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22 4

IF Wilkerson ei al. / lsospin-nonconserving particle decays

(13 n A n 37 " we summarize previous experimental work in this area, then we outline the remainder of the paper. Some remarks on the nomenclature for charge-dependent interactions') in the context of our research are relevant We have measured the proton decay of compound nuclei with Z = N + I (Z'= N' in residual nuclei), the a -decay of nuclei with Z = N (residual nuclei Z'= N'= N -2), and the proton decay of two of these latter nuclei . Therefore, whether one should use the terms "charge independence" or "charge syrnmetry" depends on both the decay mode that is investigated and on the presumed sites of the isospin mixing: in the compound nucleus, the residual nucleus, or in both . To avoid confusion, we therefore use the term "isospinnonconserving" (I NQ in discussing our measurements, consistently with the nomen2,3 clature of Ormand and Brown ) . Thus, we also do not distinguish between Coulomb (or other electromagnetic) and hadronic effects . 1 .1 . MOTIVATION

e now describe the three main areas in which recent progress has been made that motivate further measurements and analyses of INC particle decays in light nuclei . LLL Measurements of charge-symmetij, breaking in nucleon-nucleon scattering. Charge-symmetry breaking in n-p scattering has been accurately measured in polar4)] ized-beam/polarized-target experiments with 477 MeV neutrons at TRIUMF [ref. 5 and with 183 MeV neutrons at IUCF [ref. )]. Analyses of these results, especially the latter, suggest that (in addition to electromagnetic corrections) the main contribution to charge-symmetry breaking of the two-nucleon interaction, when described by the momentum-space meson-theoretic Bonn potential model '), is p-W meson mixing. Such meson mixing can also explain the apparent difference between n-n and p-p scattering lengths'). 1 .1.2. Nuclear-structure calculations with isospin-nonconserving forces . The realism and sophistication of nuclear-structure calculations with INC forces have increased significantly since the first reliable measurements ") of isospin-forbidden proton decays. The demonstration in the pioneering work of Adelberger et al. 9 ) that for proton and neutron decays isotensor (AT=2) mixing is comparable in size to isovector (AT= 1) mixing proved that a detailed treatment of the nuclear structure is necessary. (Recall that if a nucleon moves in the field of a T = 0 target nucleus only AT= I is allowed in first-order perturbation theory for a one-body Coulomb potential or for an isovector term in a nucleon optical potential .) This discovery shows the importance of many-body approaches for the nuclear-structure calculations. Later results of Wilkerson et al. ") confirmed this result . ecently there have been two types of detailed calculations of nuclear structure with INC forces . The first type consists of essentially exact, three-nucleon calculations with nucleon-nucleon interaction models that describe a wide variety of on-shell data. For example, Brandenburg et al. ") studied the 31j_3 He binding-energy

J. F. Wilkerson et al. / lsospin-nonconseruing particle decays

22 5

difference using a charge-dependent modification of the Bonn potential to fit values of the nucleon-nucleon scattering lengths and effective ranges . They showed that the experimental binding-energy difference can be explained by the charge dependence in combination with other known effects. The second type of nuclear-structure calculation has involved solving the manynucleon problem. For example, Blunden and Iqbal '') showed that anomalies in the mass-energy differences between mirror nuclei could be essentially explained by using charge-dependent nucleon-nucleon forces consistent with the TRI iJ F results. Ormand 2 ), and Ormand and Brown ;) have provided extensive analyses of INC erects in light nuclei using a shell-model approach with empirical INC interactions obtained from fitting the masses of isobaric multiplets 2.") . They '4) calculated INC corrections to isospin-allowed fl-decay Fermi matrix elements and predicted the isospin-forbidden Fermi-decay matrix elements for z4C1 and 34Ar. Ormand and Brown ;) also predicted INC nucleon-decay amplitudes, ®,NC, for nominally isospin T = 3 states to nominally T == 0 nuclear ground states . They used the s-d configuration space and an INC interaction ") that reproduces experimental isotopic mass shifts in the A > 22 region of the shell. As we describe in detail in subsect. 4.2, they estimated a range of decay amplitudes that is consistent with our preliminary report of the proton-decay amplitudes"") and with neutron-decay amplitudes . They predicted that the main source of INC nucleon decays is mixing of the T = state with many nearby T =15 states of the same JT in the compoundnucleus continuum, which is typically at S MeV excitation. Because the mixing is very sensitive to the detailed separations of the T = ; states from the T = state, which are not well described by the shell model, at least one order of magnitude variation in 0, NC is produced when the excitation energies of the states are moved through ranges consistent with model uncertainties. In order to help remedy this defect of the model, we attempted to measure for 2 ' Na the positions of these states, as described in subsect. 3 .4. Regrettably, for the nuclei considered there are no calculations of isospinforbidden a-decays of T = 2 states to T = 0 states, which measure isotensor amplitudes directly. Ormand and Brown predict that isotensor and isovector mixing at the nucleon-nucleon level are comparable, in agreement with previous schematic analyses of p-, n-, and a-decays 9 '"') . As an incentive to calculations of the decay amplitudes, we have accurately measured the a-decay widths, F,,, for the lowest-lying T = 2 states in 24Mg,2HSi and ;`'S . 1 .1 .3 . Advances in high-energy-resolution resonance experiments. INC particle decays of resonances are particularly useful in studying charge dependence because the resonance appears in a decay channel only if isospin is not conserved . Therefore, we have a very sensitivie technique. For comparison, in Fermi ß-decay the INC '4), and only corrections to isospin-allowed decays in the s-d shell are 0 .1% or less the rarely measured isospin-forbidden 8-decays can give information comparable in sensitivity to the resonance decays.

22 6

J. F tVilkerson et al. / Anspin-nonconserving particle decays

he most difficult aspect of I NC resonance experiments is the need to extract informadon about the wry narrow INC resonances that are embedded in a sea of uch broader isosomaHowednompound-nucleus resonances. In the past decade several advances in experimental techniques have greatly improved the accuracy of ur resonance measurements using the Triangle Universities Nuclear Laboratory T ) FN tandem Van de GraaR accelerator. We summarize the improvements briefly here, with more detail being given in sect. 2. First, polarized- proton beams of high energy resolution (FWHM < I keV) were 6. prepared and, for the first time, their energy resolution was measured , '7) The use of polarized proton beams to measure analyzing powers, A, .(O, E), in addition to cross sections, cr(0, E), greatly reduces ambiguities in resonance parameters, especially for narrow and poorly resolved resonances"") . Second, when desirable, an excitation function over a 10 key range and centered on a F = ? or T = 2 resonance can be obtained by applying short-term voltage ramps to the target at fixed accelerator energy and binning the data according to energy. This allows, for example, eReUs of longer-term energy drifts to be monitored and corrected. Some of the (p, p) data and all the (p, a), and (a, a) data reported here were acquired in this way. Third, wflien appropriate, ion-implanted targets at controlled implant depth and of known thickness can be prepared, as we have done for 2)Ne and 36Ar. We thus avoid the difficulties associated with controlling and characterizing gas-jet targets. Fourth, the excitation energies ef the T ;states, which are preferentially populated by (p, t) and ( p, 3 He) reactioas, have been accurately obtained by magneticspectrograph measurements using the Princeton University and Indiana University Cyclotron Facility (IUCF) cyclotrons 15,111, ' 9 ). By combining the precise resonance energies and a careful recalibration of the TUNL tandem energy scale 20), the excitation energies can be measured to an uncertainty of about 2 keV. 1 .2 . PR --.VIOUS EXPERIMENTAL WORK

1 .2.1 . T= z4 states . Systematic measurements of INC proton-decay partial widths, I'P , of nine low-lying T =~"2 states in nuclei with A- 9-41 have been reported in refs .''-';) . The available energy resolution and lack of A,. data limited the reliability of the resonance parameters obtained . Significant improvements in both of these areas were described in preliminary reports of some of the present results 10,15 ), and our work has doubled the number of proton-decay measurements . We have also extracted the INC partial widths and INC amplitudes using consistent experimental procedures and analysis techniques. INC-neutron-decay measurements of partial widths, Fn , for T=='2 states are important adjuncts to those for proton decay. A difference between n- and p-decay amplitudes measures isovector mixing, whereas their average measures isotensor

J.F. Wilkerson et al. / Isospin-nonconserving particle decays

22 7

plus compound-nucleus mixing. Neutron decay results for A = 9-25 are summarized in ref. 24) . 1.2.2. T=2 states . Prior work on the isospin-forbidden ac-decay of T=2 states in light nuclei (A = 8-44) had resulted in r,, values being assigned to eight 25-28), states with upper limits assigned for several more states'`') . Most of these states were populated by isospin-allowed ( , t) reactions 'S'`') and the decay Ranching ratios, 0/I', were determined, but F and Q could only be estimated. In refs . 26' 29) resonance experiments were performed to measure both r,, and r. In both experiments the energy resolution was poorer than that available for the present experiments. 1.3. OUTLINE OF THE PAPER

This paper is organized as follows. Experimental procedure, for identifying T = 32 and T = 2 states in light nuclei are described in sect. 2. There -. re aiso describe the preparation and characterization of high-energy-resolution proton and alphaparticle beams for resonance measurements, target preparation, and data-acquisition methods. In sect. 3 the methods used for the resonance analyses (including resolution-broadening effects) are summarized, isospin-nonconservation probabilities are extracted, and efforts to determine spins and parities of T = 21 states that could mix with a T ;state are described. Comparisons with theoretical models are made in sect . 4, then a summary and conclusions from this work are presented in sect. 5. A preliminary account of some of the results presented here was given in ref. Details of much of the present work are available as theses in refs . 15,29,30) . 2. Experimental roteures carious experimental procedures are required for measuring INC charged-particle decays in light nuclei by resonance reactions. These procedures are summarized here and are described in detail elsewhere 15,18,29,30) . The first challenge is to identify the T = ='2 or T = 2 states in the compound nucleus of interest . The methods we used to identify the states are summarized in subsect . 2.1 . In subsects . 2.2 and 2 .3 we describe the preparation and characterization of beams and targets for resonance experiments, and in subsect . 2.4 the data-acquisition methods are summarized . 2.1 . IDENTIFICATION OF 1= 23 AND T=I STATES

The only indication of the presence of an INC decay in a resonance experiment is the relative narrowness of the resonance at the appropriate bombarding energy compared with neighboring states whose decay is isospi n-al lowed. However, at the bombarding energies of interest, isospin-allowed high-spin resonances can be narrowed by small centripetal-barrier penetrabilities, so this identification method is ambiguous if the orbital angular momentum of the resonance channels is not known. Also, the excitation energy of the state should be consistent with Coulomb

IF Uilkerson et al. / Isospin-nonconserving particle deetkys

228

is lace ant energies from isobaric-analog states . The displacement energies are erturbed su ticntly by nuclear structure and INC effects ") that they cannot be sed to inpoint the very narrow states involved (1'< I keV) . he J ' the coWound state should match that of parent isobaric-analog states 102 ), using in adjacent nuclei . Our earlier measurements with polarized beams" for isospin-allows d decays "), were the first to show !hat techniques suggested analysis of high-resolution resonance (p,p) excitation functions with polarized beams measured at several angles leads to unique J' assignments. The lowest-lying -- 2 states in even-even nuclef have J ' = 0% since they are analogs of adjacent ground states, and thus must have an isotropic resonant a-decay amplitude. These 0` states do not contribute to analyzing powers in the (p, a) reactions used to opulate them . 2.1.1. 77ie (p, 1) and (p, -'He) reactions for T= 2;states . The isospin-identification techniques described above can be used as part of the resonance measurements . owever, there is suMcient ambiguity in identifying the T = 2 states that different signatures are also needed. The lowest-lying T state in a Z = N + I nucleus can usually be reliably identified from P-delayed proton decay 34 ) . For higher-lying T = -32 states this method fails because the P-decay branching ratio is too small to compete with -y-decay . We therefore turned to (p, t) and (p, - He) reactions to provide the necessary isospin selectivity. Here we summarize the method and results, the details being given elsewhere 15,18,29) . t suMciendy high energies that Coulomb energy and Q-value differences have negligible effects, the (p, t) and (p, 3 He) reactions on T = ; nuclei that form T = z hnal states are expected to proceed uniquely by enhanced AT= 1, no-spin-transfer (,,AS = 0) transitions having the same angular distribution, characteristic of the orbital angular- momentum transfer, Q in the reaction . For transitioas iv T -;- 21 staies, on the other hand, admixture with AT = 0, AS = 1, which can occur in (p, 'He) but not in (p, t), is expected to remove the agreement of the angular distributions. an experiment at the Rinceton University Cyclotron Laboratory we used 2.4 MeV protons incident on targets of `'; Na [ref. `y)], -''A1, `P, ' -"CI and 39K [refs. 15,18) ] , to produce t and 3 He which were detected at 15' (lab) by using a agnetic spectrograph with energy resolution for (p, t) of about 7keV and for 3 (p, He) of about 12 keV. Enhanced yields of similar magnitude in t and 3 He spectra were used to suggest several candidate T = ;states in each of '' Na (6 states), ''Al (6 states ), 29P (7 states ), 33C1 (5 states) and 17 K (3 states) . ngular distributions for (p, t) and ( p, 3 He) was measured at the Indiana University Cyclotron Facility (IUCQ [refs . 15,19)] in which unpolarized 59 .4 MeV proton beams bombarded an -''A1 target and 59.4 MeV polarized beams were used to bombard a 31 p target . A magnetic spectrometer provided an energy resolution for t and 3 He of between 30 and 80 iced', depending mainly on target thickness. For the candidate T = ;saws angular distribution of cr(O) and A, (0) measured from 6 * to at least 30' in about 5' steps were in good agreement with those predicted for the

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J.E Wilkerson et al. / Isospin-nonconserving particle

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229

appropriate analog states by DWBA calculations . The analyzing powers provided the most sensitivity to the .1-transfer, since the differential cross sections are mostly sensitive only to L. Several candidate T= ='2 states in 2 'AI (three probable states) 29p and (eight probable T= z assignments) were investigated by this means. In subsequent resonance measurements at TUN L no contradictions to the T 2 assignments from the (p, t) and (p, 3 He) experiments were found. 2.1 .2. Identification of T=2 states . Our primary means of identifying the lowest(24M g, 21ISi, and 32S) lying T = 2 states in the even-even nuclei that we investigated relied on previous (p, t) measurements of reactions populating the T = 2 compoundnucleus stateS25). Measurements of resonance-reaction excitation functions near these states have confirmed the exis..ence of very narrow J'=0 4- states at the 26 appropriate excitation energies in 24mg [refs . -35-37)], in 21 Si [ref. )] and in -12S [ref. -")] . Our measurements confirmed the excitation-energy and spin-pzrity assignments. 2.2 . PREPARATION OF HIGH-ENERGY-RESOLUTION BEAMS

The resonance experiments require high energy resolution in the entrance channel, but only sufficient resolution in the ground-state-decay channel of interest to resolve this state from higher-excited states and from background contaminants in the spectra. The entrance-channel requirement is particularly demanding for these INC :n 0.04 to 12 keV and average resonances, because their natural widths, F, range fro about I keV. Special techniques were therefore needed to ensure high energy resolution of the incident proton and 4 He beams. 2.2.1 . Beam preparation . Polarized proton beams produced by a Lamb-shift polarLed ion source -9) were accelerated in a HVEC model FN tandem Van de Graaff 40), then magnetically accelerator, momentum-analyzed in a dual-90' magnet system switched into a beam line containing the scattering chamber. In order to correct for accelerator terminal fluctuations of order 3 kV that occur at frequencies of up to loop 41) was used to produce 100 nA I kHz, a direct fast-feedback energy regulation polarized beams on target with a typical beam energy resolution (FWHM) of 600 eV at 4 MeV. Unpolarized proton beams of typical intensity I RA and FWHM 500 eV were obtained from ii direct-extraction ion source and th(,-'YU N L FN tandem accelerator . The electron stripping in the tandem terminal was performed in a gas, thus improving the energy resoution over that obtained with the foil stripper used for polarized beams 41 ). The way the resolution of poh6zel and unpolarized beams was measured

is described in subsect. 2.2.2. 4 An exchange ion source was used to produce He ions for injecting into the tandem accelerator. These beams were accelerated and momentum analyzed as for the proton beams. Typical beam energy resolutions of 900 eV FWHM at 6 MeV were obtained .

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J.F. Wilkersoti et al. / Isr).çpin-tionconservitig particle decays

231

normalization errors arising from drifts in beam -i ntegradon electronics and bea wander on no uniform targets. 23 . TARGET PREPARATION AND CHARACTERIZATION

For high-energy-resolution resonance measurements it is essential that beam straggling in the target contribute as little as practical to the overall energy resolution . Since the scattering yield increases proportionally to the target thickness, T, but the statistical energy straggling scales as there is a compromise thickness for each target material, beam type (p or a), resonance width, and bombarding energy . For INC decays typical natural resonance widths F are I kelf or less, and energy straggling smaller than F is desirable, therefore uniform target thicknesses of only a few hundred atomic layers are demanded. Thus, much care in target preparation and characterization was required in these experiments. of 23 24Mg, 21SSi' 3'r 32S 2.3 .1 . Target preparation. Targets Na, and were prepared by evaporation onto thin (3-10 Nag/cm') carbon backings . The `;Na was prepared from chemically pure NaCl, the 24 Mg was prepared from MgO reduced by lanthanum and tantalum, the 2XSio targets were prepared by electron-gun heating of SiO, the 31 p 32S targets were prepared targets were evaporated from red phosphorus, and the as Sb2S3- Isotopically enriched materials and precautions to a--)id contamination of the targets by contact with air were used when appropriate. of 2 Ion-implanted 2()Ne targets of 2.3 :1:0 .2 P-g/cm:! thickness "Ne on thin carbon 20 or aluminum backings were prepared by implantation of 15 keV Ne ions to produce a localized concentration of Ne atoms near the surface of the foil facing the beam. 3 The 'Ar targets were similarly prepared, but at an earlier time when we had less 36 control over the implantation process. The Ar was implanted at both 10 and 50 kail, resulting in a bimodal distribution of Ar atoms in the host carbon foils. The target gg/Cm 2, implanted thickness thus produced was estimated to be between Vg/CM' 1 .5 and 2.1 of the host material . Isotopic to a maximum depth corresponding to 13 separation of the implanted ions from the natural-abundance gases used in the implanter ion source was accomplished by the separator magnet on the implanter. (90.5% 2oNe) was used in lower-resolution measureA gas cell containing natural Ne state, ments to locate T = 2;states in the` 'Na compound nucleus near the first T ;state, as described in subsect. 3 .4. 2.3.2. Beam straggling in targets. In the very thin targets used, energy-loss fluctuations of the beam about the mean energy loss (beam straggling) produce significant contributions to the broadening of narrow resonances . If straggling is not carefully accounted for, both the line widths and the line shapes are unreliably predicted, because the straggling distribution is asymmetric about its mean, always having a long tail towards lower energies . 43,44), 0 We used the Landau-Vavilov methods 4, as included in computer programs to estimate the energy distribution of protons and alpha particles in the targets . These estimates are quite approximate, but are sufficient for our analyses described

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232

J.F IVilkerson ei al. / lso.,%piii-tiotieatiservii2g partiele decays

in subsect . 3 .1 because the FWHM of the straggling distribution was typically less ,han one-half the FWHM contribution from incident-bearo resolution . The effect of target straggling on the beam energy distribution is illustrated in fig. 1, for 2'Mg(a, 02"Mg at 6.1 MeV. 2.3.3. Doppler, lattice, and atoinic-e--ccitation broadening. The significant energy for resonance studies is the energy available in the center-of-mass (c.m.) frame. The target nuclei have a distribution of energies because of their thermal and lattice contributes vibrations, and ths distribution to broadening of the resonance. Ithough line broadening from thermal sources is well-known in slow neutron scattering 45), its effects might appear to be negligible at beam energies, E, in the eV rang, because thatmal-vi brational energies, E, are only of order 25 meV. owever, when computing Doppler shifts one first combines velocities, thence calculates energies . THs produces Doppler-broadening effects scaling as N/EE, _ 10-4 . which is the same or ier of magnitude as the beam-energy resolution, We therefore estimated the coradbudons to the overall energy resolution from thermal motions and lattice vibrations in the target atoms. Target temperatures were estimated by assuming that the targets lost the energy deposited in them by the beam only by radiation to the room-temperature environment . This is a good assumption because the targets are sufficiently thin that conduction of heat through them to the metal target holders is negligible . Typical target temperatures thus estimated were T---320-33010 . For the noble-gas ionimplanted targets used, 24 'Ne and I 'Ar, it is reasonable to use only this temperature to estimate E, since they are fairly loosely bound in the host lattice. The that al-vibrational energies, Q, for solid targets depend not only on the temperature but also on solid-state properties, especially if the Debye temperature,

H, ) , is higher than T For anisotropic materials the density of states of phonon modes *") should be used to estimate E, reliably. For example, because of the tight

in-plane binding of C in grailh4e, Q = 66 mey rather than about 25 meV that would result for free carbon atoms at the target temperature. For most of the solid targets other than carbon, we used 01) values from the literature 4h Y The resulting thermallattice contribution to resonance broadening produced a FWHM in a gaussian distribution ranging from 150 eV ("Ar+ p at 3.3 MeV) to 940 eV (12C +p at 14.2 MeV) . An example of this broadening effect, with a FWHM of 500 eV, is shown for 24 Mg( M a Y4 Mg at 01 MeV in fig. 1 . Similar effects have been extensively iwvesfigated in y-scattering 4") . tomic-excitation emens arising from sudden nuclear recoil within a target atom result in a distribution of the energies available for nuclear excitation. For a light , target nucleus (such as ' C) bombarded by a swift projectile (such as a 14 MeV proton) there is good matching of the recoil time through the atom to the K-electron eriod, prodvcing a high probability of atomic excitation . For the ligh ..est system that we studied (12C + P) we predicted an atomic-excitation distribution with FW of 6,00 eV, -3s described in detail eh.-~, where 3 '). Th,.- mechanisin of the atomic excitation and its emects on resonances have also been discussed by Brizgs and Lane 4M)

J. F Wilkerson et A / lsospin-nonconserving particle decays

233

and reviewed by Heintz 4y) . For the other resonances reported here, atomic-excitation effects are relatively negligible . The combined effects on resonance line shapes of beam energy resolution, beam straggling in targets, and of Doppler, lattice, and atomic-excitation effects, are described in subsect . 3 .1 .2. They are displayed in fig. I and also in fig. I of ref. 32) . 2 .4 . DATA ACQUISITION

The high-energy-resolution proton or alpha-particle beams, prepared as described in subsect. 2 .2, were incident on the thin targets (subsect . 2.3) placed in a 61 cm scattering chamber. The incident current was integrated in an electron-suppressed Faraday cup downstream from the target . A normalization uncertainty of less than 10% in absolute cross sections is estimated as arising from target-thickness and current-integration uncertainties. Outgoing protons or alpha particles were detected in silicon surface-barrier detectors subtending angles of about :i:1' in the scattering plane. From four to eight angles were measured simultaneously . For polarized-beam measurements, detectors were set at the same angles on the left and right Ode the beam . The beam polarization aRei die bearn had traversed the target was measured by using a 4 He(p, P)4 He polarimeter; Normalization uncertainties c°Î less than 3% were introduced into the A, data from uncertainties in the beam polarization . Standard electronics were used to shape the signal,-, from the detectors for computer processing . Some of the data acquisition was performed by the XSYS data-handling software for nuclear-physics experiments f;()) on a VAX 11/780 computer . Careful monitoring of the accelerator system parameters proved crucial to acquiring good data . Especially for the target-voltage-tamping scheme (subsect . 2.2.3), special electronic and software control routines were required, as described in detail in ref. 29). Because the conditions were quite different for high-resolution data than for the lower-resolution data also made for O'Ne(y p)'"Ne, we describe the latter measurements separately in subsect. 2.4.2. 2.4.1 . Nigh-energy-resolution excitation functions. For the excitation functions measured widi beam energy resolutions of I keV or better (subsect . 2 .2), special !teedcd to be taken. to achieve precise and reproducible beaksl energies for polarized or unpolarized protons and for alpha particles. For such measurements an energy reproducibility of the accelerator plus analyzing-magnet system of _±2 keV [ref. :!())] was achieved by systematically recyling the analyzing magnets through their hysteresis cycles . Also, during measurement of an excitation function the magnet fields were always increased steadily . )24Mg, 24Mg( P)20 23 P)23 ra, a Na , 23 Na(p, a )20Ne, For the 20Ne(p, Ne, Na(p, 31 )28Si " P(p, p)" P and P(p, a measurements, the target-voltage- ram pi ng, scheme (subsect. 2.2.3) was used to acquire excitation-function data after the resonance had been located . The magnet-control and voltage-tamping circuitry were under com-

a

23~

J.F L~ilker°.scrra et czl. / Isospira-rarrrac~raser°virï~ pcir°ticle decays

ute co t o , and con itional eve t sorting co 1 be use to iscri inate against events c wire w ile tire a alyzing- agnat et s were outside acceptable ranges `'y). or o ®line analysis of t e ig ®resolution a s acquired in the target-voltage ra ode, rogra s were writte ;") to allow is lay of 3- spectra, with ramp voltage an o tgoing- articâe energy as t e two in eperadent variables. y this ata from background-subtracted sans, ano alias in ata ca~ul e identifie an separate runs could e co i e reliably. ig -resolution measurements are shown epresentative excitatio f n~ .ào s fro )''C an in fig. 1 of ref.' ° ) for'~Si(p, p)' x Si in fags. 2-7, in fag. of ref. ~') o '' ( 'y over t e Tourt = ;state in

0.72 -+ t t,270

v.dcs ° t t,256

E P ( MeV )

° - . _._. . ~ . . _____ . ® 9 ~ .262~ ~ 11 .~266 ? 1.27D

F'ig . ~. Excitation functions of ~ and .4, near the first T = ;, Jn = i -, state in "~ for °~®(p, p)'~® at five laboratory scattering angles as a function of proton laboratory energy E~, measured by stepping the beam energy to Steps of 280 ev. The error bars are statistical. The curves show the best fit with resonance parameters in table 1 and an overall energy resolution with EV'de-I~~ 950 eV .

J. F: Wilkerson et al. / Isospin-nonconserving particle decays

235

20Ne (p,p) 2°Ne 0.7 0.6 0.5

,a C

6.868 6.870 6 872 6 874 6.876

Ep (MeV)

6 86.* 8.8700 '. 872 6.8-14

875

Fig. 3. Excitation functions of Q and A, near the lowest T = 2, J' = 2 + , state in 2' Na for (p, p) on an ion-implanted target of 2° Ne, shown at four scattering angles as a function of proton bombarding energy, measured by ramping the target energy (subsect. 2 .2 .3) in steps of 160 eV . The error bars are statistical. The curves show the simultaneous, best fit at all angles, using the resonance parameters in table 1 and an overall energy resolution with FWHM 850 eV.

20Ne(p, 2.4.2. P)'"Ne excitation functions with medium energy resolution . Excitation functions of cr and A, . were measured at eight angles in steps of about 7 keV over the bombarding energy range 6.4-7.7 MeV to provide an excitation function centered on the lowest T= state in `''lea . Three of the eight angles used correspond to zeros in the center-of-mass frame of the Legendre polynomials for L = 1 and L = 2 . A resonance of the appropriate L-value will show minimal effect in a (PL = 0) and maximum effect in A, . (PL maximum) at such angles, which should help to identify L (and thus 7r) of the resonance. The data for the gas-cell target have an overall uncertainty of about 12% for o- and about 10% for A, esona ce a

yses

The analysis of the excitation-function data whose measurement is described in sect . 2 required special care for the very narrow INC decays, for two main reasons.

Aerson et al. / Isospin-nanconserving particle decays

IE

236

23

NA

PO)2 Sa

23

Na(p,

20Ne

Is ~ M

1 .6 F4 1 .2 42 3.8 3.4

3902

3904 390 E (MeV) P

3 9"

3.0

3902

3.904 3906 EP (MeV)

3508

Fig . 4. Cross-section excitation functions for 2 'Na(p, p) 2-'Na and 23 Na(p, a ) 2"Ne near the lowest T = 2, .1 - = 0 4 , state in 2 " Mg at four laboratory scattering angles . The data were measured by target ramping in 90 eV steps, and error bars are statistical. Dot-dashed lines are linear-least-squares fits to the non resonant (r, and solid curves show best fits with resonance parameters in tables I and 2 and resolution FWHM of 1200 eV .

First, the excitation energies in the light compound nuclei involved are sufficiently low that there is no simple and reliable descrip ,cion of the off-resonance scattering. For example, there are often broad T = ' resonances nearby . Since the branching ratios of the INC decays into the proton and a-channels are often small, the resonances are usually revealed by interference, which requires describing nonresonance amplitudes as well as (r off-resonance. Therefore we used helicity formalisms (subsects. 3 .1 .1, 3 .1 .2) for describing (r and A,. . Second, the observed T=-' resonance excursions in the excitation functions are significantly broadened and distorted from the natural resonance shapes, because of the several resolution Was discussed 1 n subsects . 2.2 .2, 2.3 .2, and 2 .3.3, some of which produce distributions

Wilkerson et at / lsospin-nonconserving particle decays

237

24Mg(a'a) 21Mg

E (Me V) a

Fig . 5 . Cross-section excitation functions for 24 Mg(a, a )24 Mg near the lowest T= 2, J' = 0% state in Si at the five laboratory scattering angles indicated, measured by the target-vamping method (subsect . 123) in 180 A steps. Error bars am statistical . A linear, rather than constant, nonresonant a-variation with a-particle energy has been assumed . The solid curves show simultaneous fits at all angles, using resonance parameters in table 2 and an overall energy resolution with FWHM 1800 eV . 28

skewed about their peak values. Therefore careful convolution of resolutionbroadening effects with the resonance and resonance-interference effects was required (subsect. -3.1 .3). Searching for parameters to describe the resonances involved four resonance parameters (including the energy resolution) and three or four nonresonance parameters at each angle . The methods that we developed are described in subsect. 3.2. To estimate INC probabilities from resonaace partial widths required extracting

J.F Werson et al. / lsospin-nonconserving particle decays

238

M 80

Y

W

P)

28

Si 517

04

76

02

QO 08 0.6 04

04

4 4

20 0~ 9033

, ?04 D

1547

1

1~ ) 5

E ;D (

9D3',

v.(_ 4 r,

9047

905 ,2

eV ~

Fig. 6. Excitation functions of (T and A, M "SUN p)`'%i near the seventh é"= ~', J' _ +, state in ``'P four laboratory scattering angles, measured by stepping the beam energy in 485 eV steps. Error bars are statistical or are smaller than the points . The solid curves show best fits using resonance parameters in table I and an overall energy resolution of FWHM 1400eV. M

barrier-penetration effects and estimating single-particle reduced widths . These procedures and the results for all the states investigated are described in subsect . 3 .3. Finally in this section, we describe our attempts to assign J' to states near the lowest T= ', stw.e in 2 'Na . IL

EUCITY A

PLaUDES PLUS A SINGLE-LEVEL RESONANCE

ecause the INC resonances are much narrowcr. than other structures in the excitation functions, we can approximate the nonresonant "background" scattering

IF Wilkevson et al. / lsospin-nonconserving particle decays 32S( r, 520

P

)

239

J &S

Q04,-

61 .6 "

480

-. ~~l¢~ _~_

41~

4

ô ~ " .1;1, y

440k9r

1200

0.88

4

_

4

T

084--

on

S1

8C)P

U_ (mb/Sr) 70t- .

1402 0 1

Y

140.2 0

Am SO 070 1606

J OL 06 20 (A 3.3

1

41

I

3 372

1

3.375

(')!a 1

3 3 78 E

P

-

(_)21

337u

( MeV)

1

3.372

3

-

1

1

75

3378

:;2S(p, p) ;2S across the first T = 3 state in 3'Cl at four Fig. 7. Excitation functions of u and A, for 2 laboratory scattering angles in steps of 420eV obtained by stepping the beam energy. Error bars are statistical. The solid curves are best fits wing resonance parameters in table I and resolution FWHM of 710 eV, that were determined from the backward-hemisphere angles only . The dashed curve at 61 .6° is a ht obtained by varying only the nonresonant-scatteri ng amplitudes and using resonance and resolution parameters from the other angles .

amplitudes as constant over the energy interval of several I' within which the INC resonance elects occur. The background amplitudes are most simply characterized by complex-number helicity amplitudes, a(O) Or non-spin-flip scattering of protons or alpha particles, and NO) Or spin-flip proton scatted . Thus, at each angle (but independent of energy) there are at most four real parameters to describe the background scattering and its interference with the resonance scattering . For the

J 2 ()

. F:

~~'ilkersorr et al. ~ Ltii)Sn1r1-ril)ri('Or3 .fierL'rrl~, pai'tl(°Ie dec'ays

analyses reports ere there are si ple explicit formulas for reactions proceeding t rough a ~y~~gle-level resonance plus scattering that has a much slower energy dependence than caused y this resonance. In the search for fitting parameters to describe t e data ( subsect. 3.2 ) e ciency of co potation is important because at sac angle t ere are up to seven scattering ara stars to be çearc ed on, in addition to energy-resolution ara stars (su sects. 2.2.2, 2 .3.2, 2.3.3). 3.1.1. Cj°cuss sect~ojts ar~c afla 'zH~gg oilers ,for (p, p) on a spin-zero nucleus. For ( , ) on a s in-zero (positive-parity) target proceeding through a resonance with partial wave L, and total spin J, tT and ,. are obtained from (3.1) ,.( , E ) = 2 lm(a x b) ,

(3.2)

`here

`The background (non-resonant) complex-valued helicity amplitudes, a  and b,~, are fitting para stars for the excitation functions at each angle, and the gait-~V~~igner single-level resonance a plitudes, ak and bK , are given by J+ 2k

I';~,. EK - E +'--h K

-

in ~~~ ich J is the îorn n unrl-nucleus (Ci~i) spin, L, is the partial wave for the resonance, and the Legendre functions describe the angular dependences of the resonance arnp itud~'s . In eqs. (3.4) and (3 .5), k is the c.m . wavenumber, T; and I'~ are initial- and final-state partial widths; ~' is the iNÇ' resonance royal width, ~,~ is t e resona~ice energy, and ~ is the bombarding energy. VVe quote energies in the laboratory frame to allow easy comparison with data displays, but all widths are quoted in t e c. . frame. n eqs. (3. ) and (3 .5) the square-root factors have an associated sign, which is the relative sign of the initial- and final-state 11î1C a litu es. T e i portance of the analyzing-power data in addition to cross sections for the reliable extraction of partial widths is suggested in fig. 8, where we show essentially equivalent fits to ~( ®, ~ ) for two dii~erent values of the branching ratio ~~,/I' with t e background amplitudes adjusted to provide an optimal fit. The figure shows t at t e re icte ~.( ®, ) values are quite difïerent. Thus, constraining the fitting pare stars by using polarized-beam data significantly increases their reliability. 3.1 .Z. Cross sections .jor (p, ~c) and (~, cr) through a J ~ _ ® ~ resona~ace. `V~e require only t e for ula for the cross section of a JT = 0' resonance observed in an a, -decay c annal and decaying to a 0+ state, since the resonance contribution to the analyzing

J.F Wilkerson et al. / Isospin-nonconserving particle deca`~s

I

2 C(p,p)" -C

I

I

,

I

24 1

r -,-TI--r-,

ec .m : 1 63.5°

E lob

(MeV)

Fig. 8. Sensitivity of analyzing powers, A,, to variation of branching ratios, but with nonresonant helicity amplitudes adjusted to give the sa :e fi* to the differential cross section for the lowest T = 2 state in °; N observed by °2C (p, p)'2C . Solid curve has I' /l'=0.16, dashed curve has I' /1'=0.24. For both curves a tup d width 1'= 1000 eV and a total resolution with FWHM 1130 eV have been used.

power vanishes . The result can be parametrized as o,(0, E)=ar( ®) +Re[aR(0, E)F(0) ]+

4Ja R (O, E )12 ,

(2 . ; + 1)(2Ip + 1)

,

(3 .6)

in which o- 1,(®) is the nonresonant cross section and F(0) is a complex-valued nonresonant amplitude. Both nonresonant quantities are fitting parameters at each angle . In eq. (3 .6) 1, and 1p are target and projectile spins. The «-decay must have L, = 0, so both the resonance amplitude and resonance cross-section components in eq. (3 .6) are isotropic. 3.1.3. Convolution with resolution-broadening effects. Before eqs . (3.1), (3.2), and . (3 .6) can be used to analyze data and extract resonance parameters, 0' and TA, must be convoluted with the resolution-broadening distributions discussed in subsects. 2.2 and 2.3. For this purpose, we first convoluted the beam-energy profile (subsect . 2 .2.2) with the target straggling (subsect . 2 .3.2), and with Doppler, lattice, and [for ' 2C(p, p)1-'C] atomic-excitation broadening effects (subsect . 2.3.3), to produce an overall resolution function . The largest effect, and the main uncertainty, in this procedure arose from the ill-determined beam profile, so the total energy resolution F HM was usually considered to be a fitting parameter. In order to understand the effect of resolution broadening on a resonance, it is instructive to consider the slightly unrealistic example of a lorentzian resolution

242

IF Werson et al. / Isospin-nonconserving particle decays

function with FWHM equal to 1. When convoluted with a Brefit-Wigner resonance of natural width r and partial width r., a lorentzian resolution function produces another Brefit- Wigner resonance, but with observed width r+,d and partial width 1'P,J1 + .Iff . Thus, in this approximation, uncertainties in the resolution function contribute directly to uncertainties in r and make the observed resonance branching ratio smaller by about a factor R = -,/-I + Al r. Since the resonance-interference excursion about the nonresonant background scattering scales roughly as this branching ratio, the importance of having A as small as possible is clear. For example, for the lowest T = 2 state in "Si observed in'"g(a, «)`'4 Mg, earlier 26 measurements ) had .1 == 3 keV, whereas in the present work A - 1 .2 keV. Since we determined ~ subsect. 3 .3.7) that F = 0.13 keV, we estimate interference excursions about 50% larger in the present measurements, in rough agreement with what is observed. [Compare our fig. 4 with ref. 26) fig. 2.] In the present analyses the convolutions were computed numerically. Convolution is time-consuming, and therefore it had to be done efficiently in order to minimize the time for the resonance- peramstar search in subsect. 3.2. U. SEARC ING FOR RESONANCE PARAMETERS

The relations between the resonance parameters and observables are summarized in eqL (A)-(35), with the additional complication that o- and o-A,, have to be convoluted with the energy-resolution distribution (subsect . 3.1 .3) before comparing with data . Three real parameters describe the two complex-valued nonresonant helicity amplitudes in (p, p) at each angle, since one parameter can he eliminated by fitting o- 1, and the nonresonant region of aA,. . For (p, a) and (a, a) there are only two real parameters for the nonresonant scattering at each angle. The resonance has three or four parameters. For all analyses the energy-resolution distribution has at least one parameter, typically a FWHM value; A allowed to vary by small amounts about the value obtained by combining the beam-resolution measurements (subsect . 2.2.2) with energy- resoluti on effects from the target (subsects. 2.3 .2, 2.3 .3). Thus, there are between six and eight parameters to describe each excitation function. Further, the resonance parameters must be angle-independent, and for excitation functions measured simultaneously at several angles the resolution function must also be angle-independent, since only the initial-state energy is relevant these measurements . The estimator for a best A was the 'V 2 value in which the weigh for each point of the excitation function was the inverse of the variance of the datum at that point . e evolved the resonance-parameter search method from straightforward interactive searches 15,23), that were mostly under control of the user, into searches in the multipararneter space that were controlled by the algorithms of the generalE. 51, 532~1 ' urpose mini-ni7at ion program MINU! Kfef L these methods,

J. F Wilkerson et A / lsospin-nonconserving particle decajw

3.2.1. Grid-search method. This method was used by Ikossi et al.

24 3 23,26) to fi t 0.(

0, E), and by Wilkerson ") to fit QI E) and A,4 0, E), by varying the helicity amplitudes systematically and independently for a given choice of resonance and resolution parameters until a best fit was obtained at each angle. Then these latter parameters were changed and the fitting repeated in an attempt to reduce X2. This procedure was repeated for each 0. The resonance and resolution parameters and their uncertainties were estimated as the O-averaged values and their standard deviations . The grid-search method was instructive for understanding the interplay between of X2 fitting parameters, and it allowed the dependence on parameters to be readily examined, but it was time-consuming and it did not impose the condition that resonance and resolution parameters must be independent of 0. We therefore also used more automated search methods. 3.2.2. Automated searches using INUIT The minimization system MINUIT [ref. ")] can use a variety of search algorithms - Monte Carlo, simplex, and variablemetric - to minimize X` . In our applications to fitting the excitation functions, we found it to be robust and stable . MINUIT can also be used to estimate parameter uncertainties as standard deviations, but care must be used in interpreting such 5;'S4) . estimates A major advantage of using MINUIT was that all the excitation functions for a given reaction could be analyzed simultaneously, this assuring the consistency of resonance and resolution parameters from angle to angle. 2( 23 P)23 Na, MINUI1 was used for the analysis of the Ne(p, p)20Ne, Na(p, 23 24Mg( M g, 31 p( p, p)3' 31 )28 a, a )`'4 P and P(p, a Si data, and also for Na(p, a )2()Ne, 2( P) 2"Ne (subsect . 3.4) . the phase-shift analysis of the T=!2 resonances in Ne(p, 3.2.3. Parameter correlations. The correlations between parameters, which hindered attempts to specify them uniquely and independently, were a problem in both search methods . For example, from the discussion of resolution broadening in subsect. 3 .1-3, it is clear that I- and A are highly correlated . For the polarized-beam (p, p) data the correlation effects are different between o- and A, . and lead to I'" values with little ambiguity . Also and A were better discriminated than they were from the (p, a) and (a, a) data. Therefore, the beam-energy-resolution measurements described in subsect. 2.2.2 were important for interpreting the resonance data. For some of the analyses, the branching ratios p - or I',,/ I' could be constrained to be consistent with values obtained from P-delayed proton measurements 55 ), from 25,27,56), or from (a, y) data s'), so such information (p, t) followed by a-emission was used to minimize elects of correlations between parameters. Representative fits to the polarized-beam (p, p) data, and to the (p, a) and (a, a) 12C ( p, P ) 12C through the cross sections are shown in figs. 2-7 . The data and fits for 13 first T = i2 state in N were shown in a previous discussion of atomic-excitation 28Si( p, P)28Si through the fourth ffects on reson-ances 32), while data and fits for -/"=="2 state in 29p were presented in a discussion of the systematics of isospin mixing n light nuclei '0). All the resonance parameters used in the fits are given in tables 1 and 2.

r

r li

J. ~: l~ilkers(rra

244

/ 1.srr.spira-rac)rac~~rasert.-irag particle cïecays

~t ttl.

Tnt;t .t-. 1 Rcsc~nance parameters and 1 NC' amplitudes, ®p, for proton widths of T = z and T = 2 states by (p, p) and ( p, c~ ) reS~nances . l~idths !' and 1 ~, are in c.m . frame . Excitation energies, Ex , based on ground-state irt+ I37aSSeS from ref. ), are accurate tO aO®uj ~ ~C V , dJ t~IJI:Y,Ï3JCll CYL 1116 C.1®û Ûm .i`ta. i .' .~ ~ ::~ aâ ,. e f. zo ) . ~ntrics in the rightmost column are predictions from ref.'°) table 5 .4. The J~ for the T=2 resonances in `$ g and ; `S from the ( p, n ) rPactinn äre those of the tarbea ~~ 1 lab ) e ~% ~ ' ~ '' F -'° ~sa `s i~4g `'Al =`'

P

z`'S ;? C'I

~~ l~

E~ e®%)

14 .231 11 .264 1? .710 6 .s72 7 .125 3 .905 5 .867 5 .941 5 .839 7 .163 8 .019 8 .920 9 .044 3 .288 3 .374 4 .717 5 .273 3 .284 4.9n3

15 .®8® 11 .20~ 12 .563 8 .976 9.217 15.432 7 .903 7 .?7 : 8 .386 9 .664 iû .490 11 .360 11 .480 12 .049 5 .5as 6 .850 i .3f9

ï_ 5+ I+

a+ ~+ r

5+ i 1+

;+ S+ ;_ 5+

ï+ ï+ ~+ z S+

~i ï+

6.686

z

i~pi u

1~(e~)

l'p(e!%)

1010t30 183~ 13 2600 t 200 650$50 2800 t 350 345 t 50 105~ 18 1302 t 135 271 t 10 3065 t 54 R84 t 170 3540 t 530 1530 $ 120 40~15 100t9 356t 145 108t7

261 t 15 23t2 766 t 24 117t 10 1450 t 150 207 t 31 18t4 232 t 12 216t5 2430 t 40 99 t 11 108 t 20 370 t 20 36t16 98t9 100t 15 96t2 34t5 555 t 320

38t6

2750 t 1060

1~3

4 .1 t0.1 1 .4t0.1 7 .7 t 0.1 6.5t0 .6 13 t 1 .3 21 t 2 3 .2t0 .3 10.0 t 0 .3 11 .7t0 .1 18 .3 f 0 .2 5 .3 t 0 .3 5 .0 t 0.8 8 .2 t 0.3 4 .3~1 .0 7 .5t0.3 12t2 8 .6t0.1 17~1 .5 13 t 0 .5

~®ÎNC'I X

1® 3

+

6.1 _ 1(1 ; .s.2 .O 7.4 +IS _~ ~ 6 .4± ~' Sy 11 .1±KS~ +lu.~ 3 .6 _ z ~ +IS.7 5 .7 _ ~ .s 4 .1 ±z

s 8 .1 +Iti _ 4. ~ tlza 7 .1 _ s, ( , 2 .9±i x +4 (1 7 .1_In +0.2 0.7_os

Tnt;t .t : 2 Resonance pare eters and isospin-nonconservation amplitudes, 8~, for alpha decay of J" = 0 + , T = 2 states . The excitation energies, fi x , are based on the ground-state massES tabulated in ref. ~°') . Excitation energies, Ex , based on ground-state masses from ref. ßx), are accurate to about 2 keV, as discussed at the end of subsect . 1 .1 .3 and in ref.'-(') . The widths I' and 1 ~ are in the c .m. frame

c~~

°~® `( 'hle ` $ IVig -' xSi 3 `S ~ ( 't'a °°~Ti

EIZ ( lab ) ( ev)

3 .905 `~) 6 .112 h ) 3 .288 `~ )

Previous work (

ev)

27 .5 2" .72 16 .73 15 .~ .~2 15 .222 12 .049 11 .988 9 .34

`° ) From i p, a ) reaction .

Present work

l'(ev)

l'~(e~)

Rel:

ssoo ~ .z0oo 12I00 t 3500 2100 t 500 435 ~ 135 317 t 40 < 170 81 t 10 1 .10 ~ 0.28

11 t 22 190 t 100 130 ± 110 30t 14 222 t 22 <7 t 7 80 t 10 0.35 t 0.07

27 )

l'(ev)

Ta (ev)

z7 )

zs ) z7 ® l rb )

z7 )

z~ ) z7 )

345_ :50 130 ~ 20

19=4 105 t 10

40t 15

6.St2 .7

IeQ I x l0 3 1 .4 t 1 .4 5 .9 t 1 .5 5 .5 t 2 .3 4.Ot0 .4 18 ~ 1 8t2 (114t7) 53t5

J.F. Wilkerson et al. / 1sospin-nonconserving particle decays

24 5

The fits obtained from the searches for resonance parameters using the formalism described in subsect. 3.1 are generally of high quality. For a few of the excitation functions (for example, shown in fig. 6) the INC resonance lies on a wing of a much broader isospin-conserving resonance. The effects of the latter can be represented, to a very good approximation, by a linear energy dependence of the helicity amplitudes over the energy interval relevant to the INC resonance analysis . The appropriate modifications were made to the formulas in subsect. 3.1 and to the search Lechniques in subsect. 3.2. In addition to the 19 proton and 3 a-particle INC partial widths reported here, (p' .3He) several other candidate T 2 resonances were identified in the (p, t) and experiments (subsect. 2.1 .1), and some polarized-beam (p, p) excitation functions were measured in the neighborhood of these states . Generally, either the data were not of the uniform quality of those shown here, or the resonance analyses indicated significant problems with the data and their analysis . Therefore, we do not quote the resonance parameters from such data; they are available elsewhere ' '9). 3.3 . ISOSPIN-NONCONSERVATION PROBABILITIES

The resonance decay partial widths F,, and I,,, obtained by the analysis methods described in subsects. 3 .1 and 3.2 are not immediately useful for the study of isospin nonconservation, because their values have very large effects from the strongly charge- and angular-momentum-dependent tunneling through the Coulomb-pluscentripetal barrier. Also, the available calculations of INC effects predict INC amplitudes, rather than partial widths . We denote the partial widths generically by I'f with f = p or a, since they were observed from the final-state (decay) channel . We now summarize how we converted Ff to an INC amplitude. Then we discuss the details for each resonance state. 3.3.1 . Estimating probabilitesfrom partial widths. Using standard resonance theory for an isolated single level "), we removed penetrability effects to construct reduced widths, Y2 . To convert these to probabilities we used to estimate the single-particle 2 width the value y2 = h 2/IAR , where IA is the reduced mass in the decay channel and R is the channel-matching radius . Thus, we have finally a formula for estimating the magnitude of the INC probability amplitude Of , 10fl

R

=-

P,(R) Fqg L )

(33)

in which the penetrability PL appears . For a choice of R near the charge radius of nuclei, the value of Of has a sensitivity to R which is typically less than the uncertainty in Ff. In the present analysis we chose (31) 1 .4A "' An , where the mass number A is that of the compound nucleus. In previous

246

decays IF Wilkerson et al. / Isospin-nonconserving particle

analyses "'''s'-' ;'?6) we used A of the target nucleus . We made this relatively minor change for consistency of corn arison wit'1i the predictions of Ormand and Brown ') for p and n INC decay of co pound nuclei . The choice of R-matrix matching radius in isospin studies has bee discussed in detail previously "). The choice of made here is intended to include a region within which INC effects in the CN should be operative, but to be small enough that Coulomb mixing outside the nucleus should not be significant . In order to re-emphasize the sensitivity of the analysis of isospin-forbidden reactions for measuring INC effects, we remark (as tables I and 2 show in detail) that O f averages <0.01 . Thus, the INC amplitude in the compound-nucleus wavefunc< 10-4 . tion is < 1% and the probability is 112 Lowest T= state in "N. The measurement and analysis of this state p, P ) 12 through 12C ( C with excitation functions at four angles measured using a polarized proton beam and stepping the beam energy across the resonance, has been discussed in some detail in the context of atomic-excitation effects on nuclear 32)1 . The results in table I are consistent with our reactions [subsect . 2 .3 .3 and ref. 15). previous analysis . Further detail is given in ref. =3 3.3.3. T= - j' states in "F: The first and second T 2 state resonance parameters p, P ) 160 with polarized proton beams and stepping across were determined from 16®( the resonance by varying the beam energy "). The data for the lower state measured at five angles are shown in fig. 2. The results of the resonance analysis (table 1) are in agreement with, but are more accurate than, earlier results from unpolarized-beam

measurements 60) . 3 .3.4. T= -1, state in '' Na. Data were obtained for the first two T ;states observed in 2"Ne(p, p)"Ne using polarized beams and ramping the target voltage (subsect. 2.2 .3) . The o, and A,, excitation functions and fits at four angles, shown in fig . 3, 2 produced the resonance parameters for 'Na given in table 1 . The parameters 23,61) . obtained are more accurate than those from earlier measurements 2 [in A candidate resonance for the third T = 2~' state in 'Na with spin-parity 12 6)] agreement with ref. was observed at an excitation energy of 10 .082 ± 0.002 MeV 20 P)20Ne experiments. No selectively populated state was observed in in the Ne(P, 23 the Na(p, t)-'' Na reaction studies (subsect . 11 .1) . Therefore, we have not included this resonance in the entries in table 1 . Details of the measurement and analysis of the elastic scattering are glen in ref. ") . 3.3.5. Lowest T=2 state in 24 Mg. This J' = 0' state was formed as a resonance in 23 Na(p, P)23 Na and 23 N"(p, a ) 2()Ne by using an unpolarized beam and the target-voltage-ramping method (subsea. 123) . Excitation functions over the resonance were measured at four angles, as shown in fig. 4. The proton partial width 24M g) having A T = I and is characteristic of INC resonance formation (table I for T = 2 isospin mixing. The branching ratio T,,/ I' = 0.60 is consistent with the ranges given in refs . `'S'`') . The a-decay partial width (table 2 for 24Mg ) arises only from T - 2 . We determined Q and F with higher accuracy than previously''), as

J.F. Wilkerson et al. / ®sospin-nonconserving particle decays

24 7

indicated in table 2. The measurement and analysis of this reaction are described in detail in ref. '°). 3.3.6. T= 3 states in 'sA/. The two lowest-lying T = states in 25 A1 were studied by 24Mg(p, P) 24 Mg using polarized beams and varying the beam energy across the resonance. The very narrow lower resonance was analyzed by adopting the branching-ratio value Tp /T = 0.17 from Sextro et al. 5S), because of the ambiguity between total energy resolution (® =1200 eV) and total width (d' =105 eV) discussed in subsect. 3.1 .3 . With this constraint, good fits were obtained at five scattering angles, and for both states the resonance parameters (table 1) are more reliable than those obtained previously using unpolarized beams 23) . The details of the measurement and analysis are in ref. 's ). 3.3.7. Lowest T=2 state in 28Si. This measurement was made using 24 Mg(a, a )24 Mg and varying the target voltage to produce the excitation functions at five angles, as shown in fig. 5. The improved energy resolution compared with a previous measurement 26) produced a stronger resonance signal, as discussed in subsect. 3.1 .3. Probably for this reason and because of more extensive search techniques (subsect . 3 .2.2), we obtained significantly different (table 2) values for the resonance parameters . The measurement and analysis are discussed in more detail in ref. 3") . 3.3.8. T = ~ states in 29P. The seven lowest-lying T = states in 29e., except for the third and fifth states which were. too weak to appear in 28 Si(p, p)28 Si, were measured at four angles with polarized beams using beam-energy stepping across the excitation functions. The data and fits, of which a sample is in fig. 6, are of very good quality. Only the first state had been measured previously 23 ), and our results (table 1) are of higher accuracy than obtained from that measurement. Details of the present experiments are in ref. 's ) . The five states measured span 3 MeV of excitation energy (8.4-11 .5 MeV), but the INC amplitudes ® r in table 1 do not show a general increase with energy. This fact is discussed in sect. 4. 3.3.9. Lowest T=2 state in ;`'S. The lowest-lying T = 2, J = 0+ state in 32 S was investigated through the 3' P(p, p)3' P and 3' P(p, a ) 2 'Si reactions, using unpolarized beams and voltage ramping to step across the resonance and to produce the excitation functions at four angles, of which three are shown in fig. 6. At 82° the excitationfunction data were of poor quality, so they were not included in the analysis . Previously 27), only upper limits had been obtained for I' and F,, of this state, and our values (table 2) are consistent with these. The result for h,, (table 1) is new. Details of these measurements are in ref. 3°) . 3 .3 .10. T= ~ states in ;; Cl. The three lowest-lying T = 4 states in 33 01 were populated by 32S(p, P) 32 S with polarized beams and energy stepping across the resonances, producing excitation functions of Q and A,. at four angles . The data and fits for the lowest state are shown in fig. 7 . Only the lowest state had been measured previously 23 ), using unpolarized beams. The resonance parameters from

24 8

J. F Wilkerson et al. / Isospin-nonconserving particle decays

33CI in table 1) are consistent with, but more the present analysis of this state (entry accurate than those obtained in ref. 2). The experimental details for the present measurement are in ref. 1 ). .77 3.3.11. T= - ,3 states in K. By using an ion-implanted target of -"Ar (sect . 2 .3.1) that produced an overall resolution with FWHM of 1300 eV, we were able to measure 3 )36 polarized-beam excitation functions of' o-- and A,. at four angles for 'Ar(p, p Ar = 3 states in 37 K . This was done by stepping the beam energy over the first two to produce the excitation function . The measurements are significantly improved over the previous measurements in ref. '5 ) and produced the resonance parameters in table 1 . 37 The second T = ;state in K was located in 36Ar(p, P) 3 'Ar at an excitation energy 8158 ~ 0004 MeV, in good agreement with our (p, t) measurements . The weakness of the resonance eHects, and a strongly energy-dependent nonresonance scattering, made the resonance analysis very ambiguous. Therefore, we did not include the results in table 1 . etails of the 3' r(p, P)36Ar measurements and analysis are given in ref. 15) .

a

3.4 . J' OF T== ;', STATES NEAR 8.976 MeV (N `'Na

The spin-parity of T = ;states and their spacing from a T = -,' state of interest are, according to the shell-model predictions of Ormand and gown [ref. 3) subsect. 1 .1 .2], very important for determining the INC amplitudes, 011N , because in firstorder perturbation theory the isospin mixing in the compound nucleus (CN) is proportional to the reciprocal of the spacing. The relevant excitation energies in the CN are 5-1 0 MeV for A= 37 to 21 (table 1), which is too high for shell-model calculations to predict reliably the level spacings for each J'. We chose the ;' resonance state at 8.976 Mell excitation in 2 Na as the candidate T = ;state, partly because this region of excitation had been measured and phase-shift analyzed previously by 20 Ne(M p)"Ne [ref. 63)] . We then expended considerable effort in trying to determine the spins and parities of (presumed) T =: ;states. The data, whose toicasurement is described in subsect . 2 .4.2, were subjected to exhaustive phase-shift analyses, and fits of very high quality could be obtained with phase shifts varying smoothly with energy in each angular-momentum channel . Unfortunately, there are very many sets of quite distinct phase shifts that describe the data . This is a well-known problem in phase-shift analyses for overlapping resonances, as described, for example, in ref. "). We concluded that unambiguous results for 2"Ne(p, P)2( 'Ne, using the extensive (r and A, data we obtained, was not feasible at these excitation energies . Parisons with theoretical models The purpose of this section is to compare our INC mixing results from the above experiments and analyses with nuclear-structure calculations of isospin mixing . We

J. F. Wilkerson et al. / Isospin-nonconserving particle decays

24 9

first summarize previous calculations, then we compare our results with complete shell-model calculations that use INC forces calibrated to reproduce the energy splittings of isobaric multiplets in light nuclei 3) . 4 .1 . PREVIOUS ANALYSES OF INC MIXING

e summarize previous observations and calculations relevant to isospin mixing. 4.1.1. Importance of isotensor mixing. The early work of Adelberger et al.') demonstrated that isotensor mixing must be present in nucleon decays of INC resonances because they measured differences between neutron and proton INC decay probabilities in light nuclei . They were able to make some progress in the isospin analysis, but were hindered by lack of reliable nuclear-structure calculations for comparison . 4.1 .2. Direct Coulomb effects on isospin mixing. Auerbach ;') reviewed Coulomb effects in nuclear structure. In most of the calculations of Coulomb mixing that he summarized, one-body models for the INC hamiltonian were used and the nucleon T operator appears once ualy, so that only isovector mixing is possible . Generally, such models also greatly overpredict the mixing, as we emphasized previously ';) . Two-body (proton-proton) Coulomb interactions contain both isovector and isotensor terms (products of two nucleon T operators occur), so isotensor mixing is predicted. A complete shell-model calculation that includes a large part of the configuration space will therefore include all three mixing amplitudes. Such calculations are discussed in subsect. 4.2. 0 4.1 .3. Oscillatory dependence of mixing on mass number. In previous work ' .2;) we pointed out that the INC ground-state proton decays of T = ; states show an oscillatory dependence on A, such that decay probabilities to states with Z := N in 8n +4 nuclei (n an integer) are enhanced by a factor of about five relative to those to states with Z = N in 8n nuclei . In ref. ") this effect was included in a schematic isospin analysis and found to contribute to the T = decay amplitudes the amount 10-4Z413, consistent in size with nucleon isobar (J = , T = D amplitudes (1 .2 t 0.1) X in nuclei 6') . Although a similar result was found in the present analysis, it is difficult to judge whether the oscillations are a general feature of the INC decays, because transitions to states other than the ground states have generally not been reliably measured or analyzed, and because the neutron decay widths 24 ) are not accurate enough to discern whether the oscillations are also present there. The oscillations do not appear in the most-realistic shell-model calculations performed thus far within the limitations of these calculations that are discussed in subsect. 4.2. 4.2 . SHELL-MODEL ESTIMATES WITH ISOSPIN-NONCONSERVING FORCES

As summarized in subsect. 1 .2.2, there are now sophisticated nuclear-structure calculations that include INC forces between nucleons in addition to the INC

250

J. F. Wilkenoti et al. / lsospin-timiconserving particle decays

2,3 Coulomb interactions between protons. In particular, Ormand and Brows ) have redîcted INC proton and neutron emission amplitudes, 0 . and 0,,, in an s-d shell-model con6guration space, using an eRective I NC nucleon-nucleon interaction tailored to reproduce experimental isotopic mass shifts . We now summarize the relevant arts of their work and compare their predictions with our results. 4.2.1 èctive iweractions from the IMME. The INC effective interaction was star ins semi-empirically in the work of Ormand and Brown using Yukawa-type hadronic lus Coulomb nucleon-nucleon interactions . The ranges of the hadronic terms were chosen as those of 7r- and p- mesons, and their strengths were adjusted in each isospin channel so as to get the best ht in the shell-model basis to the isotopic ass shifts in Hght nuclei . ith these adjustable strengths, generally good fits to mass shifts were obtained. y using their INC shell-model interaction, they predicted INC nucleon-decay amplitudes for T states in A = 4n + I nuclei (21 -_ A -_ 37). In the compound nucleus, T states within 500 keV of the T state contribute significantly to the total . Unfortunately, at 8 MeV excitation the shell model does not produce the details of level positions to the required accuracy . Therefore the T= ;'i excitationenergy spectrum was shifted relative to its predicted position by up to :1--500 keV, then a range of INC amplitudes was predicted . Regrettably, a range of about an order of magnitude in ON re fictions arises from this uncertainty, as shown in table 1 .

4.2.2. Comparison qfproton decaj, amplitudes from T= -'2 to T= 0 states.

As shown

in table I (rightmost column) for the proton decay of the lowest-lying T = ;states states in 21 a, 25 Al, 29 ,, 33CI, and 37 K, the agreement between their theory averaged over their energy specru m and our analysis results is quite good for the five nuclei . The predictions in table I have been assigned uncertainties as the upper and lower r. .s . deviations of the values obtained for each energy-shifted spectrum in the ±500 keV range from the energy-averaged values 2 ). An oscillatory behavior (subsect . 4.1 .3) is not predicted, but this (if present) might be obscured by uncertainties in the spectrum of T = 'i C N states . 5. S

ry and conclusions

We have measured the hospin-nonconserving (INC) proton decays of nineteen T = ;states and the alpha decays of three T = 2 states in light nuclei by making fig -resolution resonance studies of excitation functions near these states . By using Polarized proton beams hT the (p, p) studies, improved energy resolution (typically 600 eV FWHM), measur-L:ments of the beam energy resolution, and a target-voltageram ping scherne for some of the measurements, the quality of the proton and alpha-particle excitation-5inction data has been improved significantly over previous measurements. odekindepen dent, helicity-amplitude analysis technique for (p, p), (p, a), and (a, a) single-level-resonance reactions, proper accounting for energy-resolution

J. F Wilkersun et al. / Isospin-nonconserving particle decays

25 1

effects, and the development of sophisticated parameter-search methods, have resulted in accurate determination of F, I'P , and 1'(, values . From the partial widths we have consistently estimated I 1\1C amplitudes for protonand alpha-decays of light nuclei . Comparison with recent shell-model calculations with INC nucleon-nucleon interactions adjusted to fit isotopic mass shifts in light nuclei produces overall qualitative agreement for the twelve proton-decay transitions predicted . The hadronic part of this interaction is isospin dependent, in agreement with results from recent studies of the nucleon-nucleon interaction . Future experimental studies would benefit from improved energy resolution. For example, a beam energy resolution of about 200 rather than 600 eV would allow states with smaller branching ratios to be measured. Concomitantly this would require thinner targets and beams of higher intensity without sacrificing energy resolution . If T = 7 states in these light nuclei could be reliably located, then their direct proton decay to T = 0 daughter nuclei would provide a direct measure of isotensor mixing in nucleon decays . Accurate measurements of the energies of more isobaric-mass multiplets, for example those of neutron-unstable nuclei (perhaps measured by using radioactive beams), will also be relevant . Theoretical efforts in the future should include refinements to shell-model effective interactions related to INC nucleon-nucleon interactions consistent with improved determinations of charge-symmetry breaking in the two-nucleon force. Larger shellmodel bases and more-powerful computational methods would probably improve the predictions of excitation spectra near the symmetry-breaking T = 3j and T = 2 states . By all these means the nucleus will continue to serve as a testing ground for models of the two-nucleon interaction and its charge dependence . We acknowledge help during the experiments from D.J . Abbott, R.E . Anderson, Y. Aoki, C.M . Bhat, J .E . Bowsher, B .L. lurks, E.R . Crossop, B .C . Karp, T.C. Spencer, S.A. Tonsfeldt and R.L. Varner. This research was supported in part by the U .S. Department of Energy, Division of Nuclear Physics, grant number EFG05-88ER40442 . eferences 1) E.M . Henley and G.A. Miller, in : Mesons in nuclei, vol. 1, ed . M. Rho and D.H . Wilkinson (North-Holland, Amsterdam, 1979) p. 405 2) W.E . Ormand, Ph .D . thesis, Michigan State University, 1986 3) W.E . Ormand and B.A . Brown, Phys . Lett . B174 (1986) 128 4) R . Abegg, et al ., Phys . Rev. D39 (1989) 2464 5) W.W . Jacobs et al., in : 7th Int. Conf. on Polarization phenomena in nuclear physics, Paris, July 1990, C6-511 6) B. Holzenkamp, K. Holinde and A.W . Thomas, Phys . Lett . B195 (1987) 121 7) G.A. Miller, B.M .K . Nefkens and 1 . Slaus, Phys . Reports 194 (1990) 1 8) T.A . Trainor, T.B . Clegg and W.J . Thompson, Phys . Rev. Lett . 33 (1974) 229 9) E.G . Adelberger et al., Phys . Rev. Lett . 22 (1969) 352; A.B . McDonald and E .G . Adelberger, Phys . Rev. Lett . 40 (1978) 1692

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253