3. To check the effects introduced by dif- ferential hysteresis of the analyzing magnet, two sets of scans were performed: we started with zero magnet coil current for five minutes; increased the current such that the field corresponded to Ep = 1840 keV; measured the 1840 -» 1850 keV • • .**>•* «

The 41Ca(p,7)42Sc measurements 37

to 41 Ca(p .42Sc £ = 1848 keV -| y) p i CSS at fl = 55° CD 7 o 000 • "^ - CO 3

- W - •~ co < ir X c ) r^ • 01

• CO •

1000 - - < - ; ID • « i 1- i + o CO | to in s }

\ «JU1 ..1 L o r y 1 1 1.11 600 800 1000 Ey (keV) 41 42 Figure 3.4: Low-energy portion of the Ca(p,7) Sc spectrum measured at Ep = 1848 keV. The 7-rays are labelled with their energy in keV. The 879 and 894 keV 7-rays correspond to the 1.49 — 0.61 MeV and 1.51 — 0.62 MeV transitions in 42Sc, respectively; the assignment of the remaining 7-rays is shown in table 3.2.

range; turned the magnet up to the saturation value and left it there for five minutes; went down to 1850 keV; and measured Ep = 1850 -* 1840 keV. The resulting 1843-1848 keV differences are consistent within 20 eV. The average energy difference of the 40Ca(p,7)41Sc and 41Ca(p,7)42Sc resonances is found to 41 be AEP = 5.73 ± 0.05 keV. The energy of the Sc resonance has been determined previously as Ep = 1842.66 ± 0.14 keV [11]. This value is revised here as 1842.68 ± 0.11 keV on the basis of an improved value (2882.30 ± 0.06 keV) for the 2882 keV 7-ray energy which was determined as a by-product in the present 42 experiment. The energy of the Sc resonance thus is Ep = 1848.41 ± 0.12 keV. The next quantity to be determined (see eq. (3.1)) is the excitation energy Ex of the Ep = 1848 keV resonance level. A selected low-energy portion of the 7-ray spectrum from this resonance is shown in fig. 3.4. The 42Sc 7-rays are 38 Superallowed 42ScQ3+)42Ca decay

-I Ca(p,7) Sc a m ; 10a Agm 1 Ag 1j ^

i o 1 o o

600 620 640 660 Ey (keV) Figure 3.5: Energy calibration of 42Sc 7-rays relative to 108Agm and n0Agm calibration lines.

labelled by their energies. The assignment and measured energy of 7-rays used in the determination of Ex are given in table 3.2. Calibration 7-ray energies (based on the 'gold standard') were taken from the set recommended by Helmer et al. [12]. Fig. 3.5 illustrates the calibration of a few low-energy 42Sc 7-rays. The uncertainties quoted in table 3.2 take into account statistical uncertainties in peak positions, uncertainties in the standard energies, and uncertainties due to the Doppler effect, which in the case of the primary 7-rays, amount to about 60 eV. The excitation energy of the resonance is deduced to be Ex = 6076.41 ± 0.08 keV as shown in table 3.2. When inserted into eq. (3.1) this Ex value and the Ep value from the preceding paragraph yield Sp = 4272.40 ± 0.15 keV.

3.3.3 Levels of 42Sc Although the quality of the targets is not sufficiently good for (p,7) spectroscopy up to present-day standards [13], the spectra of the four resonances mentioned in sect. 3.3 yield so much new information on the energy levels of 42Sc that it is The 41Ca(p,7)42Sc measurements 39

41 Table 3.2: Excitation energy (Ex) of the Ep =1848 keV, Ca(p,7) resonance (ft)

deduced by summing four 7-ray energies (Ey) in a cascade, and the proton separation a energy (5P) of "Sc deduced from Ex and Ep h h Initial Final Ey Er Initial Final E-, Er state state state state ft — 2223 3853.057 75 0.190 ft — 2270 3807.137 75 0.185 2223— 1586 636.833 9 0.005 2270— 1586 682.808 23 0.006 1586— 611 975.245 25 0.012 1586— 611 975.245 25 0.012 611 — 0 611.046 6 0.005 611 — 0 611.046 6 0.005 Ex = 6076.393 80 Ex = 6076.444 83

Adopted : Ex = 6076.41 8 keV 1 Conclusion : Sp = 4272.40 15 keV

"All energies are given in keV; 3853.057 75 stands for 3853.057 ± 0.075 keV, etc. 'Recoil energy. 'Calculated following eq. (3.1), with Ep - 1848.41 12keV.

worthwhile to present them here as a by-product of the present investigation. The primary decay of the four resonances is presented in table 3.3. Besides the 7-ray branching ratios, it gives in the heading (a) the excitation energies of the resonance levels deduced by summation of the energy of cascad- ing 7-rays, (b) the corresponding proton energies calculated from Ex and 5P according to eq. (3.1), and (c) the resonance strengths Sp>7 = (2J + l)rpr7/r. The latter were found by comparison of the yields of the 41Ca(p,7)42Sc reso- 40 41 nance at Ep = 1848 keV with that of the Ca(p,7) Sc resonance at Ep = 1843 keV, of which the strength is known [11] as Spn = 280 ± 30 meV. The first column gives the excitation energies of the bound states of 42Sc, deduced primarily from spectra measured at 9 = 90°, in which the recommended 7-ray calibration energies from ref. [12] were complemented by secondary lines in 42Sc, deduced from the experiments at Ep = 1848 keV, described above. Our data are in good agreement with, but roughly an order of magnitude more precise than, the previous energies [14]. The levels at Ex = 3224 and 3321 keV have not been observed previously. The earlier suggested doublet character of the Ex = 2.22 MeV level [15] is confirmed. The second column lists the spins, parities, and isospins, based on the evidence from the current investigation (branchings, lifetimes) and previous data, mainly from refs. [14,16]. In the assignments the recommended upper limits (RUL's) [21] for decay 7-ray strengths play a crucial role; for details the reader is referred to table 3.5. The primary decay of the resonances was in this particular case not used to assign spins because weak contributions of nearby resonances cannot be excluded due to the relatively large target thick- 40 Superallowed 42Sc(/?+)42Ca decay

Table 3.3: Proton energies, excitation energies, strengths, and 7-decay (in %) of four 41Ca(p,7)42Sc resonances, and precise excitation energies and some lifetimes of 42Sc bound states'" b Exf (keV) J';T rm(fs) £p(keV): 1765.8 3 1848.41 12 1948.6 3 2029.8 3 5p,7(eV): 3.3 8 Q.I 2 5.5 14 \.b4 £*i(keV): 5995.8 3 6076.41 8 6174.2 3 6253.4 2 0 611.051 6 T4 616.28 6 7+!o 1490.43 4 3+;0 7 2 72 1510.10 6 5+;0 16 2 72 1586.31 2 2+;l 8 2 2187.54 5 (2,3)+;0 2 / 30 2 2223.15 y (2,3)+; 0 >300 62 29 2 « 1 2269.13 3 1,2+;0 >100 212 21 2 2389.06 5 3+;0 17 2 2433.33 8 (3-5)+;0 >200 3 1 15 2 2486.59 13 2+;l 3 1 2650.98 8 1,2 50 50 4 1 15 2 2795.3 3 (5-7) 3/ 2815.37 6 4+;l 50 20 3/ 92 3 1 2847.6 4 (1-3)+ 4 / 4 1 2910.4 4 (3-5)+ 4 1 2995.53 7 (3-5)+ >200 72 3022.80 15 4" ss 1 12 3089.1 3 5+ >200 3 1 31 4 / 3223.82 6 3+-5+ >300 13 2 12 3321.36 10 1+-3+ >200 10 2 3322.8 3 (3-5)+ <50 82 3668.7 3 4/ 3719.3 4 (5-7)+ >100 20 2 4047.72 6 2-4;l <20 62 4 / 4468.8 4 (2,3)+ 4 1 Unknown decay : 215 42 18 5 14 5

"The notation 5998.8 3 stands for 5998.8 i 0.3, etc. 'Arguments for J*\T assignments are given in table 3.5. cThe earlier suggested doublet character of this level [14,15] is confirmed by the present J* = (2,3)+ assignment, which is different from the J* - 1+ assignment [14] to the short-lived level which was not excited in the present experiment. The 42Sc->42Ca superallowed ft value 41 ness (see sect. 3.3.1). The third column gives (limits of) the lifetimes for some levels, deduced from the attenuated Dopplershifts deduced by comparison of the 8 = 55° and 90° spectra. Because the constitution of the target is not precisely known (sect.3.3.1), a calculation of the usual F(Tm) curve is not straightforward. The known [14] lifetimes of the 1.59 and 1.89 MeV levels were therefore used as calibration points. The new (or more precise) data for the bound states are summarized in table 3.4. This table presents especially the branching ratios (with uncertainties and upper limits for unobserved transitions) deduced from spectra measured with the CSS at 8 = 55°. It may be noted (by comparison of the data presented in tables 3.3 and 3.4) that all four resonances decay via 7-ray cascades to both the Ex = 0.61 MeV, T + + J = 1 and Ex = 0.62 MeV, J* = 7 levels, whereas none of them excites one of these levels directly.

3.4 The 42Sc->42Ca superallowed ft value 42 42 The measured Sp( Sc) and 5n( Ca) values, together with a value [7] of 782.338 ± 0.010 keV for the neutron-proton mass difference, imply a Q-value of 6425.92 ± 0.17 keV for the total (/?+ and electron capture) decay energy of 42Sc. It may be noted that our (n,7) measurements of the 7-ray energies are based mainly on the 'gold standard' [12] for E~, < 2.2 MeV and on the 'mass doublet standard' for En > 2.2 MeV. The (p,7) measurements rely exclusively on the 'gold standard'. The consistency of these scales has been discussed previously [18]. The 'gold standard' was found to be 9±6 ppm higher than the 'mass doublet standard'. This situation will improve, because in Cohen's most recent evaluation of the fundamental constants [19], the wavelength-energy conversion factor (which is essential in the 'gold standard') is 7.7 ppm lower than in the previous evaluation. In first order, this may shift the 'gold standard' energies down by about 7.7 ppm, which in the present case leads to a change in the final /3-decay energy of about 20 eV, an amount irrelevant for the following discussion. A definite correction of our (and other) energies for this effect, however, better waits until the influence of the updating of all the fundamental constants on the 7-ray energy standards has been carefully evaluated. Such an evaluation s underway [20]. Meanwhile, we have increased the uncertainty in the final ?-decay energy by 20 eV, thus leading to a final value of 6425.92 ± 0.19 keV. This value agrees with, but is more accurate than, a value 6423.5 ± 2.6 keV ref. [23], updated to incorporate a change in the reference 27Al(3He,t)27Si Q- alue following the 1984 mass adjustment [7]), but differs significantly from the Jue 6423.7 ± 0.4 keV reported in ref. [17]. Both these values are deduced om (3He,t) measurements. Recently, Q-value differences between four pairs of Table 3.4: Decay (in %) of 42Sc bound states"

E%i £«i: 0 0.61 0.62 1.49 1.51 1.59 1.85 1.87 1.89 2.19 2.22 Other b J";T 0+;l l+;0 7+;0 3+;0 5+;0 2+;l 3+;0 0+;l l+;0 (2,3)+;0 (2,3)+;0 levels 1.49 3+;0 <1 100 <1 1.51 5+;0 <4 <1 100 1.59 2+;l 8.7 10 91.3 10 <1 1.85 3+;0 X X 1.87 0+;l <8 100 <8 5 i 1.89 l + ;0 100 2.19 (2,3)+;0 <2 5 2 <2 <2 95 2 2.22C (2,3)+;0 <2 <2 <2 5 2 <2 95 2 2.27 l,2+;0 20 2 22 3 <1 <5 <3 52 5 6 3 2.39 3+;0 <4 <4 <3 12 4 88 4 <2 <2 2.43 (3-5)+;0 <2 <2 <4 52 3 48 3 <2 <2 <2 2.49 2+;l <10 77 5 <5 23 5 <5 <4 <5 <5 <4 2.59 X 2.65 1,2 <3 <3 <2 <3 <3 90 3 <3 <3 10 3 2.80 (5-7) 100 2.82 4+;l <1 <1 <1 34d 2 61 5 5 3 <4 <2 <2 <2 2.85 (1-3)+ X 2.91 (3-5)+ X 3.00 (3-5)+ <3 <8 <4 <3 <7 <5 <5 80 iO 2.39 : 20 10 3.02 4~ 100 3.09 5+ 100 3.22 3+-5+ <4 <4 <2 <6 27 3 <5 <5 <3 <8 32 3 2.39 : 41 3 3.321 1+-3+ <5 50 10 <4 50 10 <6 <8 <8 <4 <6 <6 3.323 (3-5)+ <5

"All energies are given in MeV. The notation 8.7 10 stands for 8.7 ± 1.0, etc. A x indicates an observed transition of which the branching ratio is unknown. 'Arguments for Jr;T assignments are given in table 3.5. w + O 'The earlier suggested doublet character of this level [14,15] is confirmed by the present J = (2,3) assignment, which is different from the J* = 1+ assignment [14] to the short-lived level which was not excited in the present experiment. ''Compare with ref. [15]. 'Unknown decay: 50 r The 42Sc->42Ca superallowed ft value 43

Table 3.5: Arguments for the J*\T assignments to 42Sc states listed in table 3.4

Ex J';T 0 1 (MeV) previous data' present experiment conclusion* 0 3 0.61 3 1+ e 0.62 3 7+;0 + ta 1.49 1+3 3+(2) 3 ;0 1.51 1+3 1.59 1+3 1.85 3+!o 1.87 3 (0-2) 1.89 3 1+ 2.19 1+3 (1-3)+ (2,3)+; 0 2.22 1+3 (2,3)+; 0 2.27 (3) 2.39 (1-3)+ /Uh5+;fl] 3+;0* 2.43 1 (3-5)+; 0" 2.49 3 (1,2)+ 2.65 1,2 2.80 (1+3) 2.82 2.85 (1-3)+ (1-3)+ 2.91 (3-5)+ (3-5)+ 3.00 (3-5)+ (3-5)+ 3.02 0 4- 3.09 1+3 5+ 3.22 3+-5+ 3.321 1+-3+ 3.323 1+3 (3-5)+ 3.72 3 (5-7)+* 4.05 2-4;l 4.47 (1-3)+ (2,3)+

"Ref. [16]. 'As reviewed in ref. [14]. eThe notation / 5+ [—» 2+; 1] etc., means that a J* = 5+ assignment would imply either a RUL-violating decay branch to a J*;T — 2+; 1 level or an M2 or higher multipolarity decay branch competing favourably with allowed E2 or dipole transitions. ''The T = 0 assignments to the lower states are based on the absence of analogue states in the 42Ca level scheme. 'Strictly speaking, the JT = 5+ and 6+ assignments, although unlikely[14], have not been excluded experimentally. The J* — 7+ assignment, however, is quite generally accepted in the literature because theoretical calculations produce a Jw — 7+ state and no JT= 5+ and 6+ states with a low excitation energy. 'The J(1.51) = 5 assignment (see next line) implies J(1.49) = 3; see ref. [22]. »In the unlikely case that the J'(061) = 7+ assignment would turn out to be incorrect, this assignment has to be reconsidered. *The 42Ca level scheme only has a JT — 0+ analogue state in a wide energy range. 44 Superallowed 42Sc(/?+)42Ca decay superallowed /J-emitters have been determined [24] from the measured energy differences between pairs of triton groups produced in two (3He,t) reactions. Two of these differences involve the 42Sc value, namely the 26Alm-42Sc and 42Sc- 54Co Q-value differences. These doublet data, when averaged with earlier values using a coupled least-squares algorithm, lead to the value 6425.03 ± 0.28 keV [ref. [25]], which is closer to, but not compatible with, the present value. Another way to obtain the 42Sc Q-value from previous data is to start from the most accurately measured Q-value [32] for a superallowed /J-emitter, namely 14O, and use two of the doublet measurements by Koslowsky et al. [24] to boot- strap up to 42Sc. Q(«Sc) = [Q(42Sc)-Q(26Alm) ] + [Q(26Alm)-Q(14O)] + [Q(14O)] = [2193.5 ±0.2] +[1401.68 ±0.13] +[2830.32 ± 0.08] keV = 6425.50 ± 0.25 keV, which leads to a value consistent with the present experiment. The ft value of a particular /3-transition depends upon the Q-value and the partial half-life. The relevant experimental data for 42Sc and a light-mass example known with good precision, 14O, are surveyed in table 3.6. In arriving at "adopted" values we consider two philosophies. The first is the statistical procedure advocated by the Particle Data Group [26] in which a weighted average is calculated according to 1/2 x ± Ji = (£ wml £ wi) ± QTWi )- , (3.3) where to, = [ l/(6x,)2 ] and the sums extended over all N relevant measurements. In each case \2 is calculated, and a scale factor 5 is determined. 2 /2 S = [x /(N-l)Y . (3.4) If S > 1 and the 6x{ are all about the same size, then the uncertainty in eq. (3.3) is increased by the factor S, which is equivalent to assuming that all the experimental errors are underestimated by the same scale factor. This method produces the set of adopted values designated as SET A in table 3.6. The scale factors used have been noted. The alternative procedure is not to average the data at all or to average only a subset of the data. The set of values designated as SET B in table 3.6 is obtained in three cases by selecting only the most accurately determined value and in the fourth case from a selective averaging of the data. In contrast to the SET A values, the uncertainties are generally smaller. The SET B values can be considered a more optimistic set or a set based on the (most recent) high-precision measurements. A careful treatment of the uncertainties is important because ultimately they will provide the limits by which the conserved vector current hypothesis (CVC) The 42Sc-»42Ca superallowed ft value 45

Table 3.6: Q-values and half-life measurements in 42Sc and superallowed ^-decays «Sc 14O° Q-value (keV) 1. Hardy et al. [23] 6423.50 ± 2.60 Butler and Bondelid [27] 2830.58 ± 0.46 Vonach et al. [17] 6423.70 ± 0.40 Bardin et al. [28] 2832.37 ± 0.56 Koslowsky et al. [24] 6425.50 ± 0.25* Roush.et al. [29] 2832.83 ± 0.48 Koslowsky et al. [24] 6425.30 ± 0.34e White and Naylor [30] 2830.78 ± 0.37 present work 6425.92 ± 0.19 Vonach et al. [17] 2829.91 ± 0.80 Barker and Nolen [31] 2832.39 ± 0.60 White et al. [32] 2830.32 ± 0.08 Adopted SET A Average* (S = 2.0) 6425.43 ± 0.26 Average (5 = 3.2) 2830.47 ± 0.24 SET B present work 6425.92 ± 0.19 Ref. [32] 2830.32 ± 0.08 Half-life (ms) Hardy and Alburger [33] 684.50 ± 120 Alburger [35] 70480 ± 150 Wilkinson et al. [34] 680.98 ± 0.62 Singh [36] 70320 ± 120 Clark et al. [37] 70588 ± 28 Azuelos et al. [38] 70430 ± 180 Becker et al. [39] 70684 ± 77 Wilkinson et al. [40] 70613 ± 25 Adopted SET A Average (5 = 2.6) 681.72 ± 1.44 Average (S = 1.5) 70597 ± 27 SET B Ref. [34] 680.98 ± 0.62 Avg: refs. [37,39,40] 70606 ± 18

14 "For O, the Q-value for the superallowed branch was obtained using with Ex = 2312.798± 0.011 keV as the excitation energy of the lowest J*;T = 0+\l state of 14N (ref. [41]). 'From doublet measurement Q(42Sc)-Q(26Alm) = 2193.5 ± 0.2 keV and Q(26Alm) = 4232.0 ± 0.2 keV; see sect. 3.4 of the text. cFrom doublet measurement Q(54Co)-Q(42Sc) = 1817.2 ± 0.2 keV and Q(54Co) = 8242.5 ± 0.3 keV. dFrom a coupled least squares algorithm to handle doublet data as discussed in ref. [25].

is verified by the superallowed 0+ -> 0+ ^-transitions. According to CVC, the ft values for such transitions should all be identical after small radiative and Coulomb corrections have been applied. Radiative corrections have been expressed as a sum of two terms [53]: the outer radiative correction, 6R, which varies from nucleus to nucleus, and the inner radiative correction, AR, which is nucleus independent and which can be absorbed in the definition of the coupling constant. For a verification of CVC, only 6R need be considered. Significant progress has been achieved recently [54- 56] in the analysis of these corrections. The order O(a) and 0(Za2) corrections can be calculated almost without approximation, and the leading term of the order O(Z2a3) is estimated. The size of the O(Z2a3) contribution is taken as 46 Superallowed 42Sc(j9+)42Ca decay the uncertainty in the summed correction, 6R. Here Z is the charge number of the daughter nucleus in the /?-decay and a is the fine-structure constant. The recent improvement is the re-evaluation of the order Za2 correction, which differs significantly from earlier estimates [57] and which, as a consequence, has brought the highly accurate data from superallowed /9-decay into closer agreement with the predictions of CVC. The Coulomb correction, 6C, is very much nuclear-model dependent. It is a correction that tries to estimate the breakdown of isospin symmetry due to the presence of Coulomb and other charge-dependent forces in the nucleus. It is generally calculated as a sum of two pieces [42]: one, 6cl, arising from a + charge-dependent configuration mixing with other 0 states, and the other, 6c2, being due to the small differences in the single-particle neutron and proton radial wavefunctions which cause the radial overlap integral of the parent and daughter nucleus to be less than unity. Generally 6c2 > Sci, so we will discuss Sc2 first. The correction, 6c2, has been evaluated by Wilkinson [58] and Towner, Hardy, and Harvey [59] (THH), and the two calculations are generally in agreement. The correction is of order 0.5%. The idea is that the proton in the parent nucleus undergoing /?-decay and the neutron in the daughter nucleus can each be described by a one-body Schrodinger equation in which the nucleon is assumed to be moving in an average potential due to all the other nucleons. This average potential is modelled by a Saxon-Woods form with the depth of the potential adjusted so that the eigenvalue matches the experimentally known separation energy obtained from mass differences. The correction, 6c2, is the departure from unity of the radial overlap of single-particle proton and neutron Saxon- Woods wave functions. By making reasonable variations in the parameters of the potential, an estimate of the uncertainty in the 6c2 value is obtained. A theoretically more satisfying approach is to obtain the average central potential from a self-consistent Hartree-Fock calculation. This has been attempted by Ormand and Brown [60,61] (OB). They use a Skyrme interaction for the bare nucleon-nucleon interaction which has been adjusted to obtain good results for binding energies and radii of closed-shell nuclei in Hartree-Fock calculations. Five different variants of the Skyrme interaction are used to obtain an estimate of the uncertainty in the 6c2 calculation. The values of 8c2 obtained from the Hartree-Fock calculation of OB are systematically smaller by about 30% than those obtained in the Saxon-Woods calculation of THH. The reduction is due to effects arising from the different forms of the Coulomb and isospin-dependent nuclear potentials used in the two calculations. Our strategy is to take a weighted average of the two calculations, using the stated uncertainties as the weighing factors.

The other smaller contribution, 6ci, coming from isospin mixing can be esti- 42 42 The Sc— Ca superallowed ft value 47

Table 3.7: The /t values for 42Sc and 14O superallowed /J-decays 42Sc (SET A) 14O (SET A) 42Sc (SET B) 14O (SET B) Q(/?+ + i) (keV) 6425.43 ±0.26 2830.47 ±0.24 6425.92 ±0.19 2830.32 ±0.08 4468.10 ±1.00 42.701 ±0.024 4469.98 ±0.73 42.686 ±0.008 Half life (ms) 681.72 ±1.44 70597 ±27 680.98 ±0.62 70606 ±18 br » (%) 99.9932 ±0.0010 99.336 ±0.010 99.9932 ±0.0010 99.336 ±0.010

"Statistical rate function, calculated with the computer code described in ref. [42]; includes a recoil correction. 'Branching ratio, from refs. [43-47] for 14O, and refs. [48-51] for 42Sc. ^Partial half-life, includes a correction for the electron capture fraction from tables of ref. [52]. ''Outer radiative correction, from ref. [56]. 'Coulomb correction.

mated in a shell-model calculation. Towner and Hardy [62] (TH) have recently repeated their 1973 calculation [42] following the suggestion of Ormand and Brown [60,61] that the parameters of the empirical nuclear charge-dependent interaction be adjusted to reproduce the experimental b- and c-coefficients of the isobaric-mass-multiplet equation. The results of these calculations are generally small, Scl < 0.1%, and we have simply averaged the TH and OB calculated values.

The summed correction 6C = 6cl + 6c2, is listed in table 3.7. An overall model uncertainty of ±0.10% for 6C has been assigned. Because of the model dependence in these calculations and the discrepancy between the OB and THH results, it will be difficult to justify reducing the error in 6C below this level. Ultimately, this uncertainty will limit the test of CVC available from data on superallowed /3-decay. The corrected ft values for superallowed decays are denoted by Ft and the statement of CVC becomes Ft = constant, where

ft{\ - 6C) = fRt(l - 6C) = Ft. (3-5) Thus, from the entries in table 3.7, the 42Sc and 14O data are seen to be in good accord and to support the CVC hypothesis. The result, expressed as a ratio, is 48 References

R = [ :R(42Sc) - ;R(140) ] = (18 ± 27) x 10-4 (SETA) (3.6) = (13 ± 18) x 10-4 (SET B), where the bulk of the experimental uncertainty comes from the 42Sc lifetime. A re-measurement of this datum has just been completed at Chalk River [63], and the analysis is in progress. It is expected that an uncertainty of ± 0.3 ms will be achieved in the measurement. In the 14O data, the largest contributor to the uncertainty in the Tt value is the theoretical uncertainty in the 6C calculation. Ultimately, a more demanding test of CVC is a detailed comparison of all eight accurately measured superallowed /3-emitters ( 14O, 26Alm, 34C1, 38K, 42Sc, 46V, 50Mn and 54Co ). A survey of the experimental data is in progress [62] and will be available shortly.

Acknowledgements We thank E. Ormand and B.A. Brown for communicating their calculations of Coulomb corrections prior to publication. We acknowledge helpful discussions with J.C. Hardy, V.T. Koslowsky, and A.H. Wapstra. The targets were made by Micha Petek and we thank her.

References [1] S. Raman, T.A. Walkiewicz, and H. Behrens, At. Data Nucl. Data Tables 18 (1975) 451 [2] J.C. Hardy and I.S. Towner, Nucl. Phys. A254 (1975) 221 [3] D.H. Wilkinson, Nature 257 (1975) 189 [4] L. Szybisz and V.W. Silbergleit, J. Phys. G7 (1981) L201 [5] P. de Wit and C. van der Leun, Phys. Lett. 30B (1969) 639 [6] S. Raman, E.T. Jurney, D.A. Outlaw, and I.S. Towner, Phys. Rev. C 27 (1983) 1188 [7] A.H. Wapstra and G. Audi, Nucl. Phys. A432 (1985) 1 and private communication [8] J.W. Maas, E. Somoray, H.D. Graber, C.A. van den Wijngaart, C. van der Leun and P.M. Endt, Nucl. Phys. A301 (1978) 213 and G.J. Nooren and C van der Leun, Nucl. Phys. A423 (1984) 197 [9] H.J.M. Aarts, C.J. van der Poel, D.E.C. Scherpenzeel, H.F.R. Arciszewski and G.A.P. Engelbertink, Nucl. Instr. Meth. 177 (1980) 417 [10] P.F.A. Alkemade, C. Alderliesten and C. van der Leun, Nucl. Instr. Meth. 197 (1982) 383 [11] F. Zijderhand and C. van der Leun, Nucl. Phys. A466 (1987) 280 [12] R.G. Helmer , P.H.M. Van Assche, and C. van der Leun, At. Data Nucl. Data Tables 24(1979)39 References 49

[13] P.M. Endt, P. de Wit and C. Alderliesten, Nud. Phys. A459 (1986) 61 and A476 (1988)333 [14] P.M. Endt and C. van der Leun., Nucl. Phys. A310 (1978) 1 [15] N.R. Roberson and G. van Middelkoop, Nucl. Phys. A176 (1971) 577 [16] P.B. Void, D. Cline, M.J.A. de Voigt, Ole Hansen and 0. Nathan, Nucl. Phys. A321 (1979) 109 [17] H. Vonach, P. Glässel, E. Huenges, P. Maier-Komor, H. Rosier, H.J. Scheerer , H. Paul, and D. Semrad, Nucl. Phys. A278 (1977) 189 [18] C. van der Leun, P. de Wit and F. Stecher-Rasmussen, Inst. Phys. Conf. Ser.No.62 (1981) p.548 [19] E.N. Cohen and B.N. Taylor, CODATA Bulletin No. 63 (Pergamon, NY, 1986) [20] R.G. Helmer , P.H.M. Van Assche, and C. van der Leun, to be published [21] P.M. Endt, At. Data Nucl. Data Tables 23 (1979) 3 [22] F.M. Nicholas, N. Lawley, I.G. Main and P.J. Twin, Nucl. Phys. A124 (1969) 97 [23] J.C. Hardy, G.C. Ball, J.S. Geiger, R.L. Graham, J.A. Macdonald and H. Schmeing, Phys. Rev. Lett. 33 (1974) 320 [24] V.T. Koslowsky, J.C. Hardy, E. Hagberg, R.E. Azuma, G.C. Ball, E.T.H. Clifford, W.G. Davies, H. Schmeing, U.J. Schrewe and K.S. Sharma, Nucl. Phys. A472 (1987)419 [25] V.T. Koslowsky, E. Hagberg, J.C. Hardy, H. Schmeing, R.E. Azuma and I.S. Towner, in Proc. 7th Int. Conf. on Atomic Masses and Fundamental Constants, ed. O. Klepper (Gesellschaft für Schwerionenforschung, Darmstadt, 1984) p.572 [26] Particle Data Group, Phys. Lett. 170B (1986) 1 [27] J.W. Butler and R.O. Bondelid, Phys. Rev. 121 (1961) 1770 [28] R.K. Bardin, C.A. Barnes, W.A. Fowler and P.A. Seeger, Phys. Rev. 127 (1962) 583 [29] M.L. Roush, L.A. West and J.B. Marion, Nucl. Phys. A147 (1970) 235 [30] R.E. White and H. Naylor, Nucl. Phys. A276 (1977) 333 [31] P.H. Barker and J.A. Nolen, in Proc. of Int.Conf. on Nuclear Structure, Tokyo (1977) p.155 [32] R.E. White, H. Naylor, P.H. Barker, D.M.J. Lovelock and R.M. Smythe, Phys. Lett. 105B (1981) 116 [33] J.C. Hardy and D.E. Alburger, Phys. Lett. 42B (1972) 341 [34] D.H. Wilkinson and D.E. Alburger, Phys. Rev. C 13 (1976) 2517 [35] D.E. Alburger, Phys. Rev. C 5 (1972) 274 [36] J. Singh, Indian J. Pure Appl. Phys. 10 (1972) 289 [37] G.J. Clark, J.M. Freeman, D.C. Robinson, J.S. Ryder, W.E. Burchatn and G.T.A. Squier, Nucl. Phys. A215 (1973) 429 [38] G. Azuelos, J.E. Crawford and J.E. Kitching, Phys. Rev. C 9 (1974) 1213 [39] J.A. Becker, R.A. Chalmers, B.A. Watson and D.H. Wilkinson, Nucl. Instr. and Methods 155 (1978) 211 [40] D.H. Wilkinson, A. Gallman and D.E. Alburger, Phys. Rev. C 18 (1978) 401 [41] E.K. Warburton and D.E. Alburger, Nucl. Phys. A385 (1982) 189 50 References

[42] I.S. Towner and J.C. Hardy, Nucl. Phys. A2O5 (1973) 33 [43] G.S. Sidhu and J.B. Gerhart, Phys. Rev. 148 (1966) 1024 [44] R.W. Kavanagh, Nucl. Phys. A129 (1969) 172 [45] G.J. Clark, J.M. Freeman, D.C. Robinson, J.S. Ryder, W.E. Burcham and G.T.A. Squier, Phys. Lett. 35B (1971) 503 [46] H.S. Wilson, R.W. Kavanagh and F.M. Mann, Phys. Rev. C 22 (1980) 1696 [47] A.M. Hernandez and W.W. Daehnick, Phys. Rev. C 24 (1981) 2235 [48] P.D. Ingalls, J.C. Overley and H.S. Wilson, Nucl. Phys. A293 (1977) 117 [49] R.M. DelVecchio and W.W. Daehnick, Phys. Rev. C 17 (1978) 1809 [50] A.M. Sandorfi, C.J. Lister, D.E. Alburger and E.K. Warburton, Phys. Rev. C 22 (1980) 2213 [51] W.W. Daehnick and R.D. Rosa, Phys. Rev. C 31 (1985) 1499 [52] N.B. Gove and M.J. Martin, Nucl. Data A10 (1971) 205 [53] A. Sirlin, Phys. Rev. 164 (1967) 1767 [54] A. Sirlin and R. Zucchini, Phys. Rev. Lett. 57 (1986) 1994 [55] W. Jaus and G. Rasche, Phys. Rev. D 35 (1987) 3420 [56] A. Sirlin, Phys. Rev. D 35 (1987) 3423 [57] W. Jaus and G. Rasche, Nucl. Phys. A143 (1970) 202; W. Jaus, Phys. Lett. 40 (1972) 616 [58] D.H. Wilkinson, Phys. Lett. 65B (1976) 9 [59] I.S. Towner, J.C. Hardy and M. Harvey, Nucl. Phys. A284 (1977) 269 [60] W.E. Ormand and B.A. Brown, Nucl. Phys. A440 (1985) 274 [61] W.E. Ormand and B.A. Brown, preprint 1988, and private communication [62] I.S. Towner and J.C. Hardy, to be published [63] V.T. Koslowsky, E. Hagberg, J.C. Hardy, H. Schmeing and X.J. Sun, to be published Chapter 4 Investigation of the 40Ca level scheme

Abstract The 40Ca level scheme has been investigated via the 39K(p,7)40Ca reaction. A thin-target yield curve has been measured in the Ep — 0.3-2.9 MeV energy region in which over a hundred resonances have been observed. Many of these resonances had not been observed previously. The 7-decay of 25 resonances has been studied in detail and yields for bound states (including the new states at Ex = 6.931 and 8.68 MeV) and resonances (i) accurate excitation energies, (ii) branching ratios and upper limits for unobserved transitions, (iii) J*;T assignments and, as a by-product, (iv) lifetimes from the observed Doppler shifts. The reaction Q-value has been determined as Q — 8328.28 ± 0.09 keV. Shell model calculations have been performed to describe the non-normal parity states of nuclei in the A = 37-41 region in a 16O(ls,0d)A-16~n(0f,lp)n model space. Special attention has been paid to aspects of the cross-shell interaction.

4.1 Introduction The nuclide 40Ca has been the subject of a large number of experimental and theoretical investigations. Since the 1978 A = 21-44 evaluation [1] over 400 theoretical articles have been published on this nuclide, which indicates that 40Ca is of great theoretical interest. The many experiments resulted in a sizeable amount of data, but the precision and completeness of the 40Ca level scheme is not up to present-day standards. This is especially due to the fact that the Nal and small Ge(Li) detectors formerly used in the 7-decay measurements, which are one of the most prolific sources of precise and complete information,

51 52 Investigation of the 40Ca level scheme are vastly inferior to the presently available high-resolution, large-volume Ge detectors. The first comprehensive (p,7) measurements leading to 40Ca were performed by Leenhouts and Endt [2,3] who determined energies and strengths of 53 resonances in the Ep = 0.5-2.9 MeV region as well as decay modes and spins and parities of many levels. Subsequent (p,7) work by several other authors [4-6] concentrated on a few selected levels or properties. The incentives for the present work thus were, besides the theoretical importance of the doubly-magic nuclide 40Ca, (i) the possibility to improve and extend the available data by the use of modern detector systems such as the Utrecht Compton Suppression Spectrom- eter (CSS, sect. 4.2), and (ii) the need for 'overlapping' data to check the consistency of more specialized, previous experiments. In the course of the present investigation the work of Schoeman et al. [7] was published, in which a bare 3 125 cm Ge(Li) detector was used to study 14 resonances in the Ep = 1.06-1.58 MeV region, with emphasis on the disentanglement of the decay of close-lying resonances. Of the resonances studied by these authors, six are also investigated in the present experiment. In the present work a thin-target yield curve has been measured for the 39 40 K(p,7) Ca reaction in the Ep = 0.3-2.9 MeV region, which is discussed in section 4.3. These measurements form the basis for 7-decay measurements at 25 selected resonances in the Ep — 0.62-2.75 MeV region which yield precise resonance and bound state energies, 7-ray branchings, (restrictions on) spins, parities and isospins for several states, and, as a by-product, lifetimes of some levels from the observed Doppler-shifts. These results are discussed in sect. 4.4. The Q-value of the 39K(p,7)40Ca reaction, or equivalently the 40Ca proton binding energy, was obtained via precision measurements of the excitation energies (from the 7-decay) and proton energies (relative to five selected neighbouring 27Al + p calibration resonances). In sect. 4.5 shell-model calculations are discussed, based on (a variant of) an interaction devised by Warburton et al. [8], which cover the A = 37-41 region in a 16O(ls,0d)A-16-n(0f,lp)n model space.

4.2 Experimental In the yield curve and 7-decay measurements, two hyperpure n-type Ge detectors (7-X) and one Ge(Li) detector were used with volumes of 90, 95 and 126 cm3 and an energy resolution of 1.7, 1.8 and 2.0 keV at EL, = 1.33 MeV, respectively. In several measurements one of the 7-X detectors was placed in a Compton suppression shield [9]. As target material KI (39K > 98%) has been used. Although this material is highly hygroscopic and somewhat less stable than the alternatives, it allows for V

The ^K+p resonances for Ep = 0.3-2.0 MeV 53 making thin and homogeneous layers. By storing the KI targets under Ar gas the stability was enhanced considerably. Targets of KC1 and K2SO4 have been tested, but they were finally not used because of the Cl(p,7) background and the inhomogeneous structure, respectively. The KI was evaporated onto 0.3 mm Ta backings, which (before use) were heated to incandescence in order to reduce the Na content. In the experiment the backings are directly cooled with turbulent water directed towards the rear face of the backing [10]. The targets were first exposed to a weak beam (1 //A) thus heating the target [11] in order to disso- ciate the KI into K and I. After about half an hour the current was increased, to a maximum of 30 (iA on target. With a beamspot of 2.5 x 2.5 mm2 these targets withstand such a current for typically 24 hours, without deteriorating more than 20%. Protons were accelerated with the Utrecht 3 MV Van de Graaff generator, which is equipped with a 90° analyzing magnet and a corona feed- back stabilization system, and has an energy spread of « 200 eV at Ep = 1.0 MeV [10,12]. The data-taking and on-line analysis were automatized on the basis of the ANALYS-system [13].

39 4.3 The K+p resonances for Ep = 0.3-2.9 MeV 4.3.1 Measurements The yield of the 39K(p,7)40Ca reaction was measured with the two intrinsic Ge detectors placed at 6 = ±55° with respect to the beam direction. The tar- 2 get thickness was 30-50 /ig/cm (w 3 keV at Ep = 1.0 MeV) as a compromise between the requirements of energy resolution and acceptable target stability. The yield curve was measured in steps of 0.5-1.5 keV in the regions in between resonance peaks, decreasing to 0.15-0.4 keV steps at the peaks. At each measuring point a charge of 1 mC protons was collected (2 mC below Ep = 630 keV) and an 8192-channel spectrum was taken and stored. The measurement was performed downward from Ep = 2050 to 300 keV, which has the advantage of measuring the more slowly rising high-energy edge of the resonance peaks first, which makes it possible to adapt the step size in due time for the measurement of the steep low-energy edge. Subsequently, the Ep = 2050-2900 keV region was covered in upward direction in order to condition the accelerator during the measurements. Strong resonances were remeasured regularly in order to check the target condition; a target was replaced by a fresh one when the yield was down by 15-20%. For the complete yield curve measurement, which took about 120 hours, six targets were used and twenty test measurements were performed.

4.3.2 Analysis The yield curve data were used to extract the (p,7) resonance strengths [14] 54 Investigation of the 40Ca level scheme

'•*• o — fj/xiir r /r

and relative resonance proton energies. Both parameters were deduced from fits of an empirical line-shape function to the resonance peaks, as explained below. The calibration of the relative proton energies was accomplished by a precision measurement of one proton energy in each yield-curve segment (measured with the analyzing magnet always adjusted in the same direction), relative to a well- known calibration resonance.

Windows Many yield curves have been constructed off-line by setting narrow windows on 40Ca 7-ray peaks in the spectra. The windows that were finally used as representation of the (p,7) yield were chosen as a compromise between selectivity (narrow windows to reduce the yield from contaminants) and statistics (windows as broad as the peaks). Two examples of the yield curves are shown in fig. 4.1.

The upper curve a) is the sum of the background-corrected yields in narrow windows containing the 7-ray peaks from the r—»0, r—»3.74 and r-*3.90 MeV transitions (in these cases the window borders slide with the proton energy), and the 3.90—>0 and 3.74—>0 MeV transitions. The lower curve b) shows the yield in the 2.2-12.0 MeV window, with the exclusion of regions around the peaks of the 6.13, 6.92 and 7.12 MeV 7-rays arising from the (p,e*7) reaction on the ever present contaminant 19F. The lower limit was chosen to exclude the 39 36 41 38 1.97 and 2.17 MeV 7's from the K(p,a17) Ar and K(p,e*17) Ar reactions, respectively. The combination of the curves a) and b) provides selectivity and (where needed) statistics.

Resonance peaks A five-parameter empirical line-shape function was fitted to the peaks in both curves to extract the areas and the resonance positions. The (p,7) resonance shape is a convolution of three functions describing the energy distribution of the protons in the beam, their energy loss in the target and the Breit-Wigner shape of the resonance itself. The difference between the resonance peak position and the actual resonance proton energy Ep depends on the relative magnitude of the widths of the three functions in this convolution (i.e. the beam spread, the target thickness and the natural resonance width F). The fitting program used allows for simultaneous fits of an arbitrary number of (single peak) line shapes. Thus multiplets, even when only partly resolved, can be handled reliably by keeping the beam-spread and target-thickness parameters fixed for all component peaks. This method is especially advantageous in simultaneously fitting combinations 200 .-y) Ca

100 V

-H 1 1 1

b) 2000 r

1000 I

400 600 800 1000 1200 (keV)

Figure 4.1: Relative yield of the 39K(p,7)40Ca reaction as measured in two windows : a) The sum of narrow windows containing (i) the primary 7-rays to the ground state and the states at 3.74 and 3.90 MeV and (ii) the 3.74—»0 and 3.90—>0 MeV 7-rays. b) The 2.2-12.0 MeV energy region, with the exclusion of narrow regions around the 6.13, 6.92 and 7.12 MeV 7-rays from the 19F(p,o7)l60 reaction and their single and double escape peaks. The lot accumulated charge per point amounts to 1 mC. en i

56 Investigation of the 40Ca level scheme

a no 3 tr 39 4 n 600 K(p,7) °Ca

400

200 ag 0

8000 IP 6000 OS

4000 i,

2000 1

2200 2400 2600 2800 (keV)

Figure 4.1 (continued) • I .AVr* t-w V

58 Investigation of the 40Ca level scheme

of strong and weak peaks. An example of a multiplet fit is given in fig. 4.2. A more detailed description of the line shape function is given in ref. [15].

Yield cattbration The yield curves for the segments were interconnected after correction for target deterioration by means of the target-check measurements. These normalization corrections (accumulated) never exceeded 25%. The peak areas were converted into relative (p,7) resonance strengths according to the relation [14] :

where Sp,7, A, Y and Ep are the resonance strength, the proton wavelength, the measured area of the resonance peak (in units of proton energy) and the proton energy, respectively. The (p,7) strengths have been normalized on the strength Spr/ = 14.3 ± 1.0 eV (from an evaluation of all previous measurements in ref. [16]) of the Ep = 2042 keV resonance. The resulting average uncertainty in the strengths amounts to about 40%; an estimated 25% uncertainty arises from the use of yields integrated over broad windows; an accurate correction for the detection efficiency is not possible for the resonances of which the decay modes are not known in detail.

Energy calibration In order to obtain at least one calibration point per yield-curve segment, a multi-step procedure was used. In separate measurements the proton energies of five 39K(p,7) resonances were determined relative to precisely known 27Al(p,7) resonances [12]. A combined KI-A1 target was used (see fig. 4.3) and the yields from both reactions were measured simultaneously. In this way setting changes are circumvented, which are necessary when two different targets are used for the measurement of 27Al(p,7) and 39K(p,7) resonances. Two measurements were performed, one with increasing and the other with decreasing energy; the results were consistent within 30 eV. The exact relation between the proton energy Ep and excitation energy Ex of a (p,7) resonance is given by

(4.2) 2-EpmT/(mT where mi and mp denote the masses of the target nuclide and proton, respectively. As shown in table 4.1, the five proton energies, together with the excitation energies of the corresponding resonances obtained from the 7-decay measure- M The K+p resonances for Ep = 0.3-2.9 MeV 59

39K(p. )40Ca 400 7

200

I—I—I —I—I—I—I——I

6000

4000

2000

2500 2510 2520 2530 Ep (keV)

Figure 4.2: Example of a multiplet fit to a part of the yield curve in windows a) and b) (see the caption of fig.4.1) . The separate lineshape components of the resonance multiplet have identical beamspread and target thickness parameters. 60 Investigation of the 40Ca level scheme

Target spot

=-~— -^ 0.5 mm gap

Figure 4.3: Combined KI-A1 target for the simultaneous measurement of proton energies of 39K(p,7) resonances relative to neighbouring 27Al(p,7) resonances.

ments described in sect. 4.4, lead to an accurate value of the proton separation energy in 40Ca of Q = 8328.28 ± 0.09 keV. The uncertainty in this value is composed of a 70 eV statistical uncertainty, a 50 eV uncertainty due to the influence of a possible oxide layer on the Al targets [17], a 30 eV uncertainty from the calibration proton energy and the 2.6 ppm (= 22 eV) uncertainty related to the 'gold standard' calibration energies [18]. Our value deviates from the literature value of 8329.3 ± 0.4 keV [19] on the 3

4.3.3 Results and discussion In table 4.2 the proton energies, excitation energies and resonance strengths are listed and compared to previous results. Of the 112 resonances listed in this table, 36 have not been reported previously as (p,7) resonances and 26 of these correspond to 40Ca excited states that have not been observed at all previously. It should be stated that the identification as 39K(p,7) resonances of 30 out of the above 36 new resonances is based on the resonant behaviour observed in the windows discussed in sect. 4.3.2; for the other new resonances this assignment is confirmed by detailed measurements of the 7-ray spectra. As indicated in 39 The K+p resonances for Ep = 0.3-2.9 MeV 61

Table 4.1: The Q-value of the 39K(p,7)40Ca reaction, deduced from the mea- 39 sured proton energies Ep and excitation energies Ex of five K(p,7) calibration resonances" 27 6 c 39 d 39 e £P [ Al(p,7)] A£p EP [ K(p,7)] £x («Ca) Q [ K(p,7)] 773.62 4 + 9.61 ;; 783.23 12 9091.70 6 8328.22 IS 1171.82 6 -16.97 15 1154.85 16 9453.95 5 8328.22 16 1502.09 8 -13.56 16 1488.53 18 9779.49 7 8328.50 19 +74.65 16 1576.74 18 9865.15 11 8328.18 21 1799.71 9 -34.13 10 1765.58 14 10049.38 7 8328.33 15 0(x? = o.48), : 8328.28 7 Adopted-' 8328.28 9

"All energies are given in keV. The notation 773.62 4 stands for 773.62 ± 0.04, etc. "Calibration energies from ref. [12], shifted to take into account the change in the calibration energy used in ref. [12] from Ep =991.88 4 keV to Ep = 991.86 3 keV (average of 991.843 33 keV [351 and 991.91 4 keV [36]). 9 i7 'Measured Ep\? K(p,y)] - £p[ Al(p,7)], see text. ''Deduced from the measured energies of the decay 7-rays (see sect.4.4). 'Calculated from Ep (column 3) and Ex with eq. (4.2). -'The additional uncertainties are discussed in the text.

table 4.2, 25 excitation energies were derived directly from 7-ray energy measurements (see sect. 4.4); the others were calculated from the Q-value and the proton energy with eq. (4.2). In general the presently obtained excitation energies are about an order of magnitude more precise than previous results. The agreement between the present energies and the literature data is good, with the exception of the values above Ep «s 2.4 MeV, where the literature energies are systematically high. In spite of the large uncertainties the agreement of the resonance strengths with the data obtained by Cheng et al. is rather poor, especially for the resonances above Ep = 2.0 MeV. A possible cause may be the large systematic uncertainties associated with the non-standard thick target method employed by Cheng et al. [16]. The Ep a 1130, 1307 and 1577 keV doublets have been investigated recently by Schoeman et al. [7]. The new doublets at Ep as 922 and 1763 keV are discussed in sect. 4.4.3. 62 Investigation of the 40Ca level scheme

Table 4.2: Proton energies, excitation energies and strengths of 39K(p,7)40Ca resonances

(keV) E% (keV) c (eV) Present Literature" Present Literature* Present Ref. [16] 622.2C 12 c 8934.81 7 0.09 4 683.45 15 c 8994.50 11 8995.0 10 > 0.15 6 783.23 12 d 783.2 10 9091.70 6 9092.9 8 k 0.28 11 0.11 3 828.27 11 c 828.1 10 9135.66 5 9136.5 11 0.6 2 0.32 10 904.29 11c 9209.77 5 9209.9 12 ' 0.39 16 921.66 11 c 9226.69 5 9227.5 12 ' "I 0.28 1 1 922.42 12 c 9227.43 7 11 1061.01 11 c 1060.4 51 9362.54 6 9362.9 2 0.43 17 0.15 5 1076.70 18 1075.0 10 9377.8 S 9377.2 // 0.24 10 «0.05 1087.35 /7 1086.4 5' 9388.20 19 9388.3 2 0.26 10 «0.1 1095.0 3 1097.6 10 9395.7 3 9399.2 11 0.09 4 SBO.1 1104.43 77 1102.5 5' 9404.85 19 9404.0 2 0.36 U 1106.0 6 9406.4 6 0.40 10 1112.2 2 9412.4 2 0.18 7 1118.76 i« 1117.1 5' 9418.8 2 9418.2 2 0.6 2 0.36 11 1129.32 IV 1128.4 5' 9429.11 5 9429.2 2 \ «0.2 2.8 1 1 1132.8 V 1130.8 5' 9432.46 18 9431.6 2 J 11 1.6 2 1154.85 iff11 1153.8 5> 9453.95 5 9454.0 2 0.8 3 0.47 14 1202.0 15' 1200.4 10 9499.96 17 9499.4 11 0.42 17 0.07 2 1239.33 /S 1237.3 10 9536.35 16 9535.4 11 1.1 4 0.47 14 1240.9 5 1237.9 10 9537.9 5 9536.0 7/ 0.24 10 0.48 14 ; 1307.7 4 1306.3 5> 9603.0 4 9602.6 2 2.4 10 1.7 3 1309.7 4 1307.4 51 9604.6 4 9603.7 2 5 2 4.4 9 1337.2 10 9632.8 11 w0.2 ' 1346.58 13 d 1346.2 51 9640.89 7 9641.5 2 5 2 4.1 5 1361.7 9 1359.2 10 9655.6 9 9654.2 ;/ 0.22 9 «0.3 i 1368.6 2 1372.4 12 9662.3 2 9667.1 IS 0.6 2 1375.12 12 c 1373.5 10 9668.71 8 9668.2 // 2.4 10 2.4 5 1488.54 J5 * 1487.4 51 9779.49 7 9779.2 2 2.2 9 1.8 4 1494.68 iS 1493.1 10 9785.3 2 9784.8 // 1.0 4 0.9 3 1512.1 7 1509.1 10 9802.2 7 9800.4 11 0.37 15 «0.2 1516.2 10 9807.3 11 «0.2 1521.18 18 1519.5 10 9811.1 2 9810.5 11 0.27 11 «0.2 1540.11 /^ 1537.6 10 9829.54 16 9828.1 // 0.8 S «0.5 1545.79 11 1543.6 10 9835.08 19 9833.9 11 0.6 3 ss0.3 1565.76 /5 1563.4 10 9854.54 17 9853.3 11 1.1 4 «0.4 1571.1 3 9859.7 3 0.5 2 1576.75 18 d 1576.4 51 9865.15 11 9865.9 2 6 2 4.5 9 1580.9 V 1578.9 10 9869.3 4 9868.4 // 3.1 IS 2.1 4 1611.0 3 9898.6 3 0.6 2 1634.33 /« 1632.0 10 9921.4 2 9920.2 11 0.43 17 «0.3 1653.2 2 9939.8 2 0.13 5 1667.80 JS c 1666.1 10 9954.00 9 9953.4 11 1.6 6 0.7 2 1691.60 15 9977.20 17 9980 5m 1.1 4 1708.6 i5' 9993.7 15 0.5 2 V

39 The K+p resonances for Ep = 0.3-2.9 MeV 63

Table 4.2 (continued) Ep (keV) Ex (1ieV) A'PT (eV) Present Literature" Presentt Literature6 Present Ref. [16] 1756.58 13': 1757.1 15 10040.54 9 10042.1 16 0.52 swl.3 1761.9 5> 10045.7 5 1765.60 H'' 1766.215 10049.38 7 10051.0 16 4.5 19 1.9 4 1774.5 2 10058.0 3 0.17 7 1797.78 18 10080.7 2 0.94 1849.07 n 1846 2 10130.70 19 10129 2 1.46 1854.1 15 10136.7 16 w0.5 1919.3 4 1921 2 10199.2 4 10202 2 0.6 2 1925.4 8 10205.1 8 0.23 9 1931.0 2 1930.6 15 102106 2 10211.2 16 1.4 6 «0.6 1953.8 7 1953.8 15 10232.8 7 10233.8 16 1.35 0.6 2 1984.32 c 1986 2 10262.53 10 10265 2 1.3 5 1.2 2 1989.6 5n 1992 2 10267.7 5 10271 2 0.19 8 1996.9 3 10274.8 3 0.28 11 2000.1 2 2001 2 10277.9 2 10281 2 0.7 3 2007.4 3 10285.0 3 0.7 3 2042.0 4C 2042.8 15 10318.8 4 10319.6 13 n 14.3 10° 14.3 10 2056.2 15' 2059 2 10332.6 15 10336 2 1.2 5 2082.9 15e 10358.6 15 0.6 2 2085.9 15' 2089 2 103615 15 10366 2 2.1 8 2100.2 15 10375.5 15 1.2 5 2108.83 13 2110 2 10383.90 16 10386 2 2.0 8 2140.80 11' 2140.615 10415.06 6 10415.9 16 5.8 19 «4 2146.6 10 2147 2 10420.7 10 10422 2 0.8 3 2156.72 17 2158 2 10430.58 19 10433 2 3.1 10 2167.8 6 10441.4 6 2.5 8 2170.4 2 2174 2 104439 2 10448 2 1.7 5 2197.2 15e 10470.0 15 10470 gm 0.6 2 2206.1 15e 10478.7 15 1.0 4 2231.1 15e 10503.1 15 10505 4m 1.1 4 2243.1 15' 2245 2 10514.8 15 10518 2 2.5 10 2256.5 15' 2257 2 10527.8 15 10531 2 3.7 15 0.9 5 2269.0 15' 2272 2 10540.0 15 10544 2 1.0 3 2281.5 15e 10552.2 15 1.8 7 2364.04 19 10632.7 2 2.1 8 2370.61 12c 2371 2> 10639.07' 7 10641 2 11 4 4.8 10 2378.1 4 2379 Sh 10646.4 4 10649 3 1.5 6 2385.14 14 2385 2* 10653.231 16 10654 2 8 3 1.4 5 2402.8 3 2400 3» 10670.4 3 10669 3 18 7 2406.13I 15 2405 2s 10673.6£t 17 10674 3 5 2 4.6 9 2408 5» 10677 3 0.7 2 2423.9 3 2425 2' 10691.0 3 10694 2 m 3.4 14 2432.61 14' 2433 23 10699.5() 10 10702 2 10 4 4.0 8 2454.5 3 10720.8 3 10725 gm 2.1 7 2471.8 3''••' 2474 2i 10737.7 3 10742 2 4.6 18 RSl.l V

40 64 Investigation of the Ca level scheme Table 4.2(continued) Ep(keV) Ex (keV) Sp,y (eV) Present Literature" Present Literature6 Present Ref. [16] 2482.2 4C 2483 2 ' 10747.8 4 10750 2 15 6 3.9 8 2488.37 16 10753.85 18 4.5 18 2505.1 3 10770.2 3 7 3 2511.4 3 2513 2 ' 10776.3 3 10779 2 16 6 1.3 5 2516.1 3 10780.9 3 6 2 2523.1 3 10787.7 3 3.0 12 2535.7 10 2539 5h 10800.0 10 10804 3 1.1 4 2549.8 5 2552 3h 10813.7 5 10818 3 12.0 5 2566.5 6 2569 3h 10830.0 6 10834 3 2.7 10 2585.5 4 2588 3h 10848.5 4 10853 3 4.4 n 2606.3 4 10868.8 4 5.2 19 2648.6 4 2650 2 * 10910.0 4 10913 3 7 3 «2 2659.9 4 2661 3s 10921.1 4 10924 3 9 4 »s6 2691.1 4 2688 3h 10951.5 4 10950 3 16 4 2695.8 4 2695 3h 10956.0 4 10956 3 4.0 16 2716.6 12 10976.32 15 10974 7m 9 3 ss5 2728.6 4 10988.0 4 10987 11 m 8 3 2735.5 4 2735 5 * 10994.7 4 10996 5 11 4 2743.4 5 2742 3 * 11002.4 5 11002 3 2.9 12 2752.2 4C 2750 3h 11011.0 4 11010 3 14 5 2765.3 5 2767 3 * 11023.8 5 11027 3 6 2 2784.0 5 2789 3 * 11042.0 5 11048 3 6 2 2813.3 4 2817 3 ' 11070.6 4 11075 3 31 12 2861.0 5 2865 3 * 11117.1 5 11122 3 5 2 2871.4 5 2875 3 * 11127.2 5 11133 3 6 2 2910.5 4 2914 3 * 11165.3 4 11171 3 2.0 8 "Ref. [1], unless indicated otherwise. 'Derived from Ep in column 2 with Q = 8329.3 4 keV [19], unless indicated otherwise. All entries with AEX = 0.2 keV have been derived directly from Ey [7]. ^Calculated from Ex and the present Q-value of 8328.28 9 keV, with Ex derived from measured E7's via a least-squares procedure (see sect. 4.4.2). ^Calibrated on a neighbouring 27Al(p,7) resonance and used for the determination of the Q-value. 'Calibration via the analyzing magnet NMR frequency leads to a relatively large uncertainty of 1.5 keV. /Ref. [7]. »Ref. [2]. hRef. [11]. •Weighted average of the values reported in ref. [2] and ref. [11]. >Ref. [32]. * Average of Ex =9092.8 // keV (footnote ')) and Ex =9093.0 11 keV [32]. 'See sect. 4.4.3. mRef. [33]. "Average of Ex =10320.6 16 keV (footnote ')) and Ex =10318.0 20 keV [34]. "From ref. [16], used to calibrate the relative strengths measured in the present experiment. pProbably a doublet. Gamma-ray spectra 65

4.4 Gamma-ray spectra At 25 selected 39K(p,7) resonances Ge spectra have been taken. All seven resonances observed below Ep = 1.0 MeV were studied, because the previous data from these resonances are exclusively based on measurements with Nal detectors. The selection of the other spectroscopically promising resonances is based on previous data or on information from survey spectra with poor statistics.

4.4.1 Measurements The setup consisted of two Ge detectors (see sect. 4.2). For the intensity measurements the 95 cm3 intrinsic Ge detector was placed at 0 = 55° where the Legendre-polynomial term P2(c°s 0) vanishes. At most of the resonances this detector was placed in a Compton suppression shield, which provides a reduction of the Compton background by a factor of 10-20 in the E7 = 1.0-10.0 MeV region, and at the same time virtually eliminates the counting of cosmic-ray (see fig. 4.4) and room background. The energy measurements were performed with either the 90 cm3 intrinsic Ge or the 126 cm3 Ge(Li) placed at 6 = 90°, where the Doppler shift of the 7-ray energies is zero. The dispersion in the 8192-channel spectra amounted to 1.5 keV per channel. Above Ep = 1.2 MeV, lead absorbers of 2.5-5.0 mm thickness were placed between target and detector to reduce the 181Ta(p,p'7) and X-ray counting rate from the target backing. At each resonance on- and off-resonance spectra were taken with a ratio of collected charge of 2:1. Bombarding times were 12-24 h, with beam currents up to 30 fiA, corresponding to a total collected charge of, on the average, about 1 C on target per resonance. Spectra were dumped onto magnetic tape every 45 minutes.

4.4.2 Analysis In order to obtain values for peak positions, areas and widths of the peaks, the 20-30 spectra taken at each resonance were checked for baseline- and gain shifts and, when necessary, corrected before summing. Large plots (3 m length) were made of the on- and off-resonance spectra for quick reference purposes. A tailed Gaussian line shape [20] (with a negative step at the peak position and a lineair or quadratic background function) was fitted to all relevant peaks in the summed spectra in a semi-automatic interactive procedure.

Gamma-ray energy calibration First, a set of'secondary' 40Ca calibration lines was developed, based exclusively on 'gold-standard' [18] energies. Separate measurements were performed at the 56 108 m no m Ep= 922, 1155 and 1347 keV resonances with Co, Ag and Ag sources attached to the target, close (<4 mm) to the target spot. In order to average "I 1 ' 600 O o o t t 39K(p,7)4°Ca ^P =2141 keV • 7.4 5 > g u - pi - - CSS at 07 = 55°

400 - 5 - n ririri t T t «> ci O T b t S"t .7 4 t 3 t ri /, h t «s 0 s • \ // t m t " l - tj = k n

8.7 6 3 , ^ 1/ 1 t i If ?5 T B s fl 0 !| - 7.8 7 • r 1% 1 J li 1 I 3 4 5 ' i | 1 ' 1 1 ' o o o t t

5.2 G 5-6 3 - 150 6.9 1

n [S t L 00 - io — ris o Is? t t toT u 50 - u I r.4 7 "F j 5" 1 a I . i J LJulJi 0 I 2, 5" £y (MeV)

Figure 4.4: High-energy portion of the Ep = 2141 keV spectrum, measured with the Compton suppression spectrometer (CSS). The energies are given in MeV. The resonance is denoted with 'r'. The accumulated charge amounts to 1.5 C. Gamma-ray spectra

7 -X

Proton beam

Figure 4.5: Setup for calibration 7-ray energy measurements.

out the effects of possible deviations of the actual detection angle from 9 = 90°, two detectors were used, which were aligned with the target center by optical and mechanical means at 6+ = +90° and 0_ = -90° with a precision of 9+ - 0_ = 180.00 ± 0.10°. The effect of the beam not hitting the exact target center was reduced by averaging results obtained at target angles — +45° and (j> = +135° [21]; see fig. 4.5. The energy calibration curves were checked initially by plotting the differences between the known energies of the calibration lines and the energies obtained from a linear approximation of the calibration curve on an enlarged scale. The required smoothness of the residual curve, of which the shape depends on the specific combination of detector, preamplifier, main amplifier and ADC, then clearly reveals erroneously interpreted peaks, or peak shifts due to e.g. contam- ination. The important 3736 and 3904 keV 7-ray energies from the 3.74—>0 and 3.90—»0 MeV 40Ca transitions, respectively, have also been deduced from the single-to-photo peak (SF) and double-to-single escape (DS) distances, in order to check the validity of the calibration in that energy region. The deviations of 2 these distances from mec [21,31] have been determined for each detector. In the final results, however, no SF and DS distances have been used. The maximum absolute deviation of linearity was 1.7 keV over a 6 MeV 7-ray energy interval. As a next step, the energy calibration curves for all spectra were constructed from a set of lines including primarily our 'secondary' 40Ca calibration energies 68 Investigation of the 40Ca level scheme

Table 4.3: Energies of secondary calibration lines

Nuclide Ex, Exf E, Nuclide Exi Ext Ey (MeV) (MeV) (keV) (MeV) (MeV) (keV) 40 Ca 4.49 --• 3.74 754.73 5 40Ca 6.025 -< 3.74 2288.72 5 40 «Ca 2.42 -- 1.52 899.42 S ° Ca 6.29 -H 3.90 2380.69 8 40 «Ca 2.75 -•* 1.52 1227.65 3 " Ca 7.114 -H• 4.49 2622.23 IS «Ca 1.52 -- 0 1524.68 3 ' 40Ca 7.114 -H• 3.74 3376.83 13 40 40 Ca 6.29 -» 4.49 1793.64 5 Ca 3.74 -• 0 3736.48 5 40 40 Ca 5.61 -- 3.74 1876.83 4 Ca 3.90 -• 0 3904.13 5 40Ca 7.53 -- 5.61 1918.65 8 16Q 6.13 -> 0 6129.27 5 b 40Ca 5.63 -- 3.35 2276.77 10

"Ref. [37]. »Ref. [18]. obtained as described above, and in addition precisely known 42Ca and 160 7-ray energies; all are listed in table 4.3. The final 7-ray energies were obtained as weighted averages of the energies measured at several resonances. The average reduced x2 value of 0.95 indicates an acceptable mutual consistency. For the spectra measured at 9 — 55°, the set of calibration lines was restricted to include only those originating from long- lived levels.

Intensity calibration In each geometry the efficiency curves were determined with 7-rays from standard radioactive sources [23] and with (p,7) reactions on "B, 23Na and 27A1 [24]. A function of the form

(4-3) n=-2 was fitted to the data and normalized to e(Ey = 1.0 MeV) = 1.0. The estimated relative systematic uncertainty in the intensities associated with the use of eq. (4.3) is 1.25% per MeV 7-ray energy interval.

Assignment of lines A search for possible assignments of observed lines to certain transitions, and for the consequences of a particular placement, was performed with a sorting and searching program, which cross-checks the data with the gradually increasing amount of 'known' data. The methods used in the analysis of the spectra have recently been described in detail in ref. [22]. An important check on the Gamma-ray spectra 69

•t

Table 4.4: Intensity balance at the Ep = 683 keV, £, = 8995 keV, J* = (1-,2+) resonance" Feeding Decay

Ex J* primary secondary Ex: 0 3.35 3.74 3.90 4.49 In/out J*: 0+ 0+ 3~ 2+ 5~ 0 0+ 7011 291 3.35 0+ 146 4S 3.74 3~ 173 53 108 30 294 18 0.96 16 3.90 2+ 581 54 610 28 1287 66 0.93 7 4.49 5" 39 15 sal 5.21 0+ 575 46 552 26 1.04 /0 5.63 2+ 611 40 529 88 99 17 0.97 15 6.025 2- 105 20 53 11 28 10 1.3 ^ 6.29 3" 45 11 20 10 39 15 wl 6.58 3" 31 10 22 20 15 5 6.75 2" 42 7 33^ 6.91 2+ 43 10 «25 wl 7.113 1- 31 7 »20 wl Total 2237 2254 110 0.99 6

"The relative detector efficiency e is normalized such that e = 1.0 at Ey = 1.0 MeV. All energies are given in MeV. 'Sum of the intensity of all primary transitions, except for the r—>0 and r—»3.35 MeV transitions, which are not followed by 7-decay.

consistency of a decay scheme is the intensity balance between feeding and deex- citation 7-transitions for each of the levels involved. As an example the intensity + balance of the Ep = 683 keV, J* = (1~,2 ) resonance is given in table 4.4.

Excitation energies Excitation energies of resonances and bound states were obtained from a least squares adjustment to the energies of the 7-rays connecting these levels. The program used [30] applies the recoil corrections and produces the values (and uncertainties) of adjusted excitation energies , and a complete set of adjusted 7-ray energies and uncertainties between all pairs of levels (from the excitation energies and the error matrix). Inconsistencies in 'input' and 'adjusted' 7-ray energies are also listed. The formalism used in the program is detailed in the appendix of this thesis. In the calculation of the excitation energies the input data are properly weighted, and correlations are taken into account, which makes this method superior to the often used sequential fixation of excitation energies in order to obtain additional calibration energies. 70 Investigation of the 40Ca level scheme

The use of primary 7-ray energies originating from single resonances leads to additional constraints between bound-state excitation energies and well established values of the excitation energies Ex of these resonances. The advantage of (p,7) reactions, where most often one bound state is excited at many resonances, over (nth,7) reactions, with only one capture state, is apparent.

4.4.3 Results and discussion

Excitation energies The excitation energies of bound states listed in table 4.5 have been obtained from one consistent simultaneous adjustment to the connecting 7-ray energies. Our data are in excellent agreement with, but in most cases considerably more precise than, the previous energies. The resonance excitation energies are listed in table 4.2 and have been discussed in sec. 4.3.3.

Branching ratios

Bound states The bound state branching ratios, and upper limits for unobserved transitions are presented in table 4.6. Many bound-states were excited at several resonances; the branchings have been calculated by properly [22] averaging the single-resonance values. The uncertainty in a bound state or resonance branching is the quadratic sum of the statistical error and the 1.25% per MeV systematic error in the detector efficiency (see sect. 4.4.2). The branchings of levels with 'unknown' entries in table 4.6 have been normalized to the total feeding intensity.

The upper limit intensities Juj. were obtained as

where N(E^) is the background level (per channel), W{E1) the ratio of detector resolution and dispersion (or, equivalently, the width in channels of an unbroadened line) and e(E-,) the detector efficiency, all at 7-ray energy £7. Corrections to the measured peak areas for coincident summing have been taken into account, but were only appreciable in some r—>0 transitions. The effects of random summing were found to be negligible. In order to establish consistent decay schemes it was necessary to introduce new levels at Ex= 6.93 and 8.68 MeV. Gamma-ray spectra 71

Table 4.5: Excitation energies (in keV) of (semi)bound states of 40Ca determined in the present experiment compared to previous data Present work Ref. |1|" Ref. |7| Ref. [34J 3352.64 9 3352.1 3 3352.5 5 3736.67 5 3736.9 2 3736.77 10 3904.39 4 3904.5 2 3904.37 £ 4491.43 4 4491.5 2 4491.57 i^ 5211.56 11 5212.9 7 5212.4 4 5248.79 5 5248.8 3 5248.98 J6 5278.80 6 5278.8 5 5279.0 3 5613.52 3 5614.2 2 5613.59 15 5629.41 0 5629.6 ^ 5629.72 22 5628.0 10 5902.63 8 5902.3 3 5902.64 11 5902.0 J0 6025.47 5 6025.8 5 6025.38 25 6029.71 6 6029.1 ^ 6029.7 4 6285.15 4 6285.0 3 6285.17 J0 6507.87 JS 6508.4 3 6507.7 3 6542.80 9 6543.5 4 6542.5 3 6582.47 J0 6583.1 5 6582.84 20 6750.42 7 6750.9 3 6750.38 n 6908.70 # 6909.3 3 6908.67 2-f 6931.29 £ * 6950.48 7 6950.9 4 6950.8 ^ 7113.72 tf 7113.7 5 7113.79 12 7239.07 8 7238.6

"Evaluation of the results of all the pre-1978 experiments. 'See the discussion in sect. 4.4.3. Troton unbound. 72 Investigation of the 40Ca level scheme

6 S

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^ to N in in C0T-

C « w H ^J (O N W3 CO t- i ^4 ^) ^4 (^ {^ QQ [^* Vyy«yyyyi-yyyy yfNyyyvvvyM

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+ + t,

I^OJflNN«(CiO 0 g Table 4.6 (continued) I Ex] E., : 0.00 3.35 3.74 3.90 4.49 5.21 5.25 5.28 5.61 5.63 5.90 6.025 6.030 Other i + J" 0+ 0+ 3" 2+ 5" 0+ 2+ 4+ 4" 2+ 1" 2~ 3 levels' a 7.68 (2'••-5~)d 100 7.695 3" ;1 <3 <1.9 90.8 12 <2 <2 <1.7 <1.5 <1.5 9.2 12 <1.2 <8 <1.3 <1.2 - 7.702 0+ 100 <5 <5 7.77 (2---5") 66 6 34 6 7.87 2+ 70 20 U : 30 20 3 7.93 4+ <6 <4 <7 <5 50 9 <3 <6 <3 50 9 <4 <3 <4 <4 8.09 2+ 100 + 8.13 (2 -4 ) 60 SO U : 40 20 8.19 (2- 100 e + d 8.32 (1"-,2 ) 2.0 7 <0.3 59.4 19 <1.3 <0.5 <0.3 2.7 5 <0.3 <0.3 <0.4 1.1 7 15.6 10 —6.29 : 1.3 3 8.34' (2+-5-)d —6.51 : 25 6 8.37 4+ 70 20 U : 30 20 8.42 2" 1 <1.5 <6 70 4 <4 <4 <4 <4 <4 <3 <6 17 3 13 3 <3 8.44 0+ >80 8.58 2+ 100 <7 <6 <4 <4 <5 <5 <5 <4 <4 <4 <4 9 d 8.68 (1- -5") 65 IS —6.29 : 13 5 8.75 2+ >80 8.76 3" 22 6 34 10 19 7 16' 6 U : »9 9.03'1 4" <5 <5 <5 28 5 <3 <3 12 5 40 5 <3 <3 —6.29 : 10 3

"All energies are given in MeV. 'From ref. [1], unless indicated otherwise. T = 0, unless T = 1 is explicitly given. CU stands for unknown decay. ''From the present work. 'In addition : -»6.75 : 7.4 6, U : 11 2. 'In addition : —6.54 : 60 6, U : 15 8. SV : 22 16. fc In addition : — 7.695 : 10 3, and Tp = 1.5iy

••-•f a 74 Investigation of the 40Ca level scheme

Tellez et al. [27] already suggest that the Ex= 6.93 MeV level observed in their (p, p'7) experiment is a doublet. They state that the energies and Doppler shifts of the two decay branches (fifty-fifty to the Ex = 3.74 MeV, J* = 3~ and 5.28 MeV, JT = 4+ levels) are barely consistent with a single initial state. Heavy-ion experiments [28] show a 6.93 MeV state decaying exclusively to the Ex = 5.28 MeV level; it is identified as the J" = 6+ member of a (4p-4h) rotational band based on the Ex = 3.35 MeV, J* = 0+ level. This identification is supported by calculations (see e.g. ref. [26]). A J* = 6+ assignment, however, is inconsistent with the presently observed decay modes of the level (e.g. to several J" = 2+ states). A doublet at Ex = 6.93 MeV eliminates these discrepancies, if it consists of a J' = 6+ level at Ex = 6930.2 ± 0.3 keV decaying to the EK = 5.28 MeV level, observed in the heavy-ion experiments, and an Ex = 6931.29 ± 0.06 keV, J'= (3-, 4+) level (see table 4.7) decaying mainly (83%) to the Es= 3.74 MeV level. Tellez et al. then indeed excited both members of the doublet. The new Ex= 8.68 MeV level is excited at only one resonance (Ep = 1489 keV). The decay of this resonance, however, is well established and all available information, including the Doppler shifts of the two decay branches (table 4.8) is consistent with the existence of this level. In comparison to the data reviewed in ref. [1] and the more recent (p,7) data from ref. [7] the present data are on the average more precise and extend the decay information with many hitherto unobserved decay branches. To a large extent this is, especially for weaker branches, due to the low-background CSS spectra. The 7-decay of the Ex = 8.68, 8.76 and 9.03 MeV levels is reported here for the first time. Here we mention only the stronger (JilO %) branches of those that have not been reported previously : 7.45—*5.63, 7.47—>5.25, 7.53—>6.29, 7.62->5.25, 7.66->6.29, 8.42-»5.90 and 8.42->6.025 MeV. This list underlines the fact that the CSS is superior to the traditional detectors, especially for low- energy transitions. The low background also allows the deduction of many rather tight upper limits for the intensity of unobserved transitions. On the basis of these data, several previously reported decay branches, e.g. the 6.58—»4.49 MeV transition, have to be discarded. Upper limits are often as informative as observed transitions in the comparison of experiment and theoretical calculations; they also provide information for the planning of experiments.

Resonances The resonance branching ratios, given in table 4.7, were normalized to the sum of the intensities of the 7-rays feeding the ground state and the 3353 keV level; the latter level decays by pair emission. The difference of this sum and the total primary intensity sum is listed in table 4.7 as 'unknown' decay, if it is significant. The Ep = 622, 683, 904, 922 (doublet), 2505 and 2757 keV resonances have Gamma-ray spectra 75

Table 4.7: Gamma-ray branching ratios (in 9i) of 39K(p,7)40Ca resonances ^£p(keV) : 622 683 783 828 904 922 1061 1129 1155 (MeV) 2+ (1-.2+) 3";(0) 2~;0 2-; 0 doublet 3";0 3'~;0 3"; 0 0.00 0+ 29 2 74.6 16 11.8 10 3.35 0+ 8.2 8 1.6 4 39.8 12 3.74 3~ 1.1 5 1.8 3 58.1 10 58.7 9 62.2 10 5.2 4 3.4 6 12 2 3.4 4 3.90 2+ 37.6 17 6.2 6 9.4 4 8.0 4 2.9 3 0.3010 5.9 8 4.9 6 4.49 5~ 29 2 5.21 0+ 1.3 3 6.1 5 5.25 2+ 2.1 9 4.5 2 0.4917 1.6318 4.4 8 5.28 4+ 8.5 4 5.61 4- 0.30 10 12.2 9 10.2 6 5.63 2+ 1.1 2 6.5 5 2.1 2 2.5 3 5.90 1- 0.6317 1.5 2 3.0 2 10.8 3 0.29 11 6.025 2~ 6.6 7 1.1 2 2.9 5 0.2510 1.6 2 13.6 5 1.8 3 6.030 3+ 1.2 4 2.5 4 6.29 3- 0.48 12 5.1 3 13.8 4 4.0 2 3.7 2 3.910 3.46 30.3 7 6.51 4+ 6.54 4+ 6.58 3- 0.70 10 0.33 10 1.04 14 2.0316 2.2 2 2.6 3 9.216 6.75 2~ 2.1 3 0.45 6 0.57 14 0.62 9 1.9517 3.2 2 2.1 2 6.91 2+ 0.46 11 6.931 (3~,4+) 6.95 1" 2.1 3 0.46 8 2.8 2 2.1118 7.113 1~ 0.65 16 0.33 8 7.114 4" 0.55 9 1.84 12 1.6012 1..3 3 10.5 5 7.24 (3~-5~) 0.73 10 7.28 (2,3)+ 1.3 2 1.2614 0.25 4 7.30 0+ 7.45 (3,4)+ 0.7 2 7.47 2+ 0.413 7.53 2~ 4.6 4 3.7 2 13.3 7 1.0 2 7.56 3+ 7.62 (2"-4+) 0.76 9 1.2 4 1.8 7.66 4~;1 10.9 8 36 2 7.1 6 7.68 (2+-5") 7.695 3 ; 1 2.1417 5.2 2 4.6 2 41.010 7.710 22.2 7 7.702 0+ 7.77 (2--S-) 7.87 2+ 0.32 5 7.93 4+ 8.09 2+ 8.13 (2-4+) 8.19 (2--S-) 8.32 (1-.2+) 8.34 8.37 4+ 8.42 2~;1 1.019 3.3 2 1.8 3 1.4718 8.44 0+ 8.58 2+ 8.68 (1--5-) 8.75 2+ 8.76 3" 9.03 4- unknown 5 2 14 3 i0 76 Investigation of the Ca level scheme

Table 4.7' (continued) ^£p(keV) : 1347 1375 1489 1577 1668 1757 1766 1984 2042 (MeV) 2~;1 3" 3;i l;l 4+ (2,3) -;1 4~; 1 3~;0 1+;1 0.00 0+ ' 71.4 12 82.5 7 3.35 0+ 15.0 5 11.9 4 3.74 3" 39.0 5 13.2 6 5.4 6 6.5 6 2.0 a 43.7 9 11.510 3.90 2+ 47.3 5 3.6 2 14.6 9 5.1 2 36.0 10 3.4 2 4.49 5" 3.0010 2.74 16.3 4 5.21 0+ 0.46 7 0.77 6 5.25 2+ 0.7 2 0.25 3 3.6 4 5.28 4+ 5.44 58.217 5.61 4" 19.9 8 4.84 0.95 9 5.63 2+ 4.70 10 2.0 2 0.417 2.94 0.27 7 5.90 1- 0.34 5 13.8 6 2.74 6.025 2" 3.0 3 2.0 2 1.30 10 6.030 3+ 16.1 IS 6.29 3" 0.4711 44.4 6 1.26 9 6.51 4+ 0.7 2 4.2 2 6.54 4+ 1.3 5 10.6 6 6.58 3" 1.54 1.2 5 1.4 2 7.3 3 4.15 6.75 2" 2.0618 0.212 6.91 2+ 0.50 5 0.6714 4.2 5 1.1010 6.931 (3",4+) 3.0 5 6.95 1- 0.15 5 0.324 6.7 6 0.417 7.113 1- 1.4 3 7.114 4- 26.9 7 4.4 3 14.0 4 7.24 (3~-5~) 0.76 13 7.28 (2,3)+ 0.20 7 8.4 3 7.30 0+ 3.2 2 7.45 (3,4)+ 0.6811 4.74 7.47 2+ 7.9 3 0.416 15.6 9 7.53 2- 1.8315 0.93 18 7.56 3+ 18.9 8 7.62 (2~-4+) 2.3 3 1.4 2 7.66 4~;1 7.68 (2+-5") 7.695 3~;1 0.65 13 7.702 0+ 0.5318 0.717 7.77 (2~-5~) 2.37 14 7.87 2+ 5.7 5 0.215 7.93 4+ 5.3 3 8.09 2+ 0.73 8 8.13 (2-4+) 2.7 2 8.19 (2--S-) 0.515 8.32 (1-.2+) 52.6 10 8.34 8.37 4+ 3.8 5 8.42 2~;1 8.44 0+ 0.18 5 8.58 2+ 5.0 5 8.68 (1--5-) 3.34 8.75 2+ 3.4 3 8.76 3- 5.5 7 9.03 4" 11.5 5 unknown 5 2 Gamma-ray spectra 77

Table 4.7 (continued) E,, £p(keV) : 2141 2371 2433 2472 2482 2505 2752 (MeV) 3~;1 (3~-5") 3" l";0 + l 4+;0 (1,2+) 3";0 0.00 0+ 55 3 32 2 3.35 0+ 3.74 3" 7.1 4 22.5 10 8.0 17 3.2 6 3.90 2+ 3.5 S 51.3 16 84.310 4.49 5" 3.6 3 45 5 5.21 0+ 5.25 2+ 1.6917 42 2 5.28 4+ 2.7 2 4.4 4 9.2 5 5.61 4" 6.9 3 13.6 6 2.0 2 5.63 2+ 0.8116 5.5 S 12.5 9 5.90 1" 6.025 V 5.9 3 6.030 3+ 3.8 3 6.29 3~ 0.36 9 1.4 2 7.815 6.51 4+ 1.0216 4.0 2 6.54 4+ 2.90 10 2.0 2 6.58 3~ 1.3414 1.6 2 6.75 2~ i.iO 10 6.91 2+ 15.7 4 2.6 2 4.314 6.0 7 6.931 (3~,4+) 4.0 2 42.1 12 6.95 1- 7.113 1" 3.3 7 7.114 4" 1.5717 4.0 3 7.24 (3--S-) 7.28 (2,3)+ 0.88 14 7.30 0+ 7.45 (3,4)+ 17.4 4 0.9215 7.47 2+ 5.9 2 0.9 2 7.53 2~ 1.0116 7.56 3+ 1.1311 7.62 (2~-4+) 16.7 5 7.66 4~;1 7.68 (2+-S-) 72 7.695 3~; i 0.4 2 9.217 7.702 0+ 7.77 (2~-5~) 7.87 2+ 7.93 4+ 8.09 2+ 0.72 9 8.13 (2-4+) 1.3 2 8.19 (2~-5~ 8.32 (1-.2+) 8.34 (2+-5~) 12 5 8.37 4+ 1.0114 8.42 2~-1 8.44 0+' 8.58 2+ £.68 (1--5-) 8.75 2+ 8.76 3" 9.03 4" unknown 2.5 18 9.6 18 24 3 17 2 36 5 78 Investigation of the 40Ca level scheme not been studied previously. For the other resonances, the the overall agreement of the present branching ratios with the combined data from refs. [2,4,7,16] is on the average good, where comparison is possible. The present work, however, almost doubles the number of primary decay branches observed. A few resonances (doublets) require separate discussion. Ep = 922 keV. The energies of the primary 7-rays and the distribution of the final-state spins (see chapter 5) are inconsistent with a single initial state. The primaries may be roughly divided in two groups, corresponding to two resonances with AEP = 0.76 ± 0.09 keV. The primaries to the Ex= 3.35 MeV, J* = 0+ and 6.025 MeV, J* = 2~ levels may be ascribed exclusively to the higher- energy component. Otherwise, the decay of the two resonances has not been disentangled. Ep = 1757 keV. This is the only resonance where a serious discrepancy occurs between the previous and present data. The 30% branch to the Ex = 5.63 MeV level reported in ref. [2] is based on an erroneous interpretation of the Ey = 4.42 MeV line, which is actually due to the 8.32->3.90 MeV decay. The 20% branch to the Ex = 3.90 MeV level has not been observed presently. Ep = 1762 keV. Although the resonance curve (sect. 4.3) does not show it, there is convincing evidence for a resonance between the Ep = 1757 and 1766 keV resonances. The off-resonance spectrum of the latter resonance, measured at Ep ss 1762 keV, clearly shows a primary transition to the Ex — 4.49 MeV level. It can be ascribed neither to the Ep = 1757 keV resonance, since this resonance does not decay to the Ex = 4.49 MeV level (see table 4.7), nor to the Ex = 1766 keV resonance, since even its strongest decay branch (r—>3.74 MeV) is not observed in this spectrum. We therefore conclude to a resonance, primarily decaying to the Ex = 4.49 MeV level, at Ep = 1761.9 ± 0.5 keV; the latter energy is deduced from the energy of the primary to the Ex = 4.49 MeV level measured in the Ep = 1762 keV spectrum. Ep = 2871 keV. Off-resonance measurements indicate that the r—>3.90 MeV primary, attributed to this resonance in ref. [2], is due to a lower energy resonance (probably at Ep= 2364 keV).

Lifetimes As a byproduct of the measurements of the branching ratios and excitation energies, the lifetimes of several levels have been deduced from the observed Doppler shifts by comparing the 7-ray energies measured at 9 = 55° and 90°. The F(Tm) attenuation curves (fig. 4.6) for converting the observed shifts into mean lives were calculated with a program based on the Blaugrund formalism [29]. Details on the electronic and nuclear components of the stopping power that enter in these calculations are given in ref. [25]. Gamma-ray spectra 79

100 - —

K(p,7) Ca

Ep = 1500 keV K on Ta backing d = 5 yug/cm^ d = 10 fig/cm 50 d = la ,ug/cm d = oo

• \

0 •

10' 10° T» (Is)

Figure 4.6: The Doppler-shift attenuation factor F as a function of the mean life rm, for different target thicknesses d.

Apart from the 20% uncertainty involved in the use of the Blaugrund formalism, relatively large uncertainties arise in the present experiment from (i) the small Doppler shifts of the lines in the 8 = 55° spectra relative to the 0 = 90° energies, which are only 38% of the 0°-120° shift used in standard DSA measurements, (ii) the peak fitting program (sect. 4.4.2) yields peak tops instead of centroid positions; the latter should be used since the peak shape may vary with 0, and (iii) the target thickness, which is determined by weighing, and the target composition. The agreement between the present and previous lifetimes is in general rea- sonably good, but in many cases the present values have too large uncertainties to be competitive. For a few levels, however, useful new information has been obtained; it is presented and compared with previous data in table 4.8. V

80 Investigation of the 40Ca level scheme

Table 4.8: DSA attenuation factors and corresponding mean lives of some 40Ca levels

Exi -£xf Ep F{rmy F(rm)° rm [fa] [MeV] [MeV] [keV] Present* Ref. [1] 6.931 — 3.744 2371 0.088 3 0.075 24 2000 900 4.49 2371 0.08 IS 5.25 2371 0.08 6 5.63 2371 C.04 7 7.114 — 3.744 1129 0.922 15 0.70 5 110 40 1155 0.57 n 4.49 1129 0.56 SO 1155 0.78 21 2371 0.66 6 5.61 1375 0.85 14 7.47 — 0 2141 0.999 6 0.95 4 26 20 10 3.35 2141 0.74 12 3.90 1984 0.80 14 2141 0.92 11 5.25 2141 0.99 6 7.53 — 3.744 622 0.5 3 0.39 7 310 100 215 50 828 0.666 14U 922 0.299 8 1155 0.6 3 -> 3.900 922 0.366 1414 8.32 — 3.744 1757 0.73 6 0.69 4 120 40 60 SO — 6.0255 1757 0.699 9 — 6.755 1757 0.599 9 8.68 — 3.744 1489 0.85 12 0.85 // 60 50 6.29 1489 0.9 4

"All measured F-values have been converted to those for a 10 /ig/cm2 K target at Ep = 1500 keV. 'The final error contains a 20% systematic contribution in addition to the statistical error, as discussed in the text. Shell model calculations 81

Spin, parity and isospin assignments The arguments for J*;T assignments to 40Ca resonances in the 39K(p,7)40Ca reaction are given in table 4.9. No previous data were available on J*\T of the Ex = 6.931, 7.68, 8.19, 8.32, 8.34 and 8.68 MeV (semi)bound states; our J*\T restrictions are given in table 4.6. The T = 0 assignments are based on the absence of analogue parent states in the 4°K level scheme within an energy range of 1 MeV. In the present assignments the recommended upper limits (RUL's) [43] for 7-ray decay strengths have been used. The proper use of RUL's is discussed in some detail in ref. [22]. A spin assignment is rejected if it would imply (i) a RUL-violating branch, or (ii) an M2 or higher multipolarity branch competing favourably with (several) allowed E2 or dipole transitions. w As an example, we discuss the J ; T — (2,3)~; 1 assignment to the Ep= 1757 keV resonance. The resonance, with a strength of 5PT7 = 0.5 ± 0.2 eV, decays to J* = l-,2~,3~,4- levels, which yields J* — (2,3)~. The 35 meV lower limit on + the partial width for the decay to the Ex = 8.32 MeV J*;T = (l-,2 );0 level excludes Mlis (0.35 W.u.), E1IS ( 0.009 W.u.) and E2 ( 390 W.u.) character for this transition and thus implies T — \ for this resonance. 39 40 The identification of the Ep = 2472 keV K(p,7) Ca resonance with the 39 38 J" = 1~ K(p,a0) Ar resonance observed by De Meyer et al. [11], together with the presently obtained decay modes leads to a T = 0 + 1 assignment. A pure T = 0 would imply a 100 W.u. E2IV transition to the Ex = 7.695 MeV, J*; T = 3"; 1 level, whereas the strength of the transition to the Ex = 6.29 MeV, J*; T = 3~; 0 level is too large for a T = 1 assignment.

4.5 Shell model calculations 4.5.1 Introduction The current state-of-the-art shell-model description of the 40Ca spectrum is the one by Gerace and Green [44], who for the odd-parity levels considered mixing of one-particle one-hole (lp-lh) states with > ITUJ intruder states, with the latter parameterized as a deformed band. Efforts to improve significantly on their treatment have so far been unsuccessful. In the present work the attention was therefore focussed on aspects of a related major problem in our understanding of the structure of light (A < 50) nuclei, namely the interface between the (ls,0d) and (0f,lp) major shells. War- burton et al. [45,46] were quite successful in calculating properties of A Rs 40 nuclei with cross-shell two-body matrix elements (2BME's) that were generated from a potential used by Millener and Kurath [49] to describe lp-lh states of nuclei around A = 16. Some important cross-shell 2BME's were subsequently 82 Investigation of the 40Ca level scheme

Table 4.9: Arguments for J*;T assignments to resonances in the 39K(p,7)40Ca reaction E." J';T [WeV] [keV] (P.I) Hef. [1] Other Resnlting Present Previous* reactions assignment 622.23 12 8934.81 7 2+ 2+ 683.45 15 8994.5011 (1-.2+) (1-.2+) 783.23 12 9091.70 6 3~;(0) (2+-4+) 828.2V 11 9135.66 5 2" 0 (2 3^— 2";0 904.29 11 9209.77 5 2~;0 (0-3)~ 2~;0 921.66 11 9226.69 5 (1--3-) (1--3-) 922.4212 9227.43 7 (1,2+) (1,2+) 1061.0111 9362.54 6 (2,3-);0 #4-' (3,4)~;0 (3,5)-;0d 3~;0 1129.32 11 9429.11 5 (3"-5-);0 (3--5-)c (0-3)"/ 3-,0 1154.8516 9453.95 5 (2~,3);0 S-jO + l' 3-;0 1346.58 IS 9640.89 7 (1--3-) 2-.1 2";1 1375.12 12 9668.71 S 3- (0-3)" 3" 1488.54 18 9779.49 7 (3,4+) 3;1 3;1 1576.75 18 9865.1511 (1,2+) 1;1 1;1 1667.80 IS 9954.00 9 (3--5~) 4+;l* 4+ 1756.58 13 10040.54 9 (2,3)";1 (0+-4+) 3-9 (2,3)"; 1 1765.60 U 10049.38 7 (3,4)-; 1 4" 4";1 1984.32 14 10262.53 10 (2,3") (l,3)-;0 3";0 2042.0 4 10318.8 4 (1,2+) 1+;1 1+;1 2140.80 11 10415.06 6 3 3";1 3~;1 2370.6112 10639.07 7 (3--5~) (2--4+)' (3"-5") 2432.6114 10699.50 10 3 (3", 4+) 3" 2471.8 S 10737.7 S (1~ ,2+);0 + l (1~,2+) l~3,\~ ;0k 1~-0 + 1 2482.2 4 10747.8 4 (1--4+) (0-4+) (4+,5~);0 4+;0 2505.1 5 10770.2 S (1.2+) (1,2+) 2752.2 4 11011.0 4 > 3" (l,3)-',JV;0(k 3~;0

"FroDImD tabtabl!e 4.2. Reff.[2],u. [2], unles, s indicated otherwise. cRef:f. [7], dRefif. [33]. . 'The T = 0 assignment follows from the absence of an analogue state in the 40K level scheme within a reasonable energy range. In ref. [1] the 9.42, 9.45 and 9.603 MeV 40Ca states were 40 given as candidates for the analogue of the Ex = 2.07 MeV, J* = 3" K level. The 9.42 MeV level, with J';T - 3~; 1 (ref. [40]) is the most likely candidate, wheras the 9.603 MeV level is (following the trend in the excitation energy differences of analog states of 40Ca and 40K) most 40 probably the analog of the new [40] Ex = 2.291 MeV, J' = 3"; 1 level in K. >Ref. [39], /+ = 1. 'This assignment was based on assumed /9+-decay to this level; the /8-decay branch, however, proceeds to the E% = 10049 keV level [33]. 40 * J* = 2+ has recently been assigned [40] to the Ex = 2.58 MeV state in K, previously assumed to be the parent state (J* = (2,4)+; 1 in ref. [1]). •The 4+ upper limit of the J' range given in ref. [2] is based on an erroneous assignment of the 6728 keV 7-ray to the r—+3.90 (2+) transition. This 7-ray is actually produced in the decay of the Ep = 2364 keV resonance. '"Ref. [11]. *Ref. [42]. N denotes natural parity. Shell model calculations 83 varied (independently) to fit binding energies in A — 40 nuclei. A somewhat less ad-hoc approach was followed presently by optimizing the parameters of the potential via a least-squares adjustment to binding- and excitation energies of levels in the A = 37-41 region.

4.5.2 The WBMB interaction The starting point for our shell-model calculations is the WBMB (Warburton, Becker, Millener and Brown) interaction [46]. It uses an inert 16O core, the Wildenthal 'universal' ls,0d (USD) interaction [47] for the sd shell and the Mc- Grory interaction [48] for the fp shell. The sd to fp cross-shell two-body matrix elements of the WBMB were generated from the Millener-Kurath (MK) potential [49]. The MK potential has central, spin-orbit and tensor components,

V(r) = Vc(r) + VLS(I-) + VT(r), (4.4) with

VLs(r) = VLs(Aj|P13 + AJIP33)^ • Sfm(r), and 13 33 VT(r) = FT(A^?P + A^P )Si2/T(r),

where the p*r+1«2S+l are projection operators, the A2T+1>2S+1 describe the exchange mixture, V < 0 corresponds to an attractive force, and by convention Aj? = A]% = A}? = 1. All the f(r) have a Yukawa form

er/M /W = ^ (4-5)

with n = 1.4626 fm for /c and /T, and fi — 0.7313 fm for /Ls- Millener and Kurath determined the parameters of their potential from an analysis of non- normal parity states of a number of nuclei ranging from nB to16 O. The dependence of the lp-lh spectra of 16O and 40Ca on the parameters of the MK potential was shown to be similar [45], which is indicative of a cor- respondence between the shell-crossing situation around 16O and 40Ca. The original MK parameter set was therefore adopted for use in the WBMB. The eight (f7/2d3/2|V"|f7/2cl3/2) 2BME's were subsequently varied to fit experimental binding energies in A = 40. Details of these adjustments are given in ref. [8]. The WBMB was developed for use in the full 160(ls,Od)j4-16-"(Of,lp)n model space for a single value of n.

4.5.3 Calculations In the present work the parameters of the WBMB, which are essentially the strength parameters of the MK potential (eq. 4.4) and the magnitude of the sd to 84 Investigation of the 40Ca level scheme fp shell energy gap, have been optimized through a least-squares adjustment to binding- and excitation energies of 94 levels in the A = 37-41 region. Thus, the method used by Millener and Xurath in the determination of their interaction for A = 11-16 nuclei is now applied to A — 37-41 nuclei. Our calculations were performed with the programs RITSSCHIL [50] and DIAFIT [51] for the construction «f the Hamiltonian matrices, and their di- agonalization, respectively. The latter program also allows for a least-squares adjustment of the parameters of the interaction to binding energies or excitation energies, optionally in combination with other observables like electromagnetic moments. In these adjustments the total r.m.s. deviation (see below) between calculated and experimental observables is minimized. Each cross-shell two-body matrix element M is treated as a linear combination of the interaction parameters a,-,(i = l,...,m), given in column 1 of table 4.10 Af = f>,a,, (4.6) i=i The coefficients c are obtained from the evaluation of the potential V (see eq. 4.4) between the relevant harmonic oscillator basis states. The ranges \i of the radial functions (see eq. 4.5) used in the MK interaction were adopted as 1.4585 fm for the central and tensor functions /c and /T, and 0.7292 fm for the spin-orbit function /LS- NO attempt was made to fit them. With a harmonic oscillator size of b = 1.9100 fm, the matrix elements given in ref. [8] were reproduced to within 1 part in 1000. The USD, MK and McGrory interactions were joined by adopting the standard USD A'03 mass dependency for the MK and McGrory 2BME's for A < 40, and fixing the latter two at the A = 40 values for A > 40. The USD 2BME's were given a dependence of [18/(A-n)]°-3 for 160(sd)4-I6~"(fp)n configurations. In summary, 2BME's A<40 A > 40 USD x[18/(A-n)]0-3 x[18/(A-n)}0-3 MK/McGrory x[40/^]03 xl.O The three (0s,ld)-shell single-particle energies (SPE's) were fixed to to their appropriate (USD) values. The use of the McGrory interaction implies fixed relative differences of the four (Of,lp) SPE's [48]. The least-squares adjustments were therefore constrained to reproduce exactly the centroids of the Ofjp, IP3/21 41 lp1/2 and 0f5/i strengths in Ca at 0, 2.100, 3.900 and 6.500 MeV excitation energy, respectively. Additionally, the binding energy of the 41Ca ground state participated (along with the binding energy of 93 other A = 37-41 levels) in the least-squares adjustment. Shell model calculations 85

The experimental binding energies were taken from ref. [8]. For A = 40 'ideal' experimental spectra were used. As detailed in ref. [8], these spectra result for 40K from the evaluation of stripping and pick-up data, which were used to localize the lp-lh strengths. Centroid positions, based on the spectroscopic factors, are used when the strength is fragmented. For 40Ca the 'ideal' spectrum consists of the unperturbed energies of the T = 0 lp-lh states given by Gerace and Green [44]. The 'real' experimental 40Ca J* = V{, 2J" and 3J states are considered to be > lhu> intruders, and are therefore not reproduced in the present calculations. xp The experimental binding energies .EB were Coulomb corrected and calculated relative to a 16O core according; to orr xp 16 £! = EB - EC(Z) - EB( O), (4.7) 16 where £B( O) is -127620 keV, and Ef* and Ec are the Coulomb corrected experimental binding energy and the Coulomb energy, respectively. The latter was taken from ref. [8] : 16 z EC(Z) + EB( O)(keV) 17 -82929 18 -76263 19 -69363 20 -62048 The effects of spuriosity, caused by the centre-of-mass motion, were (practically) eliminated by shifting states with large spurious components significanty upward in energy. The method used is described in detail in ref. [52].

4.5.4 Results and discussion The (Of,lp) single-particle energies (SPE's) and the MK parameters of the new interaction, labeled FALSE, are compared to the WBMB results in table 4.10. Table 4.11 compares experimental and calculated binding energies for the states that participated in the least-squares adjustment. The MK spin-orbit T = 0 and tensor T = 0 and T = 1 channel parameters proved to be practically undetermined in the fits, and were therefore fixed to their original values in the parameter set designated SET A in table 4.10. In a second approach, the exchange mixtures were constrained to their MK values, which resulted in the SET B parameters. The latter lead to a significantly larger r.m.s. deviation (see below) of calculated and experimental energies than the SET A values, which were consequently adopted. The 6 and e uncertainties in a parameter a, given in table 4.10 correspond to a change in a, that would cause a 5% increase in the Q2-value of the fit; the parameters a,^, are kept constant 86 Investigation of the 40Ca level scheme

Table 4.10: Comparison of the parameters of the WBME1 and FALSE interactions" WBMB FALSE SETA' SETBC 6 t 6 £ SPE's 0f7/2 5.645 6.808 0.0U4 2.121 8.040 0.005 1.302 Ofs/2 10.161 11.373 0.007 2.496 14.033 0.008 1.360 1P3/2 5.172 4.936 0.007 2.165 6.419 0.008 0.706 5.160 5.705 0.007 2.425 7.939 0.008 0.904 MK VCA}} 31.987 67.006 0.487 28.105 37.586 VfcA?.1 26.880 -16.016 0.170 4.541 -31.571 VcAh3 -44.800 -61.641 0.169 5.096 -52.619 0.082 1.853 VfcAc? 12.813 12.097 0.053 4.362 15.033 VLSAJI -26.000 -26.000 -4.052 0.052 19.250 VKSAH -91.000 -51.986 0.163 54.990 -14.183 -16.250 -16.250 -26.334 8.324 8.745 VTAf VTAf 6.250 6.250 10.128 "All parameters are given in MeV. For definition of 6 and e, see text. 'Undetermined parameters fixed to WBMB values, see text. 'Exchange mixtures fixed to WBMB values, see text. in the calculation of 6, and are re-optimized for e [51]. The r.m.s. deviations axe calculated according to fin (4.8) where E|M is the calculated binding energy, and n the number of levels. They are compared to the WBMB results in table 4.12 for individual nuclei and for specific ranges of the neutron number N. The (sd fp| V|sd fp)T=o matrix elements are determined exclusively by states with active protons in the fp shell, and active neutron holes in the sd shell or vice versa. This implies in the present model space that energy levels in N > 21 nuclei are only sensitive to the T = 1 part of the cross-shell interaction, whereas N < 20 nuclei may pin down both T = 0 and T = 1 strength parameters of this interaction. As seen from table 4.12, the quality of the N = 21 and N > 21 results is similar in the FALSE interaction, which is indicative of a trustworthy fp interaction [8]. The N < 21 results have improved as compared to the WBMB, which may be attributed to the optimization of the cross-shell interaction parameters. The remaining FALSE deviations are comparable to the WBMB, with the exception of the A - 40 results, to which, however, the latter interaction was specifically tuned (see sect. 4.5.2). The presently calculated 8361 keV difference in the 41Ca and 40Ca ground-state binding energies is in excellent . • j+,m —

Shell model calculations 87

Table 4.11: Comparison of calculated and experimental binding- and excitation energies of levels of A = 37-41 nuclei"

£'x 6 e d >rr Nuclide J%;T Exp. Calc. AEx i?g 37C1 3/2f ;3/2 0.000 -0.228 -0.228 -234.084 7/27;3/2 3.104 2.684 -0.420 3/27;3/2 3.627 3.475 -0.152 5/27;3/2 3.741 4.207 0.466 9/27;3/2 4.010 4.140 0.130 n/27;3/2 4.546 4.380 -0.166 7/2 J ;3/2 4.273 4.482 0.209 3/2j;3/2 4.177 4.674 0.497 7/2J ;3/2 4.460 4.918 0.458 5 '2.~ ;3 '2 4.3G6 1 SIS 0.420 13/2713/2 5.271 4.753 -0.518

37Ar 3/2f;l/2 0.000 0.047 0.047 -239.243 7/27;l/2 1.611 1.470 -0.141 3.185 2.947 -0.238 3/27;l/2 2.491 2.924 0.433 5/27;l/2 3.274 3.335 0.061 7/2j;l/2 3.527 3.766 0.239 1 n/27;l/2 3.707 3.370 -0.337 3/2J;l/2 3.518 4.246 0.728 l/27;l/2 4.444 4.391 -0.053 i 9/2J.1/2 4.021 4.519 0.498 13/27;l/2 5.793 5.465 -0.328

0.000 0.074 0.074 -240.282 ! 57;2 0.671 0.484 -0.187 37;2 0.755 0.827 0.072 47 ;2 1.308 1.409 0.101 1.617 1.630 0.013 07 ;2 1.745 0.968 -0.777 17 ;2 1.692 1.622 -0.070 1.784 1.722 -0.062 2J;2 1.981 1.727 -0.254 3J;2 2.743 2.618 -0.125

^Ar Of;l 0.000 0.047 0.047 -251.082 47 ;1 4.480 4.364 -0.116 37 ;1 3.810 4.385 0.575 57 ;l 4.586 4.145 -0.441 3J;1 4.877 4.997 0.120 27'1 5.084 5.627 0.543 • • Attum —r-

88 Investigation of the 40Ca level scheme

Table 4.11 (continued) Ex d Nuclide Exp.» Calc* AEx EW" 3f;0 0.000 -0.059 -0.059 -251.286 37 ;0 2.613 2.127 -0.486 47 ;0 2.646 2.321 -0.325 2r;0 2.870 2.893 0.023 57 ;0 3.615 3.128 -0.487 67 ;0 3.420 3.233 -0.187 i7;0 2.829 3.302 0.473 07 ;0 2.993 3.377 0.384

39 Ar 7/27 ;3/2 0.000 -0.002 -0.002 -257.680 3/27.3/2 1.267 1.507 0.240 5/27 ;3/2 2.093 2.416 0.323 9/27 ;3/2 2.524 2.106 -0.418 7/2j;3/2 2.481 2.784 0.303 3/2^3/2 2.631 2.631 0.000 11/27 ;3/2 2.651 2.661 0.010 1/27 ;3/2 3.266 2.844 -0.422

39K 3/2f ;l/2 0.000 -0.143 -0.143 -264.363 7/27 ;l/2 2.814 2.710 -0.104 ll/27;l/2 3.994 3.385 -0.609 9/27 ;l/2 3.598 3.670 0.072 3/27 ;l/2 3.019 3.639 0.620 5/27 ;l/2 3.883 3.982 0.099 7/2J51/2 4.126 4.142 0.016 9/2J;l/2 4.521 4.496 -0.025 13/27,1/2 5.718 5.295 -0.423

40K 47.1 0.000 -0.103 -0.103 -272.162 37 ;i 0.030 -0.087 -0.117 0.800 1.077 0.277 57 ;1 0.891 0.860 -0.031 2J;1 2.330 1.806 -0.524 32,1 2.166 2.199 0.033 2.104 1.868 -0.236 07 ;i 2.626 2.019 -0.607 42;1 2.513 2.448 -0.065 3a ;1 3.166 3.734 0.568 Shell model calculations 89

-• Table 4.11 (continued)

Nuclide JSP Exp.8 Calc.e AEx* £g>rr 40Ca 0+;0 0.000 -0.095 -0.095 -280.007 37 ;0 3.980 4.095 0.115 57 ;0 4.700 3.949 -0.751 47 ;0 5.840 6.329 0.489 3J;0 6.950 6.699 -0.251 27 ;0 6.500 6.325 -0.175 17;O 7.000 6.586 -0.414 07 ;0 8.300 7.979 -0.321 22,0 8.050 8.280 0.230

7'2~;1'2 0 000 -0.003 n nno -288.370 3/2:il/2 2.100 2.007 -0.093 1/27 ;l/2 3.900 3.807 -0.093 5/27 ;l/2 6.500 6.407 -0.093 41 1 K 3/2f;3/2 0.000 -0.181 -0.181 -282.258 l/2+;3/2 0.980 0.775 -0.205 7/2+;3/2 1.677 1.457 -0.220 5/2+;3/2 1.698 1.614 -0.084 3/2+;3/2 1.560 2.264 0.704 9/2+;3/2 2.495 2.420 -0.075 5/2+;3/2 2.756 2.679 -0.077 11/2+;3/2 2.528 2.635 0.107 l/2+;3/2 2.682 2.810 0.128 I 7/2+;3/2 2.508 3.171 0.663 13/2+;3/2 2.774 2.838 0.064

"All energies are given in MeV. 'See the discussion in the text. "Binding energy £|M, calculated with the FALSE interaction, minus the experimental ground-state binding energy £'|orr, which is given in column 6. d£l £exp 00 Investigation of the 40Ca level scheme

Table 4.12: R.m.s. deviations of calculated and experimental binding energies

AE, mo AE, •ma n» FALSE WBMB n' FALSE WBMB 37C1 11 364 260 40Ca 9 372 77 37 Ar 11 348 387 «K 11 311 230 MC1 10 274 277 41Ca 1 93 80 MAr 6 376 378 JV = 21 29 297 266 38K 8 352 696 AT>21 11 311 207 39 Ar 8 274 230 JV>20 40 301 236 39K 9 330 485 JV<21 54 356 454 40K 10 336 107 All levels 94 333 347

"The r.m.s. deviations are given in keV. N stands for the neutron number. 'Number of levels taken into account.

Table 4.13: Comparison of the calculated and experimental 1 = 2 hole strength in 40K 2 40R state 7 S a ( ^x(keV) Exp FALSE WBMB (d3/2f7/2) 4r° 0 1.03 is 1.108 1.111 1.125 3- 30 0.86 6 0.837 0.846 0.875 27 800 0.57 7 0.549 0.592 0.625 5r 892 1.46 n 1.341 1.364 1.375 aRef. [53].

agreement with the experimental value of 8363 keV. Finally, we compare FALSE and WBMB results for the / = 2 hole strength in 40K in table 4.13, and conclude that here the FALSE and WBMB interaction produce essentially identical values, which are rather close to those obtained in a pure (d^f^) model space. The present interaction has been used to calculate strengths of selected electromagnetic transitions in 40Ca. As expected (see sect. 4.5.1), the agreement is in general poor and the results will not be presented here. The discrepancies in the dipole strengths can be attributed mainly to the fact that in A « 40 nuclei the active nucleons are mostly in d3/2 and f^/2 orbits between which operators of rank A J < 2 are forbidden. Furthermore, the enhancement of E2 strength through collective effects can not be reproduced in the current model space. Summary and conclusions 91

4.6 Summary and conclusions The present measurements yield a considerable amount of new or improved data on the 40Ca level scheme, which is summarized in the tables. Proton energies, excitation energies and strengths of 112 resonances in the range Ev = 0.6-2.9 MeV, of which 36 have not been observed previously, are given in table 4.2; precision energies of 46 bound states (2 new) in table 4.5; branching ratios of (semi)bound states in table 4.6 and of 25 resonances in table 4.7; lifetimes of a few bound states in table 4.8 and spin, parity and isospin assignments (or limitations) in tables 4.6 and 4.9. The reaction Q-value has been determined as Q = 8328.28 ± 0.09 keV. In view of the fact that 40Ca has been extensively investigated in the past, it is remarkable that simple (p,7J singles spectra can add so much new information. The high precision of the data, which is important for the detection of doublets, considerably improves the completeness (no missing levels) and purity (no misassignments of e.g. spins) of the 40Ca level scheme. An all-important positive factor is the availabily of the Compton suppression spectrometer, which yields practically background-free spectra from which precise information on especially low-energy 7-rays and tight upper limits for the strength of unobserved transitions can be readily obtained. On the other hand, the target stability is less than ideal, and the 39K target nuclide with J = 3/2 apparantly imposes an upper limit of J = 5 on the spin of levels excited in the present experiment. A rather low target spin is, however, a prerequisite for successful (future) measurements of mixing ratios, of which, remarkably enough, hardly any are known in 40Ca. A natural next step in the investigation of 40Ca would thus be the measurement of angular distibutions of 7-rays at a few well-chosen resonances. For reasons of time, this has not been included in the present work. Needless to say, some of the weaker resonances observed in the present yield-curve measurements might have high spin, and could supply interesting complementary information. The sd to fp cross-shell matrix elements of the WBMB interaction have been optimized via a least-squares adjustment to binding energies of 94 levels in A = 37-41 nuclei. Comparison of binding energies calculated with the new interaction and with the WBMB to experiment indicate that the present treatment, while being more elegant since no ad-hoc changes are made to selected matrix elements, produces equally acceptable results. Improvement is obtained foL the levels which depend strongly on the cross-shell particle-hole intera<.rion. V

92 References

References [1] P.M. Endt and C. van der Leun, Nud.Phys. A310 (1978) 1 [2] H.P. Leenhouts en P.M. Endt, Physica 32 (1966) 273 [3] H.P. Leenhouts, Physica 35 (1967) 290 [4] H. Lindeman, G.A.P. Engelbertink, M.W. Ockeloen and H.S. Pruys, Nud.Phys. A122 (1968) 373 [5] A.R. Poletti, A.D.W. Jones, J.A. Becker and R.E. McDonald, Phys.Rev. 181 (1969) 1606 [6] K.W. Dolan and D.K. McDaniels, Phys.Rev. 175 (1969) 1606 [7] J. Schoeman, J.P.L. Reinecke and J.J.A. Smit, S.-Afr. J. Phys. 11 (1988) 68 [8] E.K. Warburton, J.A. Becker, D.J. Millener and B.A. Brown, The WBMB shell- model interaction and the structure of 160(ls,Od)A~16-n(Of,lp)n levels in A = 31-44 nuclei with particular emphasis on binding energies and spectra, BNL Report 40890 (1987) [9] H.J.M. Aarts, C.J. van der Poel, D.E.C. Scherpenzeel, H.F.R. Arciszewski and G.A.P. Engelbertink, Nucl. Instr. Meth. 177 (1980) 417 [10] G.J.L. Nooren, H.P.L. de Esch and C. van der Leun, Nud.Phys. A423 (1984) 228 [11] R.J. de Meijer, A.A. Sieders, H.A.A. Landman and G. de Roos, Nucl. Phys. A155 (1970) 109 [12] J.W. Maas, E. Somorjai, H.D. Graber, CA. van den Wijngaart, C. van der Leun and P.M. Endt, Nucl. Phys. A301 (1978) 213 [13] E.L. Bakkum and R.J. Elsenaar, Nucl. Instr. Meth. 227 (1984) 515 [14] H.E. Gove, in Nuclear Reactions I, eds. P.M. Endt and M. Demeur, North-Holland, Amsterdam (1959) [15] W. Bruynesteyn, R.J. van de Graaff laboratory internal report V2995c (1969) (unpublished) [16] C.-W. Cheng, S.K. Saha, J. Keinonen, H.-B.Mak and W. MacLatchie, Can.J.Phys. 59 (1981) 238 [17] R.O. Bondelid and J.W. Butler, Phys. Rev. 130 (1963) 1078 [18] R.G. Helmer, P. Van Assche and C. van der Leun, At. Data Nucl. Data Tables 24 (1979) 39 [19] A.H. Wapstra and G. Audi, Nucl. Phys. A432 (1985) 1 [20] R.G. Helmer, J.E. Cline and R.C. Greenwood, in The Electromagnetic Interaction in Nuclear Spectroscopy, ed. W.D. Hamilton, North-Holland, Amsterdam (1975) [21] P.F.A. Alkemade, C. Alderliesten, P. de Wit and C. van der Leun, Nucl. Instr. Meth. 197 (1982) 383 [22] P.M. Endt, P. de Wit and C. Alderliesten, Nucl. Phys. A476 (1988) 333 [23] R.J. Gehrke, R.G. Helmer and R.C. Greenwood, Nucl. Instr. Meth. 147 (1977) 405 [24] F. Zijderhand, F.P. Jansen, C. Alderliesten and C. van der Leun, Nucl. Instr. Meth., to be published [25] E.L. Bakkum and C. van der Leun, Nucl. Phys. A500 (1989) 1 [26] W.J. Gerace and J.P. Mestre, Nucl. Phys. A285 (1977) 253 References 93

[27] A. Tellez, H. Ronsin, R. Ballini and I. Fodor, J. de Physique 34 (1973) 281 [28] J.J. Simpson, S.J. Wilson, P.W. Green, J.A. Kuehner, W.R. Dixon and R.S. Storey, Phys. Rev. Lett. 35 (1975) 23; A.M. Nathan and J.J. Kolata, Phys. Rev. C14 (1976) 171 [29] A.E. Blaugrund, Nucl.Phys. 88 (1966) 501 [30] B.J. Barton and J.K. Tuli, BNL report NCS-23375/R (1977) [31] R. Vennink, J. Kopecky, P.M. Endt and P.W.M. Glaudemans, Nucl. Phys. A344 (1980) 421 [32] J.A. Nolen and R.J. Gleitsmann, Phys. Rev. Cll (1975) 1159 [33] J. Honkanen, J. Äystö, M. Kortelahnti, K. Eskola, K. Vierinen and A. Hautojärvi, Nucl. Phys. A380 (1982) 410 [34] R. Moreh, Phys. Rev. Cll (1975) 1159 [35] D.P. Stoker. P.H. Barker. H. Naylor; R.E. Whitp and W.B Wood;Nnrl Instr Meth. ' 180(1981)515 [36] M.L. Roush, L.A. West and J.B. Marion, Nucl. Phys. A147 (1970) 235 [37] S.W.Kikstra, C. van der Leun, S. Raman, E.T. Jurney and I.S. Towner, Nucl. Phys. A496 (1989) 429 [38] R.G. Sextro, R.A. Cough, J. Cerny, Nucl. Phys. A234 (1974) 130 [39] H. Fuchs, K. Grabish, and G. Rösschert, Nucl. Phys. A129 (1969) 545 [40] T. von Egidy et ai, J. of Phys. G10 (1984) 221 [41] R.C. Greenwood, R.G. Helmer and R.J. Gehrke, Nucl. Instr. Meth. 159 (1979) 465 [42] T. Nakashima, J. Phys. Soc. Japan 36 (1974) 10 [43] P.M. Endt, At. Data Nucl. Data Tables 23 (1979) 3 [44] W.J. Gerace and A.M. Green, Nucl. Phys. 113 (1968) 641 [45] E.K. Warburton, D.E. Alburger, J.A. Becker, B.A. Brown and S. Raman, Phys. Rev. C34 (1986) 1031 [46] E.K. Warburton and J.A. Becker, Phys. Rev. C37 (1988) 754 [47] D.J. Millener and D. Kurath, Nucl. Phys. A113 (1975) 315 [48] B.H. Wildenthal, Prog. Part. Nucl. Phys. 11 (1984) 5 [49] J.B. McGrory, Phys. Rev. C8 (1973) 693 [50] D. Zwarts, thesis, Utrecht University (1984) [51] A.A. Wolters, thesis, Utrecht University (1989) [52] A.G.M. van Hees, thesis, Utrecht University (1982) [53] D. Roaf, F. Watt, E.F. Gorman, F. Pellegrini and P. Gauzzoni, Phys. Rev. C20 (1979) 55 Chapter 5

Statistical nuclear spin assignments

Abstract A new method for the assignment of the spin of uucleai levels, based uu a. statistical analysis of 7-ray transitions, is described. The results for the almost 200 levels of four nuclides studied so far, agree very well with the spins assigned with traditional methods. For 11 levels the present method yields more precise information.

5.1 Introduction Modern Ge detectors and especially Ge Compton suppression spectrometers (CSS) make it possible to unravel the nuclear 7-decay in hitherto unknown detail. In fact the amount of data from 7-ray experiments is becoming so huge, that statistical treatment of these data may yield information about the nuclei studied that cannot be deduced along traditional lines. Statistical methods for the assignment of spins have been used previously. These methods are based on 7-ray decay strengths [1] and on the isotropy of 7-ray angular distributions [2]. In this paper we discuss the possibility of spin assignments to nuclear levels purely on the basis of the mere existence of 7-ray decay to (fig. 5.1a) and 7-ray feeding from (fig. 5.1b) a large number of levels with known spin.

5.2 Average spins Even in the Nal era, when in most cases the observation of only a few of the strongest 7-decay branches of nuclear levels was possible, the 7-ray decay has already been used to set valuable limits on the possible spins of the decaying • • .**>*• -*

06 Statistical nuclear spin assignments

•A, tt J;i2

'fn

Figure 5.1: Limits on the spin J of a nuclear level L can be inferred from a) the spins 7ft (ib = 1,..., n) of the (final) states to which L decays, and from b) the spins Jit (k = 1,..., m) of the (initial) states that decay to L.

levels. Fig. 5.2 is presented to suggest that conclusive evidence on level spins might be obtained by statistically handling the information about the many 7-decay branches observed with Ge detectors. Fig. 5.2 is a graphical representation of the data presented in table 5.1. The second column of this table gives the spins J{ of the levels to which the 47V level at Ex = 6.48 MeV decays directly [3]. The first entry corresponds to the strongest (in %) decay branch, the second to the next strongest, etc. The third column lists the average final spin (Jf) when the n strongest branches are taken into account, thus

(5.1) n fc=i Fig. 5.2a shows a plot of (Jf) as a function of n. It can be seen to converge Average spins 97

p 7t- ) "V E. = 6.48 MeV, J -. 5/2

b) Al Ex ^ 7.464 MeV, J -- 3

c) Na K. •--- 9.61 MeV. J = 3/S

1L::

Figure 5.2: Average final spins of 7-transitions from states of 47V, 26Al and 23Na as a function of the number of decay branches taken into account (see text). 98 Statistical nuclear spin assignments

Table Ei.l: Final spin averages 47y 26 Al 23Na Ex = 6.48 MeV Ex = 7.464 MeV Ex = 9.61 MeV J = 5/2 J* = 3+ J' = 3/2+ n° 2Jf 2(Jf) Jt {Jt) 2Jf 2(Jf) 1 3 3.0 4 4.0 5 5.0 2 5 4.0 4 4.0 3 4.0 3 3 3.7 3 3.7 5 4.3 4 3 3.5 2 3.3 5 4.5 5 3 3.4 3 3.2 5 4.6 6 7 4.0 2 3.0 3 4.3 7 3 3.9 2 2.9 1 3.9 8 7 4.3 2 2.8 3 3.8 9 3 4.1 4 2.9 5 3.9 10 7 4.4 3 2.9 1 3.6 11 7 4.6 3 2.9 1 3.4 12 5 4.7 3 2.9 3 3.3 13 5 4.7 4 3.0 1 3.2 14 3 4.6 3 3.0 3 3.1 15 7 4.7 5 3.1 3 3.1 16 7 4.9 2 3.1 3 3.1 17 5 4.9 2 3.0 1 3.0 18 7 5.0 3 3.0 3 3.0 19 7 5.1 4 3.1 20 7 5.2 4 3.1 21 5 5.2 4 3.1 22 7 5.3 3 3.1 23 3 5.2 3 3.1 24 7 5.3 2 3.1 25 7 5.3 2 3.0 26 5 5.3 4 3.1 27 3 5.2 3 3.1 28 5 5.2 1 3.0 29 4 3.0 30 4 3.1 31 3 3.1 32 5 3.1 33 1 3.1 34 2 3.0 35 3 3.0 36 3 3.0

"Number of decay branches taken along in the average, see text. Weighted averages 99

rapidly to the dotted line at J = 5/2, which in fact is the spin of the Ex = 6.48 MeV level previously determined from angular distribution measurements [3]. With increasing n the value of (Jf) becomes so stable that it is hard to imagine a mechanism by which (Jf) could possibly move to one of the other dotted lines, corresponding to the neighbouring values J = 7/2 or J = 3/2, when more 26 branches are added. Similar graphs for the A1 level at Ex = 7.464 MeV [4] and 23 the Na level at Ex = 9.61 MeV [6] are given in figs. 5.2b and 5.2c as examples from other nuclides; the numerical data for these cases are also presented in table 5.1.

5.3 Weighted averages

The convergence of (Jf) to one of the possible spins, however, is not in all cases equally straightforward. This is demonstrated by the triangles in fig. 5.3. For the + 26 Ex = 7.20 MeV, J* = 1 level of A1 (see fig. 5.3b), the average spin approaches the value J = 1.5, instead of the expected value J = 1. This discrepancy may be attributed to the fact that there are considerably more J = 2 than J = 0 levels known in 26A1; this of course pushes the average upwards. In order to compensate for this unbalance we calculated the weighted average w- = IX-wi;^, (5-2) *=i *=i where f jflt is the fraction of the states known to have spin Jf *, which is populated via the n strongest decay branches. The values of (Jf)w as a function of n are presented in table 5.2 and plotted as open circles in fig. 5.3b. Similar plots for levels of 47V and 23Na are presented in figs. 5.3a and 5.3c. Obviously the weighted averages (Jf)w converge much better to the known spins than the unweighted averages (Jf). For the examples presented in fig. 5.2 the difference between (Jf) and (Jf)w is negligible, since in these cases (as in most others) the numbers of 'neighbouring' spins are almost equal. In the following discussion we will therefore skip the unweighted average (Jf) and exclusively use the weighted average (Jf)w. Several refinements of the straightforward calculation of (Jf)w have been considered. (i) Inclusion of the intensity of the branches in the weighing procedure, such that strong branches determine (Jf)w more heavily than weak branches. This possibility has been discarded, primarily because it does not work: one strong (e.g. 99%) A7 = 1 branch would imply (Jr)w ^ J and thus lead to an erroneous conclusion. This is due to the fact that such a procedure 100 Statistical nuclear spin assignments

5 - a) *7V E, = 6.30 MeV, J = 3/2

Weighted average Zw Normal average 2

ID 15 20 25

2 - b) ~Al E. = 7.20 MeV. J = 1

Weighted average Normal average

10 15

c) "'Na £; = 9.306 MeV, J = 7/2

Weighted average B, Normal average 2

10 15

Figure 5.3: Normal and weighted averages of final spins as a function of the number of decay branches taken into account. Weighted averages 101

Table 5.2: Comparison of weighted and normal average of final spins Tfi 23 47V A 1 Na E < = 6.30 MeV E, = 7.20 MeV Ex = 9.396 MeV J* = 3/2" J* = 1+ J* = 7/2"

n" 2J, 2(Jf) 2(Jf)w Jt (Jf> (Jf)w 2J, 2(Jt) 2< Jf)v 1° 3 3.0 3.0 0 0.0 0.0 5 5.0 5.0 2 3 3.0 3.0 2 1.0 0.5 5 5.0 5.0 3 1 2.3 1.8 0 0.7 0.3 5 5.0 5.0 4 3 2.5 2.0 2 1.0 0.5 7 5.5 5.8 5 5 3.0 2.3 1 1.0 0.6 7 5.8 6.1 6 3 3.0 2.4 3 1.3 0.8 5 5.7 6.0 7 5 3.3 2.6 2 1.4 0.9 5 5.6 5.9 8 5 3.5 2.8 2 1.5 1.0 9 6.0 6.9 9 1 3.2 2.4 1 1.4 1.0 9 6.3 7.4 10 5 3.4 2.5 2 1.5 1.0 5 6.2 7.3 11 5 3.5 2.7 1 1.5 1.0 5 6.1 7.2 12 3 3.5 2.7 3 1.6 1.1 9 6.3 7.5 13 5 3.6 2.8 0 1.5 1.0 5 6.2 7.4 14 1 3.4 2.5 2 1.5 1.0 5 6.1 7.3 15 5 3.5 2.6 3 1.6 1.1 7 6.2 7.3 16 5 3.6 2.7 0 1.5 1.0 17 1 3.5 2.5 3 1.6 1.0 18 5 3.6 2.6 1 1.6 1.0 19 3 3.5 2.6 2 1.6 1.1 20 5 3.6 2.6 1 1.6 1.1 21 5 3.7 2.7 3 1.6 1.1 22 3 3.6 2.7 3 1.7 1.2 23 3 3.6 2.7 24 3 3.6 2.7 25 5 3.6 2.8 26 5 3.7 2.9

"Number of decay branches taken along in the average, see text.

contradicts the statistical nature of the method in which the large number of branches rather than the specific properties of one or two of them should determine (J/)w. (ii) Inclusion of the 7-ray energies in the calculations, e.g. by dividing out the dominant E% or E* factor before arranging the decay brunches in order of strength. This refinement is closely related to the one above, and has therefore also been discarded. (iii) Inclusion of a factor that takes into account nuclear structure effects of a non-statistical nature, e.g. the fact that in light nuclei the high-spin states 102 Statistical nuclear spin assignments

have on the average higher excitation energies than lower-spin states and thus will be under-represented in the averaging procedure. In the first place this refinement in most cases apparently is not necessary (see below), and it would also (like the other suggestions) counteract our aim: the formulation of a set of rules for statistical spin assignments which is as simple as is compatible with efficacy. In this respect it should be stressed that the calculation of (Jf)w over the whole range n — 1,... ,N, required for drawing figs. 5.2 and 5.3, was only introduced to demonstrate the converging character of the procedure. For the actual spin assignments only the final weighted average (Jf)w for all N observed decay branches is relevant.

5.4 Chi-squared tests

Obviously the most likely spin for the decaying level is the one closest to (J[)w the weighted average over all the observed decay branches. A crucial question, however, is: what is the maximum difference |( Jf)w—J| still compatible with the notion of a 'unique' J-assignment. In order to answer this question we have performed a number of x2 tests on the distributions of the final spins. (Strictly speaking, they are 'x2-like' tests, because our uncertainties are not standard deviations.) In the discussion below, the symbol x2 stands for the reduced x2-

5.4.1 Resonances Fig. 5.4 presents the distributions of Jt for the decay of three randomly selected resonances in each of the reactions 46Ti(p,7)47V [3], ^(poO^Ca [5], 25Mg(p,7)26Al [4] and 22Ne(p,7)23Na [6]. For each of the twelve resonance levels, 7f (or 2Jf for the odd-A nuclides), i.e. the spin of the final states to which the 7-decay of the level proceeds, is plotted along the abscissa. The ordinate gives the fraction ijfk of all the levels known to have spin Jtk, that is populated in the decay of the resonance level studied. The error bars correspond to a unit uncertainty in the number of observed transitions. For the non-selfconjugate nuclei the distribution can in first order be ap- proximated by a three-bin wide rectangle (see e.g. the upper right section of fig. 5.4). It contains the final states with spins Jt = J — 1, J and J +1 to which the level with spin J can decay via dipole transitions. In a self-conjugate nucleus many of the dipole transitions are forbidden or retarded, and quadrupole (E2) transitions compete favourably. This typically leads to (relatively small) contributions in the J — 2 and/or J + 2 bits; i.e. the 'hat' function in the other sections of fig. 5.4. Such a 'hat' function, in which the crown/brim ratio has been deduced in lowest order from the ratio of the numbers of dipole and quadrupole Chi-squared tests 103

1 ' 100 T *'v S, = S.06 UeV J - 5/2 * V B* = 8.428 MeV. J= a/a " 50

50 .1 .

0 I t 0 . ' L

. . 1 , . . 1 . . 1 •

21,

*°Cn B, - 9.14 UBV. J - 2 -c. z.-8.985 UeV. J - 1 100 100 • 1 j 50 { r _ so -

0 0 /

^ « 7.37 UoV, j= 4

-nr

Figure 5.4: Fits of a 'hat' function to the spin distributions of the states to which the 7-decay of levels of 47V, 40Ca, 26A1 and 40Ca proceeds. The data points indicated with open circles have uncertainties of 100%. 104 Statistical nuclear spin assignments transitions observed in the decay of all levels of that nucleus, was fitted to the experimental distributions by centering the 'hat' on all possible J-values. For J = 0 the 'hat'-function was adapted to take into account the impossibility of J = 0 —» 0 7-transitions. Note that only the height of the 'hat' funtion was varied in the least-squares adjustments. Fig. 5.5 gives the values of the x2 asa function of the assumed J-value for the distributions plotted in fig. 5.4. The minimum x2 in each distribution, xiUii of course indicates the most probable spin. Since most of the \2 distributions are very steep, the precise choice of the criteria for the rejection of alternative possibilities, is relatively unimportant. We have finally chosen to reject the spin 2 values for which x > 2x£un- For a large fraction of the levels («90%) this procedure leads to a unique spin, in the remaining cases to only two possible J-values. The x2 > 2Xmia rejection criterion corresponds to the (x2 + l)-criterion for xfnj,, = 1. For the resonances studied so far, we find an overall average xj^n = 1.20. This is remarkably close to unity in view of the fact that our choice of the uncertainties (see above) is rather arbitrary. Although fitting the 'hat'-function to all measured Jf-distributions as described above is quite simple, an even simpler procedure also leads to acceptable results, i.e. the straightforward calculation of (Jf)w. If one accepts an assignment J for the decaying resonance as unique when {Jf)w falls in the range J — 1/3 to J +1/3, the conclusions are very similar to those of the x2-tests. The criterion chosen implies that the difference between {Jf)w and the next-closest spin J* is at least twice as large as the difference with J, or

|{Jf)w - J*\ > 2|(Jf)w - 7|. (5.3) If this criterion is not fulfilled the two possibilities J, J* are accepted. Chi-squared tests 105

- 7.37 U*V. J - 4 "US,- BOH U«V. 1- t

40 _r-_

\ 30 J -

_ 0 u

Figure 5.5: Chi-squared of the 'hat'-function fits in fig. 5.4 for various spin assumptions. 108 Statistical nuclear spin assignments

Table 5.3: Comparison of traditional and statistical spin assignments to resonances in the 25Mg(p,7)26Al reaction a Ev Ex N J*;T" •t (keV) (MeV) X2 Combined 317 6.61 25 3";0 3 3 3 390 6.68 26 2+;0 2 2 2 435 6.72 28 4-;0 (3,4) 4(3) 4(3) 497 6.78 34 2";0 2 2 2 503 6.79 26 3";0 3 3 3 515 6.801 25 3+;0 3 3 3 516 6.802 12 1+(1~,2-);1 (0,1) 1 1 533 6.818 20 4+;l 4 4 4 567 6.85 35 2+;l(+0) 2 2 2 591 6.87 16 l+;0 (0,1) 1 1 593 6.88 35 2+;l 2 2 2 656 6.94 22 l+;0 (0,1) 1 1 685 6.96 37 3~;1 3 3 3 723 7.00 35 2+;0 2 2 2 738 7.02 21 5+;0 (5,6) 4C C 775 7.05 32 3+;0 3 3 3 811 7.086 32 i~;l (0,1) 1 1 819 7.093 26 2+;0 2 2 2 835 7.11 27 4-;0 4 4 4 870 7.14 24 2";0 2 2 2 881 7.15 31 3+;0 (2,3) 3 3 890 7.16 31 3";0 3 3 3 896 7.17 31 4~;0 (3,4) 4 4 928 7.20 22 l+;0 (0,1) 2(1) 1 953 7.22 21 5+(4+); 1 (4,5) 5(4) 5(4) 969 7.24 39 3~;0 3 3(2) 3 986 7.25 41 2~;l(+0) 2 2 2 1025 7.29 29 4+(3+);0 (3,4) 4(3) 4(3) 1043 7.31 38 2+;l 2 2 2 1084 7.35 27 4~;l(+0) 4 4 4 1103 7.37 24 4+;0 4 4 4 1135 7.397 34 2+;0 2 2 2 1137 7.399 36 3~;1 3 3 3 1148 7.41 32 4~'l(+0) 4 4 4 1164 7.43 31 3+;0 (3,4) 3 3 1184 7.44 30 (0,1) 1 1 1196 7.455 24 l+;0 (0,1) 1(2) 1 1205 7.464 48 3+;0+ 1 3 3 3 1237 7.495 45 3+;0 + l 3 3 3 1239 7.497 29 2~;0(+l) 2 2 2 1283 7.54 32 2~;1 2 2 2 1292 7.55 17 5-;0 5 5 5 Chi-squared tests 107

Table 5.3 (continued) a Ep Ex N J*;T" •/statin 1 (keV) (MeV) (Jt)yt X Combined 1302 7.558 34 2+;0 2 2 2 1306 7.561 42 2+;l 2 2 2 1337 7.59 34 4+(3+);0 (3,4) 3 3 1342 7.596 29 4+;0 4 4 4 1351 7.604 43 2-,0(+1) 2 2 2 C 1370 7.62 25 l+;0 (1,2) ¥ 2 1375 7.63 28 5+;l (5,6) 4(5) 5 1515 7.76 34 3";0 3 3 3 1525 7.772 44 3+;0 3 3(2) 3 1568 7.81 27 i+;0(+1) (0,1) 1 1 1580 7.82 28 4~;0 4 4 4 1587 7.83 32 4+;0 4 4 4 1622 7.865 29 2+;0(+1) 2 2 2 1632 7.874 35 3+;0 3 3 3 1637 7.88 19 l+;0 +1 (0,1) 1 1 1649 7.89 35 4+jl 4 4 4 1680 7.92 19 5+;0 (5,6) 5 5 1699 7.94 43 3+;l 3 3 3 1714 7.95 44 4+;l 4 4 4 1744 7.98 45 2+;l 2 2 2 1763 8.00 35 1~;1 (0,1) 1 1 1771 8.008 44 2+;0 2 2 2 1774 8.011 20 5~;1 5 5 5 1800 8-04 20 l~;0 (0,1) 1 1 1811 8.05 40 3";0 3 3 3 1829 8.06 37 2 2 2 1833 8.067 19 5";1 5 5 5

"Number of decay branches taken into account, see text. »Ref. [4]. eSee sect. 5.4.1.

Table 5.3 compares the conclusions reached along the two lines discussed above with the spins assigned on the basis of the combined evidence of all the available data from many different reaction studies in which, of course, the 25Mg(p,7)26Al reaction also plays an important role. The resonances in the reaction 25Mg(p,7)26Al of which the decay is known in insufficient detail (JV < 10) have been excluded. Comparison of columns 4 and 7 shows that for almost all levels the present and previous assignments are in perfect agreement. The double-valued entries (some in column 4, some in column 7, some in both) are not conflicting. Only for the Ep = 738 and 1370 keV resonances the two columns give 108 Statistical nuclear spin assignments conflicting spins. The previous spectroscopic assignments to these levels have been confirmed by detailed comparison with shell-model calculations [4]; there can be no reasonable doubt as to their correctness. Therefore, as explained below, these cases therefore reveal the limitations of the new method. The J = 4 assignment to the Ep = 738 keV J* = 5+ resonance can be ascribed to the neglect in the present method of the strong influence of isospin on the decay modes of 26A1 levels; the T = 0 resonance in the present analysis only seems to fail to decay to low-lying J* = 5+ and 6+ levels, since there are no 26 such levels with T = 1 in A1. In this context we should also mention the Ep = 1337 keV, J*;T = 4+(3+);0 resonance, which is assigned 7 = 3. The Ep + = 1370 keV resonance decays very weakly to the J*;T = 3 ; 1 states at Ex = 4.19 and 4.60 MeV. The decay strengths of 2.0 and 1.0 W.u., respectively, are compatible with the 10 W.u. recommended upper limit for E2iv transitions in 26Al. Apparently, in this case the exclusion of decay strengths from the analysis plays an important role. It should be emphasized that the above two resonances represent quite extreme cases of their respective kinds. The new method in its current simple form, is thus in fact remarkably robust. Futhennore, in non- selfconjugate nuclei like 23Na and 47V, the influence of isospin on the decay modes is negligible. Comparison of columns 5 and 6 of table 5.3 shows that the simplest method (weighted averages) is less selective than the x2 analysis. This is mainly due to the fact that the latter method also distinguishes between J = 0 and 1 for levels of this even-A nuch'de (and similarly also between ,7=1/2 and 3/2 for odd-.A nuclides). Therefore we will skip the weighted averages in the following discussion and use the \2 analysis exclusively. The resonances in 23Na, 40Ca and 47V with 10 or more observed decay branches will not be discussed in detail here. We only mention that for all 45 resonances the present and previous assignments are in agreement. The levels for which the present method yields more information that the classical analyses will be discussed in sect. 5.5.

5.4.2 Bound states The fact that the levels discussed in sect. 5.4.1 are (p,7) resonance states and therefore unbound, is not relevant in the above discussion of statistical spin assignments. In principle, the method can thus be used just as well for J- assignments to bound states. The problem is, however, that the number o\ bound states for which a sufficient number (> 10) of 7-decay branches is known is very limited. In the 17 cases where this condition is fulfilled (all in 26A1), the statistical spin assignments confirm the spins determined previously, except foi 2 the Ex = 5.51 MeV level, where the present x analysis gives J = 3 instead o New assignments 109 the previous value J = 4. This may be ascribed (see below) to the fact that nuclear structure effects of a non-statistical nature axe more important in the lower part of the spectrum than in the resonance region. For the bound states, however, another procedure is more fruitful in which not the final spins Jt of the decay 7-transitions but the initial spins Jj of the feeding 7-transitions are statistically treated in the way discussed above. As an example we consider the bound states of 26A1 that axe excited in the decay of 10 or more higher states (resonances and bound states); they are listed in table 5.4. For none of the 64 levels, including the Ex = 5.51 MeV level (!), the statistical spin assignment conflicts with the previous assignment. Similar tables have been prepared for the bound states with n > 10 of the other nuclides. No disagreements have been found.

5.5 New assignments The statistical spin assignments to almost 200 nuclear levels of 23Na, 26A1, 40Ca and 47V that have been discussed above, confirm in practically all cases the previous assignments based on the combined evidence of all the available experimental information. For only two levels we have found conflicting spins. On the basis of these results, we estimate the confidence level for the statistical assignments outlined above at least 99%. It therefore seems worthwile to summarize the levels for which the present method yields more information that the classical methods. These new assignments are listed and compared with the previous assignments in table 5.5.

5.6 Summary and conclusion The spins of almost 200 levels of the nuclides 23Na, 26A1, 40Ca and 47V with at least 10 decay or feeding 7-ray transitions to or from states with known spin, have been assigned via a \2 analysis of the distributions of the spins of the final and initial states, respectively. The method is essentially based on the combined evidence of many 7-ray transitions instead of on specific properties (like strength, angular distribution, polarization) of one or a few transitions. For the cases studied so far, the results coincide very well with the spins assigned along traditional lines. The confidence level of the new spin assignments is estimated at >99%. The 11 levels for which the present method yields more information than the previous one are listed in table 5.5. The method may also be used for a consistency check of previous J-assignments. 110 Statistical nuclear spin assignments

Table 5.4: Comparison of traditional and statistical spin assignments to 26A1 bound states a a Ex(MeV) N J';T" J.tati.t" £x(MeV) n J*;T" J»t*t\ttc 0 31 5+ 5 4.71 33 4+;l 4 0.23 22 0+ 1(0) 4.77 31 4+ 4 0.42 60 3+ 3 4.940 19 1- 2(1) 1.06 34 1+ 1 4.941 12 5+ 5 1.76 49 2+ 2 4.95 30 3+ 3 1.85 34 1+ 1 5.007 19 2" 3(2) 2.069 43 4+ 4 5.010 14 1+ 1 2.070 42 2+;l 2 5.13 28 4+;l 4 2.072 31 1+ 1 5.14 41 2+;l 2 2.37 56 3+ 3 5.20 13 0+;l 0 2.55 53 3+ 3 5.25 31 4+ 4 2.66 42 2+ 2 5.40 24 4" 4 2.74 35 1+ 1(2) 5.43 13 1- 1(2) 2.91 49 2+ 2 5.457 30 3- 3 3.07 45 3+ 3 5.495 27 2+ 2 3.16 54 2+;l 2 5.51 20 4+ 4 3.40 20 5+ 5 5.55 31 2+;l 2 3.51 12 6+ 6 5.59 13 1- 1 3.60 46 3+ 3 5.60 22 3- 3 3.67 34 4+ 4 5.67 17 1+ 1 3.68 47 3+ 3 5.68 11 4" 5(4,6) 3.72 29 1+ 1 5.69 34 3" 3 3.751 46 2+ 2(3) 5.73 29 4+;l 4 3.754 13 0+ 0 5.85 21 2+ 2 3.96 51 3+ 3 5.88 11 3+ 3(2) 4.19 56 3+;l 3 5.916 25 2~ (2,3) 4.21 37 4+ 4 5.924 24 4+;l 4 4.35 44 3+ 3 5.95 12 1- 1 4.43 33 2~ 3(2) 6.03 13 l(+) 1 4.55 41 2+;l 2 6.08 13 5+ 5 4.60 52 3+;l 3 6.28 14 3+ 3(2) 4.62 32 2" 2 6.36 21 3+;l 3

"Number of feeding branches taken into account, see text. *Ref. [4]. 'Based on the x2 method, see text. References 111

Table 5.5: Statistical spin assignments to 47V, 2SA1 and 40Ca leveb

c Nudide £p(keV) Sx(MeV) N" J*;T* J,t,ti«t 47y 2.77 10 1/2" (3/2-) 1/2 3.77 12 1/2 (3/2",5/2-) 1/2 4.2718 10 1/2 (3/2") 1/2 4.35 12 1/2 1011 6.16 16 5/2C+) (3/2+) 5/2 1045 6.19 26 3/2 (5/2") 3/2 1127 6.27 26 n ft\(—\ SE lt\\ 3/2 o/4x } \vf2) 1209 6.350 33 3/2 1232 6.374 13 1/2 (3_/2)_ 1/2 26A1 516 6.802 12 1 <°Ca 2371 10.64 9 (3--5-) 4

"Number of branches taken into account, see text. *Ref. [3] for 47V, ref. [4] for 26A1 and ref. [5] for 40Ca. 'Based on the x2 method, see text.

References [1] P.M. Endt, P. de Wit and C. Alderliesten, Phys. Lett. B173 (1986) 1799 [2] C. van der Leun and G.J.L. Nooren, Phys. Lett. BQ8 (1981) 26 [3] H.P.L. de Esch and C. van der Leun, Nucl. Phys. A454 (1986) 1 [4] P.M. Endt, P. de Wit and C. Alderliesten, Nucl. Phys. A476 (1988) 333 and P.M. Endt, P. de Wit, C. Alderliesten and B.H. Wildenthal, Nucl. Phys. A487 (1988) 221 plus private communication [5] S.W. Kikstra, J.G.L. Booten, A.G.M. van Hees, A.A. Wolters and C. van der Leun, Nucl. Phys., to be published (see chapter 4 of this thesis) [6] E.L. Bakkum and C. van der Leun, Nucl. Phys. A500 (1989) 1 Appendix

The calculation of nuclear excitation energies from 7-ray energies. Given a set of n 7-transitions with recoil corrected energies E^ and uncertainties ?^, which cascade via m nuclear levels, one finds,

(i = 1... n, with m

K = Eb*"1?*** and (A.2) AEi = JlA-W, (A.3)

where S

Ml /us Samenvatting

Dit proefschrift beschrijft een experimenteel onderzoek naar de structuur van de atoomkernen 9Be, 40Ca en 42Sc. De resultaten zijn voor zover mogelijk vergeleken met theoretische berekeningen. De 40Ca en 42Sc kernen zijn onderzocht door ze in een aangeslagen toestand te brengen via vorming in een protonvangstreactie en de uitgezonden 7-straling bij het verval naar de grondtoestand te bestuderen. Hierbij wordt een trefplaat beschoten met versnelde protonen die geproduceerd werden door de Utrechtse 3 MV Van de Graaff versneller. Deze '(p,7)' reacties bieden de mogelijkheid zeer nauwkeurige informatie te krijgen over de eigenschappen van kernniveaus : (i) de resultaten zijn model-onafhankelijk omdat het reactieproces en de electromag- netische wisselwerking vrijwel exact bekend zijn, (ii) a) door de kleine energie- spreiding en de hoge intensiteit van de protonbundel kunnen selectief toestanden worden aangeslagen en bestudeerd, ook wanneer de werkzame doorsnede laag is, en b) de eigenschappen van de huidige 7-detectoren zijn uitmuntend, vooral in combinatie met een Compton-onderdrukkingsschild dat de invloed van storende achtergrondstraling belangrijk vermindert. Het idee achter de experimenten die beschreven worden in hoofstuk 3 is vervat in de volgende vergelijkingen

+ Sa, + Sp, en dus is = Sa — Sp + mp — mn, waarin m staat voor massa (in eenheden van energie), en 5„ and 5P respec- tievelijk de bindingsenergie van het neutron in 42Ca en de bindingsenergie van een proton in 42Sc zijn. De totale vervalsenergie van de JT;T = 0+; 1 —• 0+; 1 ß+ overgang tussen de grondtoestand van 42Sc en 42Ca is nauwkeurig bepaald door metingen van 5P in Utrecht en van S„ in Oak Ridge. De resultaten zijn ge- bruikt voor verificatie van de 'CVC hypothese', die impliceert dat de ft waarden van alle ^-overgangen van bovengenoemd type gelijk zijn [1]. Hoofdstuk 4 beschrijft een spectroscopisch onderzoek van 40Ca via de 39K(p,7)40Ca reactie. Dit resulteert in een aanzienlijke hoeveelheid gede-

MhJ 115 116 Samenvatting tailleerde informatie over zowel nieuwe als reeds bekende toestanden. De resultaten zijn vergeleken met schillenmodelberekeningen in het A = 37-41 mas- sagebied. Twee toestanden in 9Be zijn onderzocht via résonante absorptie van 7- straling. Een van deze niveaus kan niet worden bestudeerd via de conven- tionele techniek waarbij voor de energievariatie gebruik wordt gemaakt van de Dopplerverschuiving van de energie van 7's die worden uitgezonden door een, in een vangstreactie gevormde, bewegende kern. Het is echter aangetoond dat ook brede resonanties in een vangstreactie kunnen fungeren als bron van mono energetische 7's van variabele energie. De energie van deze 7's varieert met de energie van het inkomende proton, waarbij het energie-scheidend vermogen voornamelijk wordt bepaald door de dikte van de gebruikte trefplaat. De laagste twee T = 3/2 toestanden in 9Be zijn op deze wijze onderzocht. In hoofdstuk 2 worden deze experimenten beschreven en worden de resultaten vergeleken met analoog ß-verval en met schillenmodelberekeningen. De hoeveelheid betrouwbare experimentele informatie die tegenwoordig be- schikbaar is maakt het mogelijk nieuwe informatie indirect af te leiden via statistische analyses. Een nieuwe methode voor het toekennen van spins van toestanden op basis van het bestaan van 7-verval naar, en vanuit, toestanden met bekende spin wordt beschreven in hoofdstuk 5. Hoofdstuk 3 is gepubliceerd in Nuclear Physics [1], de hoofdstukken 2 en 5 zijn voor publicatie aangeboden aan dit tijdschrift [2,3], en hoofdstuk 4 zal worden aangeboden nadat de resultaten van enkele aanvullende metingen zijn opgenomen [4].

Referenties [1] S.W. Kikstra, C. van der Leun, S. Raman, E.T. Jurney en I.S. Towner, Nud. Phys. A496 (1989) 429 [2] S.W. Kikstra, S.S. Hanna, A.G.M, van Hees en C. van der Leun, Nud. Phys. (wordt gepubliceerd) [3] S.W. Kikstra en C. van der Leun, Nud. Phys. (wordt gepubliceerd) [4] S.W. Kikstra, J.G.L. Booten, A.G.M, van Hees, A.A. Wolters en C. van der Leun, Nud. Phys. (wordt gepubliceerd) Nawoord

Deze laatste bladzijden proefschrift zijn een belangrijk stukje proefschrift, niet in de laatste plaats omdat het in de meeste boekjes de enige beduimelde pagina's blijven, maar vooral om de illusie weg te nemen dat ik het allemaal alleen heb opgeknapt. Ten eerste wil ik mijn ouders bedanken, die mij de gelegenheid hebben gegeven te studeren en om dat te doen wat mij het beste leek. Mijn promotor, Cor van der Leun, wist de juiste balans tussen betrokkenheid en geboden vrijheid te vinden, hetgeen ik altijd als een belangrijk rustpunt heb ervaren. De ongedwongen wijze van samenwerken, en zijn grote inzet bij het afronden van het werk heb ik zeer gewaardeerd. I'd like to thank Stan Hanna for his active participation in the experiments and interpretation of the results described in chapter 2. I really had a great time during his visits to Utrecht. I remember quite vividly the 90 mph drive to Petten (to pick up the Be) in Aart's DS, where Stan didn't say anything about feeling just a little uncomfortable with the speed until we safely landed back in Utrecht. Liesbeth and I are still trying to reproduce the curry dinner he prepared for us in our student-home kitchen. S. Raman's visit to Utrecht also has left some tracks in my memory. The combination of writing up the results of chapter 3 and painting Utrecht red at night was very pleasant indeed. One image that comes to my mind now, is Ram calmly tearing the 'I want to fool around for a couple of years in the hope of discovering something by accident' cartoon off a theorist's door to add it to his collection, leaving question-mark faces behind. Ik heb de indruk dat veel mensen van Aart Veenenbos nog wel wat kunnen leren. In de eerste plaats heeft hij gedurende deze jaren de versneller en schrijver dezes 'on the road' gehouden. Onze detector-transportondernemingen naar Straatsburg en Munster waren behalve nuttig ook zeer aangenaam. De opgedane smokkel-ervaring kwam ons later goed van pas bij de invoer van soort- gelijke 'tot leering ende vermaeck'-apparatuur. Cees Alderliesten wil ik bedanken voor de voortdurende interesse in het werk, de stimulerende discussies en de nuttige suggesties betreffende de hoofdstukken

117 118 Nawoord

4 en 5. Met toenemende bewondering heb ik Prof. Endt aan het werk gezien voor zijn nieuwe A = 21-44 overzichtsartikel. Ik heb dankbaar gebruik gemaakt van zijn tussenresultaten. De samenwerking in ons 'Münster project' was voor mij een leerzame ervaring. Verder bedank ik hem voor het kritisch bekijken van de hoofdstukken 4 en 5. Aan inzet en interesse heeft het Flip van der Vliet nooit ontbroken. Altijd present en niet te beroerd om wat kritische noten te zingen, was hij altijd bereid de versneller en opstelling uit het slop te halen. Adri Michielsen maakte de vele trefplaatjes, waarvoor mijn dank. Op zijn onvermoeide pogingen om het Friese volk in een kwaad daglicht te zetten ben ik niet ingegaan, en zal dat ook niet doen totdat hij in smetteloos Fries 'Op 't Snitser skip der siet in skrie, dy 't mei de bek it spek fan 't spit óf snie; der wie gjin skrie dy 't dat sa die, dat er mei de bek it spek fan 't spit óf snie.' voordraagt. De studenten Ruud Geraets, Eric Kirchner, Bas Peeters en Maarten Slijkhuis hebben veel meet- en analysewerk verzet, vooral voor hoofdstuk 4. De discussies met Ruud en Eric, meestal over het nut van het heelal, waren hoogst onbevredi- gend. Heren van THEOTUE, woorden schieten mij tekort. De smaak van het Primus bier lijdt gelukkig nog steeds niet onder het feit dat er elke tweede week over hetzelfde wordt gediscussieerd. Ik ook nog niet. De wil Albert, Föns en Lex graag bedanken voor hun hulpvaardigheid en initiatieven bij de schillenmodelberekeningen van hoofdstuk 2 en 4. NiCO bedank ik hartelijk voor zijn onvermogen zijn enthousiasme over lAT^X te onderdrukken. De gezamenlijke sport-, eet-, film- en zeilactiviteiten zal ik niet gauw ver- geten. Ik kijk op een plezierige en leerzame tijd bij de vakgroep Kernfysica terug. Ik weet niet wie ik moet bedanken voor Liesbeth. Curriculum vitae

De schrijver van dit proefschrift werd op 14 augustus 1962 te Almelo geboren. Na het behalen van het diploma atheneum B aan het Christelijk Lyceum te Almelo, begon hij in September 1980 aan de studie natuurkunde aan de Rijksuniversi- teit Utrecht. Het doctoraalexamen experimentele natuurkunde met bijvakken wiskunde en informatica werd in September 1985 cum laude afgelegd. Vanaf oktober 1985 is hij als wetenschappelijk medewerker in dienst van de stichting voor Fundamenteel Onderzoek der Materie (FOM) werkzaam geweest bij de vakgroep kernfysica aan de Rijksuniversiteit Utrecht. De resultaten van het onderzoek dat in deze periode is verricht zijn in dit proefschrift beschreven. Een gedeelte van de tijd werd besteed aan het begeleiden van natuurkundepractica voor scheikunde en biologie studenten. In September 1987 werd de 'Sixth Cap- ture Gamma-ray Spectroscopy' conference in Leuven bezocht. Als kandidaat voor de hijksuniversiteit Utrecht werd hem in februari 1989 de Shell studiereis 1989 toegekend. In September 1989 werd hij door de stichting FOM in staat gesteld aan een managementtraining op Nijenrode deel te nemen.

119