Magnetic moments and nuclear structure

Progress in Particle and Nuclear Physics Progress in Particle and Nuclear Physics 49 (2002) 91-154

PERGAMON

http://www.elsevier.com/locate/npe

Magnetic Moments and Nuclear Structure K.-H. Ilnstitutfir

SPEIDEL,’

0. KENNI

and F. NOWACKIZ

Strahlen- und Kerphysik, Universitiit Bonn, Nussollee 14-16, D-53115 Bonn. Germany Wzstitut de Recherches Subatomiques, F-67037 Strasbourg Ceder, France June 26, 2002

Abstract We review the principal advances and results of magnetic moment measurements for short-lived nuclear states. The latest variants of the technique combine projectile Coulomb excitation in inverse kinematics with intense transient magnetic hypefine fields. A number of the measurements described were performed in collaboration with the Rutgers University group. Data spanning the range from neon to neodymium nuclei have provided new insights into the nuclear structure of the states in question. The results are discussed in terms of advanced microscopic calculations within the framework of the nuclear shell model. The highly improved experimental accuracy permits critical testing of appropriate configuration spaces of valence nucleons, as well as effective NN interactions derived for these spaces. The applicability of the method to next-generation g-factor measurements on exotic, radioactive projectiles and their eventual theoretical implications is assessed. PACS number(s):

1

21.10.Ky,

21.60.-n,

25.70.De

Introduction

The prominent role of magnetic moments for determining specific components of the two constituents (protons and neutrons) in wave functions of atomic nuclei, which had already been emphasized in the early days of nuclear physics, is baaed on the fundamental difference in the magnetic moments of the elementary nucleons. The very first measurements of the anomalous magnetic moments of individual protons and neutrons, associated with their spin, by Alvarez, Bloch and Purcell [l, 2, 31 indicated that the two types of nucleons are less elementary than electrons, suggesting a most likely internal substructure known today as quarks. The anomaly of the magnetic moments of the two nucleons has been fairly well explained in the framework of the constituent quark model, which yields in its simplest representation the ratio ~(v)/P(?T) = -2/3, very close to the most precise and upto-date experimental values for neutrons (v) and protons (7r) [4]: P(V) = -1.91304272(45)

and

P(K) = +2.792847337(29).

It is this marked difference of the values, in both sign and magnitude, which permits unveiling the nucleonic species in nuclear wave functions, provided the experimental precision is sufficiently high, Magnetic moments also provide important information on the fundamental forces in particle physics. This aspect has been most impressively demonstrated by means of the extremely precise measurements for trapped electrons by Dehmelt et al. [5] and recently for muons at the Brookhaven synchrotron [6] which test the standard model.

0146-6410/02/$ - see front matter Q 2002 Elsevier Science BV. All rights reserved. PII: SOl46-6410(02)00144-8

92

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

In nuclei, where protons and neutrons are bound by the nuclear force their magnetic moments experience characteristic deviations from their bare values, as the nucleons also possess an orbital component in addition to spin and they interact with other nucleons as well. For a nucleus consisting of A nucleons, the magnetic moment operator can be written as,

with effective g factors of protons and neutrons associated with their spin s’ and orbital angular mo-0 mentum 1. These values, corresponding to the bare nucleon g9,1r gsf’ee(?r) N +5.586 95’“‘(v) N -3.826

and and

g:‘ee(7r) = 1 gfree(y) = 0

and

91eff N 1.1 or -0.1

are gZff(n, V) N 0.75 *gF(r, for protons and neutrons, angular momentum ji

respectively

V)

[7]. From the coupling of the angular momenta [and

s’to total

one derives for an odd-mass nucleus, neglecting interactions of the odd particle with the even core, the well-known Schmidt values: gj = gl f $-$ for j = 1* i (2) which are generally a valuable basis for the estimation of nuclear moments. Considerable deviations from these values have their origin in residual interactions between the odd valence nucleon and the nucleons of the core. There are two major effects which cause significant corrections: (u) meson exchange and (b) core polarization [8, 91. Meson exchange gives rise to a renormalization of the bare nucleon spin and orbital g factors, whereby the orbital part gl is more affected than the spin part g9. The first order core polarization or configuration mixing effect is approximately linear in the number of valence protons and neutrons. As first emphasized by T. Yamazaki [lo], and later by Y. Yamauchi et al. [ll] mesonic effects can also be pictured in terms of quarks, since both nucleons and mesons consist of quarks. Quark degrees of freedom in the effective magnetic moment are generally masked by appreciable contributions from particlehole configurations, which make it very difficult, if not impossible, to disentangle subnucleonic effects in measured magnetic moments. It can be postulated that only in specific nuclei with simple and well understood particle configurations, might high precision measurements yield information on quark degrees of freedom as well (see also e.g. [ll]). A certainly more moderate objective, but of no lesser importance, is the sensitivity of the magnetic moment to the different role of protons and neutrons in the structure of nuclei and their subtle interplay with collective degrees of freedom. This feature has been traditionally pursued over decades in many experiments using different techniques. For ground states as well as long-lived excited states resonance techniques can be applied and have yielded g factors of such high precision that theoretical calculations were often overreached. However, the situation changes completely when dealing with short-lived states, where comparable experimental precision is unattainable. The reasons are generally of technical origin, For states with lifetimes in theps (lo-” s) range, intense magnetic fields of IcTesla strength are required, whose origin are in hype&e interactions of complex structure, so that the field strengths cannot easily be determined from ab initio calculations. Hence, one has to rely on empirical parametrizations which often require carefully selected and appropriate recalibrations, in order to account for specific experimental conditions. The technique of free hyperfine interactions for ions recoiling into vacuum or gas [12] has the merit of generating strong magnetic hyperfine fields. However, in measurements with the time-integral version

K.-H. Spcidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

93

of perturbed y-angular correlations (PAC) [12] these turned out to yield rather inaccurate results. In contrast, the time-difleerential version of PAC allowed to obtain in a few selected cases rather precise g factor values. The best result was established for “0 ions in their single-electron charge state with excitation of the 1” = 3- octupole state at 6.13 MeV and lifetime r = 26.6 ps. In this case, the ion conditions are characterized by a well-defined hyperfine frequency, WlS= (21+ 1)$%(O) tL associated with a well determined 1s electron Fermi contact field of&(O) = 8.55 kTeslo. The measurements were carried out without charge state selection of the oxygen ions, employing the recoil distance technique with plunger distances in the easily controllable pm range [13, 14). Prom the precise g factor value obtained, g = 0.556(4), expectations as to the applicability of this technique to many nuclei arose. However, this hope was not really fulfilled. The main reason for the failure has to do with the,fact that the hype&e frequency increases with .Z3, which restricts the method to nuclei with 2 5 40 since the plunger distances become unfeasibly small. On the other hand, the frequency pattern of ions in lower charge states is extremely complex, so that precise g factor data can not usually be derived. This hyperfine interaction regime has been studied in a few cases with only moderate success (see e.g. [12]). Parallel to these activities an alternative technique was developed, which subsequently became the most powerful method for measuring magnetic moments of nuclear states with ps lifetimes. In 1975, the phenomenon of transient magnetic fields (TF), already known since 1968 from several PAC experiments, in which the recoiling and excited nuclei were slowed down in ferromagnetic media, was placed on a new physical basis with respect to its origin and dependence on relevant parameters [15, 161. The novel and most spectacular feature of the TF strength consisted in its increase with ion velocity. This behaviour of the field was not at all expected from early data and from the formerly rather convincing interpretation of its origin by the Lindhard-Winther mechanism [17], which actually predicted a decrease of the field strength with increasing velocity. The new features of the TF method with some selected applications to nuclear g factors were reviewed in 1980 by Benczer-Koller et al. [18]. Only the new experiments furnished clear evidence that TF arise from spin polarized electrons in ns shells of the moving ion. The required high efficiency of polarization transfer from the magnetized electrons of the ferromagnet to the traversing ions could be explained in terms of cross sections for single electron ions [19, 20). These experimental findings, combined with a better microscopic understanding of TF, led to a real breakthrough in the application of this method to nuclear g factor measurements. Only then did it become possible to study nuclei in high and low angular momentum states, which are populated in heavy ion reactions, accompanied by high recoil velocities of the nuclei. Coulomb excitation and heavy ion fusion are the most commonly used reactions for preparing the nuclei in the states of interest under conditions favourable for spin precessions in the TF. However, for the determination of a g factor of a particular state, its population has to be well-defined, particularly avoiding feeding from unknown precursor states, which contribute to the precession of the state, thereby giving rise to uncertain corrections. Thii situation can be easily handled in Coulomb excitation, since the feeding pattern is well understood. In fusion reactions, however, the feeding pattern is generally rather complex due to sidefeeding channels. This difficulty can, however, be overcome by combining the TF technique with the recoil distance method, whereby the recoil distance of the ions causes a well-defined time delay of the decay pattern of states before the nuclei enter the TF region. The distance is adjusted for a selected state, so that a direct feeding transition refers to 7 rays exhibiting both a stopped and a flight component. The latter, detected via its Doppler-shift, signalizes that the nucleus enters the TF when the state in question is formed. For observing the precession of this particular state, the depopulating y ray is measured in coincidence with the Doppler-shifted feeding component. Moreover, any other subsequent transition in the 7 cascade will exhibit the same precession, associated with the selected state, if its lifetime exceeds the transit time of the ions through the ferromagnetic layer. Measurements using this technique were pioneered by Lubkiewicz et al. [21] and Birkental et al. [22]. The great success of this technique in conjunction with large y-detector arrays was first demonstrated by Jungclaus et al. [23]. Nevertheless,

K.-H. Speidel et al. / Prog. Pan. Nucl. Phys. 49 (2002) 91-154

94

precise determination

of g factors remains a challenging objective under these experimental

conditions.

Improved precision of experimental g factors became particulary desirable when large-scale shell model calculations provided more realistic and reliable results. Another new challenge is posed by measurements on exotic nuclei far from stability, which are at present (and in future on a broader scale) accessible as low intensity beams at various laboratories (e.g. REX-ISOLDE at CERN, and SPIRAL at GANIL, Caen). The fulfilment of both these goals requires a substantially higher efficiency of the TF technique. Projectile Coulomb excitation in inverse kinematics combined with the TF method appears to meet these requirements in an almost ideal way [24]. In the last three years, many measurements have been performed with various beams of isotopically separated stable nuclei, yielding a new class of quality data in terms of both precision and reliability (see e.g. [25, 261). Moreover, the technique has also been tested with stable beams under conditions relevant to radioactive beams, in which the use of a zero-degree annular detector for detection of light target nuclei, kinematically focussed in the forward direction, is an important component [27]. This set-up allows to stop the radioactive beam in a down-stream Faraday cup, avoiding accumulation of radioactivity in the target region. These striking achievements in the experimental technique were accompanied by a comparable progress in theory. For sd and fp shell nuclhi shell model codes were developed, which allow highdimensional diagonalizations of the hamiltonian on large computers. In these calculations, a large configuration space, beyond an inert closed-shell core, is combined with effective residual nucleon-nucleon (NN) interactions [7, 28, 29, 301. The core nucleus and the corresponding size of the configuration space of the valence nucleons are intimately linked to shell gaps, associated with magic nucleon numbers, cornerstones in the nuclear structure. In addition, Monte Carlo techniques in various versions [31, 32, 33, 34, 351, as well as highly advanced mean-field approaches [36, 371, have also been used to explore diverse issues of nuclear structure. These calculations are particularly useful for the description of heavy nuclei, where the underlying large dimension of the shell model space cannot be handled with the presently-available computer capacity without severe truncations. Altogether, both developments in theory and experiment herald a new era of nuclear structure understanding.

2

The Technique of Transient Magnetic Fields

Both the empirical methodology and the hyperfine interaction regime operative in TF measurements exhibit features characteristic of the atomic environments that prevail in the polarized ferromagnetic medium. Some of these features are specific to the particular choice of beam and probe ions, the reaction kinematics that governs the nuclear reaction between them and the ensuing ensemble of spatially and temporally correlated high-velocity, nuclear-excited ions that traverse the ferromagnet. The distinction between probe ions and beam ions stems from their different role and effects in the experiments. The probe ion is the one that usually experiences the TF, whereas the beam ion may cause attenuations of the TF strength at the probe ion. In this respect, one has also to distinguish between certain beam ions that generate the probe ion in a nuclear reaction but have no effect on the TF, and other beam ions which do not directly participate in the nuclear excitation but are responsible for the attenuations. Thus, in order to perform reliable and precise measurements under these conditions, all these features need to be taken into account in the data analysis. This is customarily done via an appropriate characterization of the perturbed angular correlation process from which magnetic moment information is extracted. Over the past two decades, the latter has been itself the subject of intensive systematic investigations. The database accumulated, as well as the understanding of the prevalent mechanisms gained thereby, represent prima facie conditions for extracting g-factor information with the accuracy required to be of relevance for up-to-date theoretical models. In the following, we shall describe the main elements of the technique by summarizing what is known and has been discussed in various preceding papers, with special emphasis on new insights and experience.

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

2.1

95

General features of transient magnetic fields

T F are experienced by nuclei when ions move through ferromagnetic materials. They arise from the

polarization of unpaired electrons in ns shells of the moving ion following spin exchange with the magnetized electrons of the ferromagnet. The fact that this interaction is the governing mechanism for the formation of the field and that its orientation is parallel to the external magnetizing field, has been shown in calculations for single electron ions in ferromagnetic iron. In this case spin exchange scattering is considered between the electron occupying the ls state of a H-like ion and the polarized 3d electrons of iron [19, 201. Due to the atomic nature of T F of the Fermi contact type their strength scales with (Z/n) a, where Z is the atomic number of the ion and n the principal quantum number of the electron state in question. Hence, hyperfine fields ranging from k T e s l a strengths for light nuclei (Z "~ 8) to M T e s l a strengths for heavy nuclei (Z "" 90) are expected. On the same atomic footing the T F of a particular ion should increase with velocity, since the ionization proceeds from outer (high n) to inner (low n) shells. However, this dependence eventually leads to a maximum in the field strength, which is attained when the ion velocity approaches the Bohr velocity of the Is electron state. Above this recoil velocity, the T F decreases with increasing velocity, finally reaching a zero value corresponding to a fully-stripped ion. This velocity dependence was first verified in measurements of 12C ions in the T F of iron with excitation of the first 2+ state at 4.43 MeV and lifetime r = 6 5 f s [38]. On these grounds the T F can be paxametrized as:

B~F = ~p,,(v,o., Z, host), q,,(v,.., Z). ~,,(Z)

(4)

n

where qns, pn8 are the ion fractions with single ns electrons and their degree of polarization, respectively; B,8 are the Fermi contact fields at the nucleus. The individual components of hyperfine fields associated with corresponding ns electron states reach their maximum values at ion velocities close to the Bohr velocity with ion fractions q,s - 0.5 (see also [39]). In the particular case of single electron ions the parametrization reduces to a single field component, which also represents the largest field attainable for a specific ion:

BrF = p~,(,, Z. host), q~,(v, Z) . B~,( Z)

(5)

The hyperfine field at the nucleus can reliably be calculated: B , , ( Z ) = 16.7. n ( z ) . Z ~ [Testa I

(6)

whereby R ( Z ) is a relativistic correction that is approximately given by [39]: R ( Z ) z 1 + (Z/84) 5/2.

(7)

Only under these ion conditions can T F measurements provide information on the degree of polarization Pls, which cannot otherwise be easily obtained due to the complex structure and analysis. Such data can then be compared with model calculations (see e.g. [19, 20]). pl~ values were experimentally determined for a series of ions from carbon (Z = 6) to chromium (Z = 24). As shown in Fig. 1 the data have a tendency to fall off with increasing atomic number, resembling a 1 / Z dependence. This relationship is derived by combining the microscopic formula of the T F (eq. (5)) with the empirical linear parametrization as given by [26, 40] (see also Sect. 2.2): BTF = a . Z . v/vo

(8)

where a is the strength parameter of the T F , depending on the ferromagnetic host; Z and V/Vo are the atomic number and velocity (in units of the Bohr velocity vo = e2/h) of the probe ion in the ferromagnet. From eqs. (5) and (8) one derives for H-like ions at velocities v = Zvo, where the T F has its maximum value with ql~ -~ 0.5: a[Tesla] 1 1 Pl~ = 16.7[Tesla]. R ( Z ) . q l s " -Z ~ Z

(9)

K-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

5

10

20

30

50

70 90

Atomic numberZ Figure 1: The degree of polarization pl, (left) and the transient field (right) as function of the atomic number of H-like ions recoiling through Cd. Dashed curve represents a l/Z dependence of pl, (see text and [56]) and the solid line the 2’ dependence of the field strength.

Evidently, the pIa(Z) data show that TF should also exist at extremely high ion velocities and the field strength increases with atomic number as 2’ (Fig. 1). This extrapolation clearly supports the expectation that the TF technique is applicable to nuclear species at relativistic beam energies. So far no nuclear g factor has been measured in these conditions. However, in the context of exotic nuclei produced by fragmentation at relativistic beam energies, ion velocities become extremely high and H-like ions are most likely to be formed. Under these conditions g factor measurements, employing projectile Coulomb excitation in forward scattering, are the subject of intensive investigation (see e.g. [41]). The excitation cross-sections for the first 2+ states of nuclei are large and spin alignment is expected to be stronger than at intermediate energies, both favourable conditions for g factor experiments [42].

2.2

Transient

field parametrizations

With the exception of H-lie ions the TF strength cannot in general be calculated from first principles, as the relevant parameters of the microscopic description (eq. (4)) are not known. Hence, for the determination of g factors one has to rely on carefully selected and appropriate field calibrations which imply nuclei and nuclear states with known g factor in most similar ion conditions in terms of the electron configurations responsible for the TF. This means, that only isotopes or neighbouring nuclei are good candidates for calibration. It also means that the calibration g factor generally refers to a longer lived state, where other experimental techniques can be applied. Besides this preferential and most reliable procedure for the determinations of the TF strength, empirical field parametrizations are also used. These are baaed on experimental data obtained with the above-mentioned calibration method. They describe the overall dependence of the field strength on three key parameters: (i) the atomic number 2 of the probe ion, (ii) the ion velocity in units of the Bohr velocity v/u0 and (iii) the polarization of the ferromagnetic host represented by strength parameters a, a’ and a”.

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154

In the literature

97

one finds three different parametrizations: Linear parametrization Rutgers parametrization

Chalk - River parametrization

[40] : BLIN

= a *2 . $

(10)

v 0.45 . A4 [43] : BRUT = a’ . Z’.’ . G ( >

(11)

[44] :

BCR

= a” . Z . $ . e-8”~“0

(12)

All three parametrizations imply essentially the same linear Z dependence but differ in their explicit velocity dependence. In comparison to BLIN , both the BRUT as well as the BCR parametrizations show a substantially weaker velocity dependence. Moreover, in the B R”T formula the magnetization M of the ferromagnetic target layer is explicitely quoted and usually determined in magnetometer meaaurements as function of temperature. In the other parametrizations small differences in the magnetization are accounted for by the error of the strength parameters. For Fe layers only small variations in the saturation magnetization have been observed, whereas for Gd layers the magnetization depends on the preparation procedure. Best results were obtained with evaporated Gd layers [45]. It has been demonstrated in early measurements using the 160(3;) state as probe, that the TF strength follows rigorously the magnetization of the ferromagnetic layer (Fig. 2) [46]. Therefore, precision measure-

8

External Field [mT]

Figure 2: Transient field precessions of the W(3;) state in iron as function of the external magnetizing field. The data follow fairly well the magnetization curve of the iron layer (continuous curve) [46].

ments require the highest attainable magnetization of ferromagnetic target layers. The specific values of a,a’,a” and /3 refer to calibration nuclei of different mass regions. Both the BRuT and the BCR parametrizations are mainly applied to medium-mass and heavy ions at moderate velocities (U/Q 5 5), in particular for nuclei in the rare earth region. In practice today, the BLrN with distinct modifications (see sect. 2.3), and the BRUT, are the most frequently used parametrizations for g factor determinations.

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

98

2.3

Ion beam induced TF attenuations

Whether ion beams, when penetrating ferromagnetic layers, will affect the magnetization and thereby the TF strength has been questioned from the beginning of g factor measurements using this technique. Clear evidence of ion-beam-induced deterioration effects, which ought to have consequences for the parametrizations as well, came relatively late. Only with the novel technique of projectile excitation in inverse kinematics, in which precessions of both the projectile ion as well a8 the target ion (two probe ions) are measured simultaneously, has the influence of the ion beam on the magnitude of the TF on both ions been unequivocally identified; distinct dependences on several parameters have been found in many dedicated measurements [47]. Evidently, the beam ion that, in collisions with the target nucleus generates the probe ion, which can be either the projectile itself or the target ion, is not the cause of the TF attenuation at the probe ions. In fact, the attenuation occurs in an uncorrelated manner to all the

_ 8

A

111

‘il

0.8

B g

0.6

Mg

u 0.4

i

t

Figure 3: Transient field attenuations of various probe ions (specified at data points) in Fe and Gd hosts vs. the stopping power of beam ions. The reduced velocities of the probe ions, v/.Zv~, are grouped in ranges according to the data. The dashed lines are drawn to guide the eye.

K.-H. Speidel et al./Prog. Part. Nucl. Phys. 49(2002)91-154

99

other beam ions and is of dynamic nature; meaning that no irreversible damage to the ferromagnetic structure has been observed. Hence, the degree of attenuation must depend on the beam intensity, as indeed has been observed. The attenuation is defined as the ratio of the measured TF, BTF, to its value determined by the linear parametrization (eq. (10)): G &am

BTF = -BLIN

This convention was chosen as the other parametrization3

(13) most likely reflect such attenuations

when

Si

0.2 '6

I III1II11101 III IIIII1I1I/IIIII1III/IHI_ 1.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Reduced Ion Velocity

v/zvs

Figure 4: Transient field attenuations at a specific energy loss of beam ions in Gd host as function of the reduced velocity of probe ions (specified at data points). The dashed line is drawn to guide the eye.

various heavy ion beams are used for calibration. As pointed out in [48], there is evidence that the BCR parametrization incorporates attenuation data expressed by the exponentially damped velocity dependence of the TF strength. In the case of the BRuT parametrization the situation is not entirely clear, although here too the velocity dependence, different from that of BLfN, indicates an influence of the ion beam employed (see also discussion in [49]). The key parameter in the attenuation scenario is the energy loss dE/dx of the beam ions in the ferromagnetic layer, whereby heavy ions cause larger attenuations than light ions. This dependence, however, is by no means a simple thermal phenomenon, in which the deposited energy drives the temperature close to or even beyond the Curie temperature of the ferromagnet, causing a breakdown of the magnetization and the TF as well. Such a thermal spike scenario has no relevance for the general experimental layout of TF measurements, as the temperature rise is localized at the ion track and extends radially only over a few nanometers. This limited spatial extension of the temperature spike will not affect the TF at the probe ion, as the latter moves (due to its uncorrelated production) on trajectories far removed from those of the beam ions in both space and time (see above). This feature has been studied in several dedicated measurements (see e.g. [48]). An additional and sensitive parameter for the magnitude of the attenuation is the velocity of the probe ion, which also determines (via its charge state) the ns electron nature of the TF. The higher the velocity, the larger the attenuations [50]. Th’IS well-established behaviour suggests that the attenuation depends on the screening of the probe ions by the host electrons via which the electron-polarized ions are shielded against perturbations caused by the distant beam ions. The screening efficiency depends on the ion velocity and is largest at velocities below and close to the Fermi velocity of the host electrons.

100

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

The effect of ion screening in solids has been specifically discussed in the context of the energy loss of ions [51, 521. Its relevance for the attenuation mechanism is an open question. The available information on this aspect of the TF technique is summarized in Fig. 3. The attenuation G is plotted, separately for Fe and Cd layers, as function of the mean energy loss of the beam ions in the ferromagnet. The data are grouped according to the reduced velocity of the probe ions, v/Zvo, which also characterizes the ns electron nature of the TF. The dependence clearly exhibits thresholds in dE/dx, at which the attenuations set in. Furthermore, the attenuations saturate at different levels depending on the probe ion velocity. Saturation is also seen in the velocity dependence for a constant energy loss (Fig. 4). The data are useful for interpolation and extrapolation at given experimental conditions and have been used with great success in the analyses of many CJfactor measurements. Efforts have been made to obtain a better understanding of the attenuation mechanism, with the objective of avoiding or at least minimizing these disturbing effects. For targets with magnetized Fe, segmentation of the ferromagnetic layer of nm dimensions has yielded substantial reductions of the attenuation [53]. This result is a direct consequence of the implied long range of the perturbation. The preparation of such micro-structured layers requires techniques which are best known from chip technology. Similar results were obtained using ferromagnetic iron-boron compounds with their well-known amorphous structure [54]. The disordered structure has obviously the same effect on the attenuation as layer segmentation, i.e., it hampers the propagation of the ion-beam-induced perturbation emanating from its original track to the distant tracks of the probe ions. As a final remark, a more profound understanding of the attenuation process is a challenge to theory to explain the dynamics and rather complex structure of the ion interactions with magnetized solids. Nuclear structure studies may benefit from any progress in this endeavour, but the underlying physics itself is highly interesting and justifies a major effort.

3 3.1

The technique of projectile Coulomb excitation The PAC technique

The interaction of the magnetic moment of an excited nuclear state with the transient magnetic field during the short nuclear lifetime is generally observed via the rotation of an anisotropic angular correlation of y rays emitted from the excited state. This well established technique of perturbed angular correlations (PAC) allows to messure rotation angles of the order of 2: 1 mrad with high reliability. These are determined from r-ray intensity ratios which are independent of detector efficiencies and beam current fluctuations throughout the measurement. The 7 rays are usually measured in coincidence with particles populating the nuclear state of interest in a nuclear reaction. Hence the particle-y angular correlation is given by [54]

w(Q,) = 1 + xArp.

Pk[cos(Q,)]

(14)

k

where Arp = Gk . Qk * Aikh”” and Age“’ are the experimental and theoretical correlation coefficients, respectively. The A;*“” coefficients refer to the maximum spin alignment of the excited state and Pk are the Legendre polynomials. The coefficients Qk and Gk represent attenuations of the correlation due to the finite size of the r-ray detectors and the acceptance angle of the particle detector, respectively 1541. The rotation or precession angle 9 of the correlation W(e,) is derived from double ratios R = (Ni t ‘Nj J.)/(Ni .J *Nj 7) of counting rates Ni,j with an external magnetic field, alternately in the “up” (t) and “down” (4) direction, perpendicular to the y-ray detection plane. The indices i,j represent a pair of y detectors symmetric to the beam axis (see Fig. 5). For small precessions the angle is given by, w . de,

pN = g. in_

s

tout t,, &F(wn(t))

. eet”dt

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (iW2) 91-154

101

Figure 5: Schematic view of -y- and particle-detectors with target and beam stop under Rotated y-angular distributions refer to an assumed precession measurement conditions. (2: --t 0:) E2 r-transition.

where g is the g factor of the nuclear state and BTF is the transient field acting for the time (tout - t,,) during which the ions traverse the ferromagnetic layer; the exponential accounts for the nuclear decay. According to eq. (15) the effect ratio R strongly depends on the slope value, S 3 l/W. dW/dQ,, of the angular correlation: large anisotropies of the correlation are therefore essential for measurements of small precession angles. The targets generally consist of three layers: in the first layer the nuclear reaction takes place in a collision with the beam ion and the state of interest is populated; in the second layer, which is the ferromagnetic medium, the probe ion experiences the TF, through which the precession occurs in the nuclear state; and finally, in the third layer the probe ion is stopped in a hyperfine-interaction-free environment to avoid any additional changes of the angular correlation. The beam itself traverses the whole target and is stopped either in a down-stream Faraday cup or in a stopper foil placed behind the target (see also Fig. 5).

3.2

Projectile

Coulomb excitation

in inverse kinematics

Coulomb excitation at beam energies below the Coulomb barrier has proved to be a powerful reaction for controlled population of nuclear states of collective nature. Besides large excitation cross-sections, high spin alignment is also achieved, clearly favouring this type of reaction for precession measurements. These desirable conditions are accomplished in the conventional version of this technique by targetnucleus excitation with backscattered beam ions. In this geometry the probe ions emerge from the target with high ion velocity into the ferromagnet and the lighter beam ions are backscattered into an annular detector. Most g factor measurements with TF were performed in these conditions. Only recently was a new version of this technique implemented, in which the role of target and beam ions is simply exchanged. In this inverse kinematics the heavier projectile ions are excited by lighter target nuclei. As a result, both ions move after the collision with high velocities in the general direction of the projectile ions. Due to the difference in mass and atomic number the lighter target ions can pass through the composite target and be detected in a 0” particle detector, whereas the excited heavy projectile ions are stopped in the target backing. Clear emergence of the target ions into the forward detector while the beam ions are safely brought to rest in a foil placed in front of the detector can be realistically accomplished. These requirements have to be fulfilled for the particle-y coincidence

102

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

measurements in the precession experiments. The general scheme of the technique is shown in Fig. 5. The merits of this inverse kinematics for g factor measurements are threefold: (i) kinematic focussing in the forward direction substantially enhances particle detection in the 0” detector and thereby the coincidence detection efficiency of 7 rays; (ii) the excited projectiles evolve with high velocities providing intense TF which generally increase with the ion velocity; (iii) the isotopically-pure beams of interest can be produced from unenriched materials in the accelerator ion-source, due to the mass selectivity of the beam-transport system. With this powerful method, experiments can be performed using a single target for excitation, obtaining precession of several isotopes by varying the projectile species, thereby enormously increasing the efficiency and reliability of g factor measurements. These experimental improvements are unequalled by any other current technique.

Figure 6: Scheme of target and particle detector arrangements with radioactive beams [27].

for precession measurements

The same technique can be applied in future experiments dealing with radioactive beams, where the handling of radioactivity as well as of low intensity ion beams are the major ingredients. The problems, associated with the stopping of the radioactive nuclei in inverse kinematics, can be solved by using an annular detector for the detection of the target ions, while the beam ions are steered through the central hole of the detector to a down-stream Faraday cup, where the accumulation of radioactivity will not disturb the measurement. The feasibility of this detector arrangement for g factor measurements has been shown in a recent measurement with stable 50Ti beams [27] ( see Fig. 6). With carefully chosen target thicknesses the effect of straggling on the beam extensions can be kept to a minimum, so that almost complete transmission of the beam to the Faraday cup is accomplished. This is an essential feature for g factor measurements in order to reduce activity counting rates, thereby reducing random coincidence rates.

3.3

Projectile excitation at high energies

Projectile Coulomb excitation is also very efficient at energies far above the Coulomb barrier [55]. In this case the beam ions are scattered at forward angles below the grazing angle in peripheral collisions with heavy target nuclei. The de-excitation y rays are measured in coincidence with the scattered ions, which are detected in an annular position-sensitive particle detector placed in the forward hemisphere. The beam itself and light charged particles from central collisions emerge through the central hole of the ion detector to be stopped in a down-stream Faraday cup. Such an experimental set-up was tested in measurements with 52Cr ions at an energy of 752 MeV [56] (Fig. 7). Ion beams were provided by the UNILAC accelerator at Darmstadt. The 5zCr(2:) state at 1.434 MeV was Coulomb-excited by scattering the ‘*Cr beam from a thick Gd target which also served as magnetic medium for the precession. The mean velocity of the forward recoiling Cr ions in the ferromagnetic Gd layer was 21vo, the highest ever investigated in a TF measurement, giving rise to

103

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Ge -detectors

Figure 7: Schematic view of the annular position-sensitive particle detector and a pair of Ge detectors at fixed angles in the y-detection plane symmetric to the beam axis, for simultaneous measurements of angular correlation and precession.

I -100

I 0

Azimuth

100

angle

200

300

I#+ [degree]

Figure 8: Particle-y-angular correlations measured for ‘*Cr nuclei as ratios of y-intensities of two Ge detectors as function of the azimuth angle of particle detection (see Fig. 7). The diamonds and squares refer to forward and backward detector pairs, respectively. The solid lines are fits to the data, the dashed lines represent the unperturbed correlations [56].

a significant precession. The annular detector for measuring the scattered and Coulomb-excited 52Cr nuclei was a parallel plate avalanche gas counter with position sensitivity in both polar and azimuthal angular directions, whereby the polar angle ranged from 11” to 24” below and equal to the grazing angle. The position sensitivity of the particle detector permitted correcting for the large Doppler-shift of the emitted 7 rays, but also allowed measuring particle-y angular correlations. With four Ge detectors, placed in pairs at fixed angles symmetric to the beam axis in forward and backward hemispheres, both particle-7 angular correlations as well as spin precessions were measured simultaneously. The angular

104

K.-H. Speidel et al. / Prog. Pan. Nucl. Phys. 49 (2002) 91-154

correlation was obtained from coincident y-intensity ratios of detector pairs as function of the azimuth angle defined by the particle detector (Fig. 8); precession angles were determined from conventional “up/down” counting rate ratios (sect. 3.1, eq. (15)). This highly efficient procedure, which is free of systematic errors, as it relies on intensity ratios, can also be applied to much higher energies of 2 100 MeV/u. In this case, only detector pairs in the forward hemisphere are required due to the Lorentz boost of the 7 distribution. Experiments in such conditions have been recently proposed for SIS ion beams at Darmstadt. It is noted, that g factor measurements on nuclei far from stability at relativistic energies are of great interest, as these nuclei are produced by projectile fragmentation with large cross-sections. With fragment separators such as the FFtS at Darmstadt, these nuclei can be separated and Coulomb excited on a down-stream target with subsequent precession of the excited states. The technique of TF is very well adapted to the kinematics at these high energies. From the measurements with 52Cr ions it was clearly demonstrated, that the high TF strength associated with the 1s electron Fermi contact field of 240 kTesla is indeed operative.

4

Magnetic moments of low spin states in various nuclei

We will now review specific g factor measurements on numerous nuclei and series of isotopes in different mass regions, which have recently been performed with the experimental techniques described by our own group or in collaboration with other groups. The new g factor results, which are characterized by substantially higher precision in comparison with earlier data, are discussed in the framework of microscopic model calculations, highlighting the salient features of the nuclear structure.

4.1

The g factor puzzle of the “ONe(47)state at 4.25 MeV

The sd shell IV = 2 = 10 nucleus 2oNe represents a focal point for nuclear structure studies and is considered to be among the best understood of all nuclei. The relevant part of the level scheme is displayed in Fig. 9. 20Ne has pronounced collective features exhibited by rotational bands, which also exhibit rather strong backbending beyond an angular momentum of I = 6+, whose origin is in the rotational alignment of both protons and neutrons in sd shell orbits [57]. All these specific properties are generally well described in nuclear model calculations. For this self-conjugate nucleus there are rather simple but reliable predictions for the magnetic moments of the ground state band members. Assuming good isospin T = 0, the g factors of the nuclear states are determined by the isoscalar values as given by [58, 591: go(I)

=

da) +91(u)+ 9.(r) + !h(v) 2

2

1 .- (Ss) I

where gr+ represent the single-nucleon orbital and spin g factors and (Ss)/Z is the expectation value of the spin density in the state in question. For collective excitations built on I = O+ ground states the angular momentum stems exclusively from vibrational and rotational degrees of freedom with (1s) = (La) or (Ss) = 0, which immediately implies that gs(1) = 0.5. Deviations from this value result from configuration mixing within major oscillator shells, as well as higher order mixing involving particlehole configurations. Thus, for T = 0 states, (Ss) is determined by the competition between the collective component which has (Ss) = 0, and a two-particle component with -1 5 (5s) 5 fl [60]. In addition, isospin mixing can also cause deviations, as eq. (16) is only valid for pure T = 0 states. As for 20Ne, the lowest T = 1 and T = 2 states are at excitation energies of 10.2 and 16.9 MeV, respectively, so that only small admixtures to the dominant T = 0 component are expected in the wave functions of low spin states. In view of these well-founded arguments on the nuclear structure of 2oNe, it was very surprising that the first g factor measurement of the 20Ne(4{) state yielded a value considerably smaller than predicted, g(4:)= - 0.10(19) 1611. In this experiment, the TF in iron layers was employed. The

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

105

360(25) fs 7000 VT 6000 5000 4000 3000 2000 1000 0

Figure 9: Level scheme of 20Ne with relevant y transitions

known g-factor value of the first 2+ state, g(2:)=+0.54(4), which is indeed in good agreement with expectations, was used to calibrate the field. As the 4: state has a very short lifetime (T = 93 fs), compared to the 2: state (T = 1.02 ps), the measured ‘zero’ precession of the 4: state could also have resulted from a not-fully-developed TF in the relevant time window. However, this conjecture was not confirmed, as full TF strengths were observed, even for considerably shorter times of the order of 10 fs [38, 621. In addition, the magnitude of the 4: lifetime was confirmed in an experiment employing the Doppler-Shift-Attenuation-Method (DSAM), in order to ensure that the lifetime is not shorter than quoted. In a series of subsequent measurements, in which the experimental sensitivity was considerably increased by using gadolinium instead of iron layers as ferromagnetic host, the original findings were confirmed. In these new experiments precessions of the 2: and 4: states were measured at two different velocities of the Ne ions of w 7~s [63] and 3~s [64]. A common fit of the g(4:) value to the data yielded g(4:)=+0.13(15) in good agreement with the Fe result. In parallel to these studies, a group from Rutgers University performed a similar measurement under almost identical experimental conditions [65]. Unfortunately, their result on the g(4:) factor was of low accuracy and thereby in both consistent with our small value and with the theoretical expectation. In conclusion, the 2oNe(4:) g factor appeared at that time to be small, contrary to all existing theoretical predictions. 4.1.1

New measurements

and results

In view of the far-reaching theoretical implications of this result, if true, a new series of measurements wits recently embarked on to clarify this issue. In these new efforts the same nuclear reaction was applied as in the former measurements, since the fusion of 12C(12C,cr)20Ne occurs at moderate energies with strong resonant population of both the 2: and 4: states, far from achievable in other reactions. High counting rate efficiency is very important for a meaningful result as the 4: precession is small due to the short nuclear lifetime. In the

106

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Figure 10: Particle- and y-detector arrangement for *‘Ne precession measurements. NaI(T1) scintillators are placed in pairs symmetric to the beam axis and a Ge detector at 0”. Two annular Si detectors detect LYparticles emitted at backward and forward angles. Two iron cones shield the ions from the stray magnetic field of the polarizing electromagnet (see text).

new measurements two new ingredients were introduced as an essential improvement over the former experiments: (i) targets of much higher quality and (ii) a second particle detector placed at 0” to the beam axis. The two-detector arrangement (0” and 180”) allows simultaneous measurements of IY particles, emitted at backward and forward directions from the same resonant states of the compound nucleus. This detector set-up (Fig. lo), turned out to be a powerful improvement, as the o-particle intensity was at forward angles substantially larger than at backward angles. In addition, 20Ne nuclei, associated with backward emitted low-energy cy particles, experience a high velocity TF (nion N 7vs), whereas the forward geometry with high-energy cr particles select low velocity Ne ions (vh N Zve). Hence, TF precessions were measured simultaneously at two diflemnt ion velocities. As shown in Fig. 10, both particle detectors were of annular type, whereby the carbon beam was steered through the central holes of the detectors onto the target and to a gold stopper placed behind the 0” detector. Both particle detectors were Si counters of 300 mm* area with thicknesses of 100 pm and 500 pm for backward and forward emitted a! particles, respectively. The 7 rays were detected with large volume NaI(T1) scintillators. In addition, a Ge detector was placed at 0” to the beam axis to measure the lifetimes of the 2: and 4: states employing the DSAM technique. The multilayered target consisted of a 150 pg/n* natural carbon layer, deposited on a 3 mg/cm* gadolinium layer, which had been evaporated onto a 1 mg/cm* tantalum foil, followed by a copper layer for stopping the *‘Ne recoils. In contrast to the earlier measurements, the Gd layer was produced by ultra-high vacuum deposition on a Ta substrate heated to a temperature of 800 K. It was found that this preparation technique yielded layers with substantially higher saturation magnetization in low external fields [45], which is crucial for precessions during femtosecond lifetimes. In magnetometer measurements it was found that the saturation magnetization was close to its theoretical value. Measurements were performed at carbon beam energies of 32.6 Mel/ and 34.5 MeV, which refer to maximum yields of the 2: and 4: states. They were determined by detailed studies of the excitation function of the fusion reaction. At both energies good spin alignments (highly anisotropic angular correlation) were also achieved, prerequisites for sensitive precession measurements. At 32.6 MeV mainly the 4: state was populated whereas the 2: excitation was very weak; at 34.5 MeV the population

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

XI02 SOW

8eeeo

0” detector, 32.6 MeV 4:

1800 detectw, 32.6 MeV

70600

4OW

60000

” 200

Ek

107

300

0” detector, :SMeV

300

400

IJW

IIW

400 300 Energy [Channels]

16W

Energy [Chmnelsl

i

500

Energy [Channels]

9owo ~

13W

180” derecror, 34.5 MeV

l4W

ISW

1

I6W

Energy [Channels]

Figure 11: Particle coincidence spectra from Si detectors located at forward (0”) and backward (180”) angles (Fig. 10). The respective a peaks refer to the *‘Ne(2:,4:) states populated at two different “C energies.

of both states was almost equally strong. Particle- and y- coincidence spectra obtained in these conditions are shown in Figs. 11 and 12. Evidently, the states of interest are well resolved in both particle and 7 spectra, allowing unambiguous determinations of y-photopeak intensities for the precessions. For each particle detector and beam energy three independent precessions were measured: (a) the pure 2: precession from the direct pop ulation (2: cr peak), (b) the (2: via 4:) precession from the population of the 4: state feeding the 2: state (4: cy peak) and (c) the pure 4: precession from the direct population (4: cr peak). The precession results obtained in three different runs are summarized in Table 1. Two features (in addition to an appreciable increase of precision) should be emphasized in comparison to the results of former measurements: (i) all four precessions of the 4: state exhibit non - zero effects; (ii) the 2: and the (2: via 4:) precessions agree within their errors. The latter finding is a strong indication, that the 4: 9 factor has the same magnitude as the 2: 9 factor (in case of equal CJfactors the two precessions should be the same). This test could not be done with the earlier data, mainly because of the much lower accuracy and/or due to missing precessions (see e.g. [64]). Beside precessions, the lifetimes of the 2:- and in particular of the 4:- state have been redetermined with much higher precision by analyzing

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

108

0” detector, 4;pmtick

5ooo

2: +

4500 4.900 3500 3000 3 3

2500 2000 IS00 1000 500 0

window

0:

mo

L

4000

4: i

300

200

3000

2;

2000

1000

J

0 400

300

Energy [Channels1

1800 detector,

Energy

400 [Channels]

1800 detector, 2;pam’cle window

4~particle window

ZOO

3500 3000

2000 mm

1000

I

2000

s

1500 1000

500

JO:

R

300 Energy

400 [Channels]

200

300 Energy

400 [Channelsl

Figure 12: y-coincidence spectra of 20Ne with particle windows on the 2: and 4: cy peaks of 0” and 180” Si detectors at a beam energy of 34.5 MeV.

Table 1: Measured precessions in Gd of 2oNe nuclei excited to the 2: and 4: states at beam energies of 32.6 MeV and 34.5 MeV with o-particle detection at forward and backward angles. precession Wzp [mrad] I”

via N 2~0: ‘forward’ (Y’S ‘uionN 7710:‘backward’ o’s 32.6 MeV

34.5 MeV

32.6 MeV

34.5 MeV

2+ 1

_

7.5(5)

9.7(2.2)

10.3(6)

2: via 4:

7.4(5)

6.9(9)

10.0(6)

8.8(1.3)

4:

1.2(5)

2.8(10)

1.5(6)

3.4(1.6)

K.-H. Speidel et al. / Prog. Part. Nud Phys. 49 (2002) 91-154

109

the Doppler-broadened lineshapes of the 1.63 MeV (2: --t 0:) and 2.61 MeV (4: -+ 2:) y lines in the concidence spectra of the 0” Ge detector. The lifetimes deduced ~(2:) = 1.02(5) ps

and

~(4:) = 93(4) fs

agree very well with values quoted in the literature. In the final analysis the known g factor of the 2: state was used for determining the TF strength. In addition, differences in the TF, resulting from small distinct differences in the ion velocities of the 4: and 2: precessions, were accounted for by the linear parametrization BLfN (see eq. (10)). A common fit to all precession data, for low- and high-velocity ions yielded with excellent x2 a g factor value: g(4T) = +0.43(9). In comparison with the earlier data, this is indeed a very surprising result. Contrary to the former value it now agrees very well with g(2:) = +0.54(4), consistent with the theoretical expectation for pure isospin T=O. In fact, shell model (SM) calculations, including full configuration mixing within the major oscillator sd shell, yield [60]: g&2:)

= +0.510

and

g,&4:)

= +0.511,

both of which agree with the experimental values. More details about the new measurements on the z”Ne(4:) g factor, including a comprehensive discussion on obvious shortcomings of the earlier data and its analysis, will be presented in a forthcoming publication [66]. A brief comment offering a possible explanation for the earlier finding of the small g(4:) factor [64] is added. From our present knowledge and appreciably broader experience with TF precession measurements beam-bending corrections can usually be neglected. With external fields of the order of 0.05 Tesla for magnetizing Gd layers, combined with a very effective shielding against stray fields, no sizable bending of incoming and outgoing charged particles is expected. In former 20Ne measurements the 4: precessions were probably over-corrected due to an overestimation of beam-bending. Following this suggestion we have reanalyzed our former precessions [64] assuming a negligible beam-bending correction. This yields a value g(4:) = +0.55(21) which is no longer in contradiction with the present result. It supports our assessment that the earlier measured precessions were correct but the magnitude of beam-bending corrections applied was too high.

4.2

Magnetic moments of fp shell nuclei and their nuclear structure terpretation

in-

Nuclei with valence nucleons in the fp shell have been intensively studied and discussed in recent years following the striking progress in the development and refinement of large-scale shell model calculations (see e.g. [67]). High quality computations of energy levels, electromagnetic transition strengths and moments as well as p-decay properties have been carried out systematically for series of isotopes of several elements for mass numbers 46 < A < 70 (see e.g. [7]). Th ese calculations are based on the largest manageable configuration space in combination with the most advanced effective NN interactions, which are linked to the original Kim-Brown (KB) interaction [29]. Modified versions of the most realistic residual interactions are found in the literature as KB3 [68] and KBJG [7]. Another frequently discussed effective interaction is FPD6 [69] which generally leads to similar results as KB3. However, there are also distinct differences, related to the explicit mass dependence of the two-body matrix elements of FPDG, which is not included in KB3. This has direct consequences, for instance for the energy gap of 56Ni, which is better described with FPD6 than with KB3 (see also [7]).

K.-H. Speiakl et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

110

It is evident that improved calculations with refined predictions of nuclear properties require better experimental data in terms of reliability as well as precision. This challenge to enhance the quality of the experimental data was the principal motivation for a new generation of measurements of g factors and nuclear lifetimes of entire series of isotopes, in order to study and explain subtle changes in the nuclear structure by comparing such data with most advanced model predictions. In these calculations the choice of an inert core, which determines the valence space of active nucleans, is a rather important point as it has direct consequences for the calculated observables. Thus, the exclusion of core excitation substantially reduces collective degrees of freedom which are sensitively probed by E2 transition rates. In fact, for lower fp shell nuclei, such as Ti, Cr, Fe and Ni isotopes, 40Ca has been considered as the most prominent core, whereas for the upper fp shell, such as Zn isotopes, due to the spin-orbit force, the ge/r intruder orbital can play a role and several possible valence spaces can be considered and equally be discussed. Specific experiments can clarify the relevant degrees of freedom. In the following we review speciiic experimental details as well as appropriate results of shell model calculations for each series of isotopes of fp shell nuclei. For determining salient features of the nuclear structure with respect to the competition between single particle and collective degrees of freedom, shell closures appear as lighthouses and play the dominant role. To this end, both g factors and E2 transition strengths have been determined, as these have very different inherent sensitivities. 4.2.1

g factors

and E2 transition

rates of even - A Titanium

isotopes

The structure of the Ti nuclei is essentially determined by their valence particles outside the 40Ca core consisting of two protons and several neutrons, all occupying the 0frj2 orbit. For the stable isotopes 46Ti 48Ti and 50Ti, the effect of gradual filling of the f,lz shell on the wave functions of the first 2: and ‘4: states has been studied by means of g factors and B(E2) values and compared to results from large-scale shell model calculations. Subtle changes are expected, since at “Ti the f,,z shell closure is accomplished.

3000 -

2ocu -

1000 -

L o+

t :iTi

24

0’

t :iTi

26

Figure 13: Low-lying states of 46,48,50Tiwith relevant y transitions. are supplemented by those of ref. [72].

The new lifetime values

In the measurements, isotopically pure Ti beams were provided in their natural abundances in the ion source of the Cologne tandem Van de Graaff and accelerated to energies between 110 and 120 MeV.

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

111

Ions of 46Ti, 48Ti and 50Ti were Coulomb excited to the 2:, 4: and 2: states (Fig. 13) by scattering from a carbon layer of a multilayered target. Carbon was ideally suited for clean y spectroscopy, as target excitation to the first 2+ state at 4.43 MeV was weak, thereby contributing no noticeable background from de-excitation y rays in the coincidence spectra. Moreover, the large mass asymmetry between titanium projectiles and carbon nuclei gave rise to efficient focussing in the forward direction of the inverse kinematics. The composite target, consisting of 0.75 mglcm2 “W deposited on a 3.6 mglcm2 Gd layer, evaporated on a 1 mg/cm2 Ta foil and backed by a 3.6 mg/cm2 Cu layer, was cooled to liquid nitrogen temperature and magnetized to saturation by an external field of 0.06 Tesla. The y rays emitted from the excited states were measured in coincidence with forward scattered carbon ions. Large volume scintillators of NaI(T1) or BaF2 were used for y detection, and the ions were registered in a Si detector of nominal 100 pm thickness and 300 mm2 area. The detector was operated at very low bias, thus reducing the effective depletion layer, in order to separate the carbon ions from light particles, such as protons and alphas from fusion reactions between carbon and titanium nuclei, via their different energy loss (Fig. 14). It was found that among various heavy ion reactions the a-transfer reactions 12C(46*48s50Ti, 2cy)50352*54C~ were particularly strong, with a pronounced population of the first 2+ states. This striking selectivity of particle transfer to low spin states has been exploited for g factor measurements, as discussed before. The low bias operation of the particle detector was beneficial for obtaining high quality -y-coincidence spectra. This is shown in Fig. 15 with spectra from a Ge detector placed at 0” to the beam axis. The pronounced Doppler-broadened lineshapes were analyzed to determine the nuclear lifetimes of the states in question.

Figure 14: Particle spectrum in coincidence with 7 rays, with the Si detector operated at low bias. The o particles refer to the a-transfer reaction and the 12C ions to Coulomb excitation of Ti nuclei. Particle-y angular corrrelations (anisotropies) as well as precessions were measured for the (2: + Or)and (4: + 2:) 7 transitions of each Ti isotope. The beam-induced attenuation (eq. (13)), Gkam= 0.83(4), was determined from experimental data included in the systematics shown in Fig. 3. The lifetimes of the excited states were determined by analyzing the shapes of the emitted y-ray lines detected with the 0” Ge detector. The shapes exhibiting pronounced structures, which result from the slowing down of the ions in the different target layers, were fitted to Monte Carlo simulations that take into account the second order Doppler effect, as well as the finite size and energy resolution of the Ge detector. For the 2: states, the feeding from the 4: and 2: states was taken into account. The computer code LINESHAPE [70] was used in the analysis. The high quality of the fits obtained is shown in Fig. 16.

112

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

800

1000

1200

1400

1600

1800

2ooO 2:

Figure 15: Ge 7 spectrum in coincidence with (Y particles and “C ions bombarding the carbon target with 110 MeV 50Ti projectiles (Fig. 14). Prominent y lines are assigned. The decay-in-flight component of the 2: state is clearly visible.

*Ori 27 ‘T=I .62(7) ps

Figure 16: Fit to the Doppler-broadened “OTi(2: + 0:) y transition.

lineshape of a) the 46Ti(2:

+ 0:) and b) the

The g factors derived from the measured precessions and calculated @‘IN/g values, along with the B(E2) values deduced from the measured lifetimes, have been compared with results from full fp shell model calculations (FSM) (Table 2). These were carried out with the computer code ANTOINE [71] using the effective interaction KB3 (see sect. 4.2), in which the gap around 56Ni is reduced and a density dependence is included, and effective gl and gs values (sect. 1). The general trend of the present data of g(2:), g(4:) and B(E2)’ s with increasing number of neutrons in the jr/r shell is rather well described by the calculations (Fig. 17). Evidently, the slight but distinct decrease of g(2:) from 40Ti to QBTiand its subsequent rise at 50Ti are well reproduced, whereby the large g(2:) value of “Ti reflects the N = 28 shell closure. The pattern observed is clearly associated

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

113

Table 2: Newly-measured g factors and lifetimes with deduced B(E2)‘s in Weisskopf units, and full fp shell model (FSM) calculations for the 2: and 4: states in 46*48?7’i isotopes [25, 26, 27). r(Z) [PSI Nucleus 4sTi

48Ti

“Ti

I”

[721

new [25, 271

B(E2) J- [W.U.]

!?(I)

Expt.

FSM

Expt.

FSM

2;

7.4(6)

8.1(4)

18.5(9)

11.7

+0.496(27)

+0.285

4:

2.4(2)

2.3(2)

20.5(18)

15.7

+0.58(17)

+0.244

2:

6.2(4)

5.7(2)

15.0(5)

9.1

+0.392(19)

+0.211

4:

1.8(4)

1.1(l)

18.4(17)

13.7

+0.54(13)

+0.472

2:

1.54(22)

1.62(7)

5.1(6)

8.4

+1.444(77)

+1.203

I

01

I

I

I

24

26

28

I

I

24

26

I

I

i

28 Neutron number

Figure 17: Comparison of experimental B(E2)‘s and g factors with fp shell model calculations represented by solid (2: state) and dashed (4: state) lines. The values g = Z/A are predicted by the collective model. The experimental B(E2) of the (4: + 2:) transition in “Ti (triangle) refers to [72].

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

114

with excitations of protons and neutrons from the Ofr,z orbit to the lpslz, O&z and the lprlz orbitals. Similar good agreement is found for the B(E2) values for both transitions (2: + 0:) and (4: -+ 2:). The fact that the numerical agreement is not better - the calculations generally underestimate both the g and the B(E2) values - might indicate that the collectivity of the nuclear states is larger than implied by the theory. It was also argued that the deviations (which are larger for &@Ti than for 5oTi) might be attributed to a systematic underestimation of the TF strength. This conjecture was, however, ruled out by the results of calibration measurements with the known g factor of =Fe(2:). Hence, the discrepancy must have its origin in certain shortcomings of the model calculations. If the KB3/KB3G interactions are indeed sufficient, the configuration space has to be enlarged by including excitations of the “Co core. It has long been known, that intruder states in the neighbourhood of %‘a are present and can affect properties of the low-lying states. Furthermore, a larger configuration space below 40Ca, equivalent to an increase of collectivity, has recently been considered [73]. Their systematic influence on g factor properties is currently under investigation. In conclusion, shortcomings in the calculations compared with present data clearly call for more extensive calculations and a critical assessment of both the effective interactions and the active model space as well. 4.2.2

g factors

and E2 transition

rates

of even-A Chromium

isotopes

New measurements of g factors and E2 transition rates for the chromium isotopes 5oCr, 52Cr and 54Cr [25, 26, 741 were found to be in line with the investigations for the titanium nuclei [26] as one would expect, since their nuclear structure is closely related. In fact, 46Ti and 5oCr are cross-conjugate nuclei, where the particle-hole symmetry predicts identical wave functions and spectra, provided that the valence nucleons are indeed in the Ofrlz shell. In this case, the g factors follow a simple linear relationship (Fig. 18) [75]. The picture, however, changes when this symmetry is broken through excitations of Of,,2 particles to the remaining fp shell orbits. The main motivation for the measurements was, however, driven by two other important aspects: (i) existing g factor data for several spin states from 2: to 8: of 5oCr [76] were in severe conflict with shell model calculations [77], suggesting a dominant collective nature contrary to the predictions, and (ii) with 52Cr, the isotone of 50Ti, effects of the N = 28 shell closure as well as core polarization [78] could be studied. The latter case will be discussed in greater detail in section 4.2.3. 1.0

o-

0 “Ti

48Ti

%r

“Ti

48Ti

93

Figure 18: Calculations of the spin dependence of g factors, g(Z), of low lying states of 48,4*Ti and 5oCr, assuming (a) all valence nucleons are in the Of,/2shell; (b) one valence nucleon is allowed to occupy the (l~s,~, 0f5i2 and lpilz) orbits [75]. Lines are drawn to guide the eye.

As shown by the 2: excitation energies of the three isotopes (Fig. 19), the N = 28 shell closure is well pronounced: the 2: energy of 52Cr is considerably higher than that of the neighbouring isotopes. The g factors and the B(E2) values are expected to show the same behaviour. In a simplified picture of the nuclear structure one expects for the g(2:) value of 52C~ a pronounced increase associated with the (f,,-J4 as dominant proton configuration in the wave function (with its Schmidt value g(n f,,2) = f1.655, assuming bare gi and gs values). Lower g values are expected for the neighbouring isotopes 5oCr and 54Cr due to significant contributions from two neutron holes in the f,j2

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Et4 WI

1’15

7fPSl

3Q30

2ooo i

1

t

A-a

2

0:

7

0+

::cr,*

0'

Xr30

Figure 19: Low-lying states of 50,52,54Crwith relevant y transitions. are supplemented by those of [72]

Figure 20: BaF2 y-ray coincidence (4: -+ 2:) transitions in 5oCr.

shell and two neutrons are negative:

spectrum

in the ps/2 orbit, respectively;

g(v A/z) = -0.547

displaying

The new lifetime values

well separated

(2:

+

0:) and

the Schmidt values of both these configurations

and g(u pa/z) = -1.275

This expectation was indeed verified by the new results (see below). The measurements were carried out at the Cologne and Munich tandem accelerators under conditions almost identical to those on the Z’i nuclei. Isotopically pure “0, 52Cr and 54Cr beams with energies of 120 MeV and 115 MeV, respectively, were Coulomb excited to the 2: and 4: states by scattering from carbon of the same multilayered target used for the Ti measurements. The y rays were detected in coincidence with forward-scattered carbon ions registered in a 0” Si detector. The application of the inverse kinematics was the decisive improvement for the g factor measurement of the “‘Cr(4:) state, where the cross-section is small and high detection efficiency is thus required. A typical T-coincidence spectrum for 5oCr of a BaF2 scintillator is shown in Fig. 20.

116

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Table 3: Newly-measured g factors and lifetimes with deduced B(E2)‘s in Weisskopf units, and full fp shell model (FSM) calculations for the 2: and 4: states in 50~52%~ isotopes [25, 26, 741. r(1) [PSI Nucleus

I”

1721

new [25, 741

B(E2) _1[W.U.] Expt.

g(I)

FSM

Expt.

FSM

= “Cr

52Cr

54Cr

2:

12.8(7)

13.2(4)

19.2(6)

18.3

+0.619(31)

+0.568

4:

3.2(4)

3.2(7)

14.6(32)

26.0

+0.78(13)

+0.742

2:

1.02(4)

1.13(3)

10.3(3)

12.4

+1.206(64)

+1.172

4:

1.5(5)

9.6(‘;:)

10.3(34)

11.7

-

+I230

2:

11.4(4)

14.5(6)

15.0

+0.840(56)

+0.625

0.4-

26 28 30 Neutron number N

0.4-1

26 28 30 Neutron number N

Figure 21: Comparison of experimental B(E2)’ s and g factors (circles and squares; triangles refer to [72]) with fp shell model calculations (lines).

K.9.

Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154 I

I

I

117

I

1

2+

4+

6+

a+

Spin I [ fi ]

Figure 22: Comparison of experimental g factors for 5oCr with shell model calculations which allow t valence nucleons to lie outside the OfTIpshell [79]. Closed circles refer to the new measurements whereas open circles are data obtained by A. Pakou et al. [76].

The (4: + 2:) 7 line is well resolved from the (2: + Or) 7 transition. With the superior energy resolution of a Ge detector, it was verified that no disturbing background 7 lines were present in the energy region of interest. The determination of 7 intensities was therefore straightforward. The g factors and lifetimes deduced are summarized in Table 3. The experimental data are rather well accounted for by results from large-scale shell model calculations. These are based on a 4oCa core and all particle configurations of the fp shell valence space, that consists of the Ofrp, lps/z, Of5,2 and lpllr orbitals. The effective NN interactions were the same as used for the Ti isotopes. Evidently, for the 50,52Cr nuclei, in the upper half of the Of,/2 shell, the g factors as well as the B(E2) values agree very well with theory. Such excellent agreement was not achieved for the Ti isotopes 46*48Ti,although the same quality of calculations was maintained. A small discrepancy is found for “Cr, for which the calculation underestimates the experimental value. Better agreement was obtained in a truncated configuration space [30, 741. The same behaviour is found in the data for the N = 28 isotones, which is not yet fully understood (see also discussion sect. 4.2.3 below). As shown in Fig. 21 and consistent with expectations, the g(2:) value peaks at 52Cr and the B(E2) exhibits a minimum, both data strongly supporting the N = 28 shell closure. Evidently, the minimum in the B(E2) of 52Cr, relative to the values of the neutron-deficient and neutron-rich neighbours, reflects the reduced collectivity expected for a semi-magic nucleus. This feature is quite well explained by the model calculations. The importance of including the whole fp shell in the description of the structure of these nuclei is further highlighted by the calculations of g(Z) for “OCr [79], for which an increasing number of valence nucleons denoted t, is excited from the Of,/2 orbit to the other fp shell orbits using the KB3 interaction (Fig. 22). The prediction of g(4:) > g(2:) is indeed confirmed. Evidently, the best agreement between theory and the new data on the 2: and 4: states [25, 261 1sachieved for t 2 2. The same conclusion has been drawn by Nakada et al. [30] and Caurier et al. [80, 811 in their descriptions of the structure of heavier fp shell nuclei. Fig. 22 also displays the previously measured g factors of 5oCr states [76] up to angular momentum I = 8+, which disagree with all existing microscopic calculations. This striking discrepancy beyond the 2: state appears to exist no longer, at least on the basis of the new data. The reason for the previous deviations might stem from the complexity of the feeding pattern, characteristic

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

118

of fusion reactions in general, which were applied in that work. Without knowing the precise time history of the feeding paths, it is virtually impossible to determine the precession of a particular state. It is therefore likely that all precessions observed were dominated by the same unknown feeding state(s). Only the combined technique of recoil distance and transient fields circumvents this problem. Such an uncertainty is, however, absent in Coulomb excitation measurements. The comment, recently put forward by A. Pakou [82] in defense of these conflicting results, may kindle false conclusions. The claim of gross compatibility with a collective Z/A prediction of the hydrodynamical model is only valid for her own data on 5oCr and 46Ti but not for the new high precision data. In fact, the improved accuracy of this data, including also the new results for &Ti, exhibits for the first time rather subtle structure effects, which differ significantly from Z/A values, but are well reproduced by present large-scale shell model calculations. This spectacular achievement of the new high-precision data represents a real breakthrough in the understanding of the structure of these fp shell nuclei. 4.2.3

Core polarization

of N = 28 isotones

Core polarization is an effect which involves the excitation of an otherwise closed shell or inert core by valencenucleons. In general, such interactions result in a quenching of the magnetic moment of nuclear states, a pattern observed in many g factor measurements [83, 841. The phenomenon was first predicted by Arima and Horie [78, 851 in their early work on core polarization, in particular with respect to the N = 28 isotones of fp shell nuclei, that spans a region (above the 40Ca core) from scandium (4gScZi) to cobalt (55Cosr) where the ground states of the odd proton members have spin and parity 7/2- corresponding to a (frls)” proton configuration. In the pure shell model, all configurations (fr/$’ of the N = 28 isotones should have the same g factor, given by the Schmidt value g(sfr,s)=+1.655 assuming bare gs and g1 values. Specifically, this prediction should hold for the (7/2)- ground states of the odd-A nuclei and for the 2: states of the even-A nuclei. Any deviation from this result would provide evidence for core polarization in terms of configuration mixing and/or the presence of mesonic exchange currents. The extremely precise experimental g factors of the ground states of 51V, 53Mn and 55Co [86] are in fact significantly lower than the Schmidt value and furthermore, they differ from each other. As already emphasized by Arima and Horie [78], core polarization accounts for both these features. Calculations in first order perturbation theory predict substantial quenching of the g factors, although their magnitude is expected to increase linearly with the number of proton holes in the f7i2 shell. Consequently, the g factor of 55Co (with one proton hole) is predicted to be smaller than that of 51V (with five proton holes), consistent with the experimental data. For the even-A nuclei 50Ti, 52Cr and 54Fe the situation was for a long time unclear, mainly because of low precision of the experimental data. This has changed with the new g factor measurements that employ the technique of projectile Coulomb excitation of the first 2+ states in inverse kinematics, For 54Fe and 52Cr the g(2:) values are found to be substantially lower than those of their odd-mass neighbours (see [88, 891 and sect. 4.2.2). Furthermore, the value for “OTi(2:) [89] is found to be considerably larger than that of the heavier isotones. As in the case of the odd-A isotones, the g factors increase with the number of proton holes in the f7/2shell, consistent with core polarization predictions. The experimental g factors of all N = 28 isotones are displayed in Table 4 and Fig. 23. The data show unambiguously that the g factors of the (7/2)- g round states of the odd-A and of the 2: states of the even-A nuclei vary, to an excellent approximation, linearly with 2, albeit with different slopes. This strikingly different behaviour of the odd and even mass nuclei is difficult to understand in the framework of large-scale shell model calculations. Shell model calculations involving diverse effective interactions which are frequently used for fp shell nuclei were carried out for the N = 28 isotones. The results obtained with the FPD6 interaction, where only one particle is excited from f7,2 to ~313, f5,2 and pi/p orbits, are shown in Fig. 23 and Table 4. Evidently, in this truncated configuration space, the calculated g factors and in particular the effective slopes of g vs 2 are in good agreement with the experimental values for both odd and even-A nuclei. If the analysis were concluded at this point, it would appear that the experimental data are

119

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Table 4: Comparison of measured g factors of N = 28 isotones with the results of shell model calculations, performed .assuming the excitation of t valence nucleons of the Of,,2 shell to the other fo shell orbits and two freauentlv used effective interactions 17. 891. g factor

Nucleus Exp. I = g-

I = 2:

FPDG(t=l)

1FPD6@=3)(“)

KB3G(t=3)@)

KB3G(full)(“)

49sc

-

+1.469

+1.457

$1.514

+1.526

5’V

t1.4710579(4)

+1.437

+1.397

-cl.443

+1.426

53Mn

+1.435(2)

+1.400

+1.394

+1.403

55co

+1.378(l)

$1.337

+1.374

+1.357

“Ti

+1.444(77)(b)

+1.484

+1.245

+1.325

+1.26

52Cr

+1.206(64)

+1.335

+1.190

+1.335

+1.25

54Fe

+1.049(60)

+1.162

+1.130

+1.280

@J)see ref. [7] (*I average of [89] and [27]

1.4

$3

i= + ”

1.2

%

i

Figure 23: Comparison of the experimental (large symbols) and theoretical (small symbols) g(2:, 7/2;) factors of the N = 28 isotones. Results of calculations with the FPD6 interaction, assuming the excitation oft = 1 and t = 3 nucleons from the OfTI shell to the other fpshell orbits (Table 4), are displayed. The solid lines are linear fits to the data; the dashed (t = 1) and dot-dashed (t = 3) lines are drawn to guide the eye.

120

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

well understood. However, the calculations depend not only on the choice of effective interaction, but even more sensitively on the degree of truncation of the model space. As shown in Table 4, with the excitation of three particles from the f7,2 shell to the other fp shell orbits, the difference in the slopes for the odd and even isotones vanishes for both FPD6 and the KBBG interactions. The same reduction in slope is predicted with full fp shell calculations [7, 891. Evidently, large-scale calculations do not explain the observed experimental slopes for these nuclei. The calculated g factors for the ground states of the odd nuclei and the 2: states of 52Cr and 54Fe are in fair agreement with experiment, but the theory fails in predicting the large observed g(2:) value of 50Ti. It is noted that the same calculations also underestimate the experimental values for the other even-A titanium isotopes. Moreover, it was argued [89] whether the calculated g factors depend strongly on the choice of the effective interaction. Table 4 shows that on the t = 3 level, there are only small differences among the g(2:) values calculated with the most frequently used FPD6 and KB3G interactions. It appears that all interactions allow too much configuration mixing. In order to obtain better insight into the core polarization phenomenon, the problem was examined using first order perturbation theory. This approach yields somewhat different results from those of the shell model diagonalizations. The difference arises in part because, in perturbation theory, only the excitation from f,,z to f5/2 contributes to the change in g and only linear terms in the interaction are retained. The difference Sg, between the g factor of a nucleus with two protons coupled to angular momentum I and that of a one-proton nucleus in the I = 712 ground state, &I =

g(50Ti)'=2

_ g(49,Q)'=7/2

(17)

is given by: &J’=X’.M’

(18)

where M’ = (91 - 9s). < (jj)‘l&l(jj’)’ with j = 712, j’ = 5/2 and the energy denominator For X’ one obtains: f ,

x2+=; J

>I,T=l

AE = ~(f~,~) -

x4+=3,

E(f5,2)which is around -6 MeV.

x6+,2.

7fi

(19)

7&

(26)

It can be shown, that 6g1 has opposite signs for two holes and two particles. Thus gf48’e)’ = g(Wo) 69’ [89]. A relation between the effective slopes m of even - even and odd - even nuclei may be written in terms of 69’: ml=2 = 312. mI=7f2 - 112. 6gIc2 (21) where m is defined as, mz+(even - A) = [(g(54Fe)‘=2 - g(50Z’i)‘=2)]/4

(22)

and m7i2- (odd - A) = [(g(55Co)‘=‘/2 - g(51V)‘=7/2)]/4 (23) It is evident from eq. (21) that even when 69’ vanishes, even and odd-A nuclei have different slopes. In the case of I = 2, the key matrix element is < (jj)21Vl(jj’)2 > 1=2,~=1 with j = j7/2 and j’ = f5/2. Its value varies for different interactions (see also [89]). Hence, one expects different 69’ values and thereby different slopes and g factors. In first order perturbation theory, the g factors of the 2: states of even-A and those of the (7/2)ground states of od&A nuclei not only lie on straight lines with different slopes but should intersect at midshell, namely at 52Cr [89]. This prediction is obviously not supported by experiment, showing that calculations beyond first order perturbation theory are important. As seen in Fig. 23 truncated shell model diagonalizations give rise to an intersection point, which is displaced towards lower-Z, N = 28 isotones. On the other hand, large-scale calculations do not yield any intersection which is rather surprising. In conclusion, the theoretical situation in the core polarization framework indicates that configuration mixing calculations with excitation of a single particle display many of the qualitative features

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

121

of the experimental results. In particular, the calculations reproduce the linear 2 dependence and the corresponding slopes of the g factors of even-A and odd-A N = 28 isotones. However, the question remains why full fp shell model calculations do not yield the same good results. 4.2.4

g factors

and B(E2)

values of even-A

Early attempts to describe heavy fp calculations [30, 92, 931 emphasized, shell across the energy gap at 2, N = the structure of isotopes heavier than B(E2) measurements [90, 911.

Nickel isotopes

shell nuclei in the iron, cobalt and nickel region by shell model that the excitation of both protons and neutrons from the f7/2 28 is essential for explaining nuclear properties. In this context, 56Ni in this shell has recently been tested by new g-factor and

ZIPS1

Erevel

ZiPSI

ZbSl

[keel 3ooc-

2000 -

1000 -

%i30

%i,,

%i34

%i,,

(T,n = 6.1 d) Figure 24: Low-lying states of even-A Ni isotopes and relevant y transitions. Also displayed is the low-level scheme of radioactive %Ni. The new lifetime values are supplemented by those of ref. [95].

The experimental data on the Ni isotopes was rather sparse prior to our new measurements [90, 911. Magnetic moments of the first 2+ states had been determined for all stable even-A isotopes (Fig. 24) with moderate accuracy, employing the conventional TF technique of Coulomb excitation of isotopically pure nickel targets [94]. Much higher precision was achieved for the g factors of the 3/2- ground states and 5/2- excited states of odd isotopes [SS] (due to the longer lifetimes), which permitted the application of high precision resonance techniques. The shell model interpretation of the odd nuclei suggests distinct single neutron configurations, ~312 and f51z,in the wave function of the states, which are responsible for the observed negative g factors of the 3/2- and positive g factors of the 5/2- states, respectively. The picture changes completely for the 2: states of the even isotopes: the experimentally known g factors [94] increase from a possibly negative value for %Ni to positive values for the heavier isotopes ‘*@Ni, a trend attributed to an increase of collectivity towards g = Z/A = 0.44. It was clear that, for a better understanding of the nuclear structure the precision of the data had to be increased. This was achieved with the new technique of projectile Coulomb excitation in inverse kinematics. Isotopically pure Ni beams were accelerated to energies of 155 Mel/ and 160 Mel/ and Coulomb excited to the first 2+ states by scattering from carbon of the same multilayered target used in the precession measurements for the titanium and chromium isotopes (sects. 4.2.1 and 4.2.2). The arrangement of 7 detectors and the 0” particle detector, as well as the overall performance of the measurements for the whole series of isotopes, were identical to those described in the preceding experiments. Typical

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

122

22so~ (b)

‘54

‘200(a)

2050 1000

/

I

1500

L

400 200 0-

‘9%

I

1750

800

6600

2:+0;

I

4; + 2;

800 1000 1200 1400 1600 1800 2000

d

Figure 25: (a) BaFs and (b) Ge -y-ray coincidence spectra displaying 7 transitions from Coulomb excitation of 155 MeV “Ni projectiles. The (2: --t 2:) transition, quoted in the literature, has not been observed (see arrow, Fig. 24 and [91]).

y-coincidence spectra are shown for the case of 5sNi in Fig. 25. The g factors were derived from experimental precessions by determining the effective TF strength on the basis of the linear parametrization (eq. (10)). The attenuation factor (sect. 2.3), in the specific conditions of the Ni ions, was determined asGben,,,= 0.69(6). This value fits nicely into the systematics shown in Fig. 3. Besides the g factors, the lifetimes of the 2: states, as well as those of higher excited states, were simultaneously measured using the DSAM technique. It is noted that all new lifetime values for the 2: states are slightly larger than those quoted previously in the literature, the exception being 62Ni, for which the known value was confirmed [90, 911. The results, including B(E2) values deduced from the measured lifetimes. are summarized in Table 5. The resulting g factors and B(E2)‘s were compared with large-scale shell model calculations in the fp shell configuration space using the same shell model code ss for the lighter fp shell nuclei. The general trends of g factors and B(E2) values are very well described by the calculations. As shown in Figs. 26 and 27, agreement with the data is considerably improved by including excitations oft nucleons (protons and neutrons) from the Ofr/s shell to the lps/2,Of5/2 and lpi/s orbits. The most striking result emerges for 5sNi(2:) with its small but definitely positive g factor, as the calculations evolve from negative values for t ‘= 0 and t = 2 to positive values, finally converging for t = 5 to precisely the experimental value (Table 5). For all other isotopes, the agreement between theory and experiment is essentially of the same quality, but, since the g factors are all positive, the dependence on the number of excited particles is much reduced and less dramatic than for 58Ni. A full calculation was made for 64Ni(2:) which agrees very well with the t = 5 result. It is also evident from Fig. 26 that for an inert “Ni core, equivalent to a closed jr/r shell (t = 0), the calculations generally overestimate the g factors. This observation supports a strong coupling of the valence particles to an excited * Ni core. The same conclusion has been drawn by Otsuka et al. [34, 351 in Monte Carlo shell model calculations. The B(E2)‘s exhibit a similar behaviour, as seen in Fig. 27. The magnitude of the B(E2) values increases steadily with the number of particle-hole excitations, but finally converges at t = 5, as shown by the full calculations for 64Ni. This dependence of B(E2) on t is much stronger than for the g factors, reflecting the differing sensitivity of the two quantities. The missing strength at the t = 5 level

123

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Table 5: Newly-measured g factors and lifetimes with deduced B(E2)‘s in Weisskopf units, and shell model (SM) calculations for excited states in even-A Ni isotopes [90, 911 compared to previous data [94, 951

T

r

B(E2) _1[W.U.]

T(I) [PSI

r

g(I)

L

Nucleus

I”

WI

’new [90,911

Expt. =

jsNi

2:

0.96(4)

1.27(2)

7.4(l)

60Ni

4+ 1

21.4

5.4(6)

11(l)

2:

0.075(14)

0.108(10)

1.4(4)

2:

0.050(4)

0.076( 12)

5(4)

2:

1.03(2)

1.31(3)

10.6(2)

WI

SM,,6 -

new [90, 911

6.60

-0.06( 12)

+0.0378(85)

-

-

_

-

-

-

-

-

+0.09(12)

+0.158(28)

9.39

S&s to.0350

-

+0.0975

_

4:

1.6(4)

4.8(1.5)

6(2)

62Ni

2+ 1

2.09(3)

2.01(7)

12.5(4)

10.02

+0.33(12)

+0.167(24)

+0.2050

64Ni

2+ 1

1.27(4)

1.57(5)

7.7(3)

5.33

+0.45(12)

+0.184(31)

+0.1405

6.6(1.0)

-

4+ 1

20.45

-0.11

2.5(4)

, %i

,

, %i

_

i

,

, 62Ni

,

,

1

64Ni

Figure 26: Comparison of experimental g factors of 2: states in even-A Ni [90, 911 isotopes with results from shell model calculations in which the number t of the nucleons excited from the Ofrll shell into the remaining fp shell orbits (open points) was varied. The result of a full calculation for s4Ni is denoted by a star. Lines are drawn to guide the eye.

might be attributed to an underestimation of the effective charge for protons and neutrons used in the calculations, e,,!(a) = 1.5 e and e,ff(v) = 0.5 e, or to a lack of quadrupole strength in the residual interaction. As seen from Fig. 27, the general trend of the B(E2) ‘s with increasing number of neutrons

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

124

O@0

I

58Ni

_

,o-

I

60Ni

--o-

_

-0

I

62Ni

I

@Ni

Figure 27: Experimental B(E2) ‘s of (2: + 0:) transitions in Weisskopf units (solid points) (90,911 are compared with shell model calculations in which the number t of nucleons excited from the Ofr,r shell into the other fp shell orbits (open points) is increased. The result of a full calculation for 64Ni is denoted by a star. Lines are drawn to guide the eye.

is very well reproduced by the calculations. It is noteworthy that the new measurement for mNi brings the experimental data much closer to the calculated value. Hence, the peaking of B(E2) is shifted from 60Ni to 62Ni, in accordance with the calculations. The inherent collectivity attains its maximum precisely at the middle of the fp shell (beyond the “Ni core), where the number of particles and holes is equal. As a consequence of the gradual filling of the fp shell orbits, the amount of collectivity is reduced as shown by the decrease of B(E2). This is supplemented by the recent results for the neutron-rich ‘j6Ni and ssNi isotopes [96]. In conclusion, the new g factors and B(E2) values of all stable even-A Ni isotopes are well reproduced by the large-scale shell model calculations. This excellent agreement was achieved only by abandoning the concept of an inert “Ni core through the excitation of more than two particles from the f,/z shell. This scenario is particularly striking for explaining the positive g factor of the 58Ni(2:) state. It shows again that the present understanding of the nuclear structure of the Ni isotopes results from high precision data obtainable only with the novel technique of projectile excitation in inverse kinematics. 4.2.5

g factors

and B(E2) values of even-A

Zinc isotopes

In sequel to our investigations on the nickel isotopes; including the outstanding description by largescale shell model calculations, we have studied the zinc isotopes [106], the next series of even-A nuclei in the fp shell. For the Zn isotopes sz~sa~66~68~702 n (Fig. 28), with two protons and 4-12 neutrons outside the doubly magic N = 2 = 28 shell, transitions from single particle to collective degrees of freedom are expected. This scenario has previously been discussed in several theoretical papers based on the nuclear shell model and in the framework of particle-core coupling calculations [97, 98, 99, 1001. With respect to the shell model frame, the restriction of the configuration space to the fp shell orbits must be relaxed by including particle excitations to the Oga/z orbit. In odd mass isotopes gs/r single particle states are clearly identified at low excitation energies. There is further evidence that, particularly

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154

125

2.3

2.32

4 30 62h2

‘“Z %o 30

Z%

(T,,= 9.2 h)

Figure 28: Low-lying states of even-A Zn isotopes and relevant y transitions. lifetime values are supplemeted by those of ref. [107].

The new

for the heavier Zn isotopes, the wave functions of low energy states have strong components of this configuration. Furthermore, in a recent study of high spin states in “Zn, superdeformation was found for the first time in this mass region [loll. This phenomenon was expected from calculations predicting large shell gaps in the single particle energies for proton and neutron numbers N, Z = 30 - 32 [102, 1031. Because of the small number of valence particles in these nuclei and their proximity to the N = Z line, nuclear structure studies are of particular interest, since reliable microscopic calculations are available to examine the influence of isospin symmetry and neutron-proton pairing correlations.

16OOp

I

1 ' I ' r 2;+0;

I

”

’

I

@Zll

900

I

27-o;

I

I

62Zll

1400 : 1200: ,lOOO~ 5

u

8OOk 600 ;

699

8991OCiI

1299 1409 Energy [keV]

1000

1500 2000 Energy [keV]

Figure 29: Ge y-ray coincidence spectra of Coulomb excited 64Zn, and 62Zn from o-transfer reactions to 58Ni projectiles (see Fig. 30).

For these isotopes g factors of the first 2+ states were known, from two independent measurements (with the exception of the unstable 62Zn) with relatively low precision and slightly conflicting results [104, 1051. From these data an overall collective Z/A behaviour was concluded; a more refined interpre-

126

K.-H. Speidel et al. /Prog.

12ooo

urpalticlesfium F*C(‘%i, 2a) %n’

’

1000

1500

2000

Part, Nucl. Phys. 49 (2002) 91-154

‘*Cions from “C(‘%i, “C) Yd

2500

i

3000 3500 Energy [Channels]

Figure 30: Particle spectrum in coincidence with y rays for ‘*Ni beams. The cr particles associated with s2Zn are well separated from the carbon ions corresponding to Coulomb excited 58Ni.

tation did not make sense at the time. In the new measurements [106], precision determination of both 9 factors and lifetimes of the 2: states were therefore the main goal. Another experimental challenge was to obtain the analogous information also for ‘j2Zn. In the experiments with stable nuclei, isotopically-pure Zn beams were provided in their natural abundances as ZnO- ions by the ion source of the Munich tandem Van de Graaff and accelerated to energies of 160 MeV with intensities of about 20 nA. For the least abundant “Zn isotope (0.6 %) the beam current of 2 nA on the target was sufficient ,to obtain accurate data, due to the high detection efficiency of the experimental technique. The beam ions were Coulomb excited to the 2: states by scattering from carbon of the same multilayered target as used in the former measurements for Ti, Cr and Ni isotopes. The y rays emitted from the excited states (Fig. 28) were measured in coincidence with forward scattered carbon ions using large volume BaFz scintillators. A conventional Si detector with 100 pm thickness and 300 mm’ area served as ion detector. The resulting y-coincidence spectra were extremely clean without any noticeable background (Fig. 29). In the absence of a radioactive “Zn (Ti,z = 9.2 h) beam, this nucleus was produced in an o-transfer reaction. For this purpose the same carbon target was bombarded with a 58Ni beam of 155 MeV with the subsequent reaction, 5sNi(‘2C,8Be)62Zn in which the 2: state of 62Zn was predominantly populated. Because of this selectivity towards low spin states, the absence of substantial feeding from higher excited states ensured a clean and well-controlled precession measurement of the 2: state. In order to obtain this information, the Si detector was operated at very low bias (see sect. 4.2.1) for separating the two (Y particles (from the decay of ‘Be) from carbon ions associated with Coulomb excitation of 58Ni (Fig. 30). Particle-y-angular correlations and precessions were measured for all Zn isotopes. It is noted that the y anisotropy of the 62Zn(2T --f 0:) transition was relatively small but still large enough to determine a significant precession. The lifetimes obtained with high precision from DSAM measurements provided B(E2) values for all (2: --t 0:) transitions and in addition a first result for the 62Zn(4: + 2:) transition as well. The experimental results are summarized in Tables 6 and 7. More experimental details are reported in [106].

127

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Table 6: Comparison of the newly-measured literature [104, 105, 1071.

r Nucleus

I”

r

T(I) [PSI new [106]

PO71 P

62Zn

g factors and lifetimes [106] with data from

9cm

%ef. [104]

new [106]

1 Ref. [105]

P

_

_

+0.371(99)

-

-

+0.46(10)

+0.445(46)

2+ 1

4.2(3)

4.3(3)

4+ 1

0.76(+;;)

1.2(l)

64Zn

2:

2.60(6)

2.70(8)

66Zn

2+ 1

2.38(9)

2.43(5)

+0.399(41)

68Zn

2:

2.18(9)

2.32(7)

+0.436(47)

rOZn

2:

4.2(4)

5.3(3)

+0.378(42)

+0.42(8)

Table 7: Comparison of the new experimental g factors and B(E2) values in Weisskopf units with results from large-scale shell model calculations LSSM I, LSSM II and LSSM III (see text and [106]).

B(E2) Nucleus

I”

new [106]

LSSM I

J [W.U.] LSSM II P

62Zn

2+ 1

17(l)

16.9

11.4

1 4+

16(l)

19.8

15.0

64Zn

2:

20.7(6)

15.4

11.3

66Zn

2:

17.5(4)

12.5

10.5

68Zn

2+ 1

14.7(4)

9.0

9.7

“Zn

2+ 1

16.5(g)

4.4

9.7

g(W LSSM III

new [lOS]

LSSM I

LSSM II

+0.46

+0.467

$0.57

+0.448

+0.445(46)

$0.48

+0.448

+0.399(41)

+0.48

+0.578

21.2

+0.436(47)

+0.58

$0.733

17.7

+0.378(42)

+1.70

+0.633

! I _

+0.371(99)

128

K.-H. Speiakl et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

The new g factor and B(E2) data (1061 were compared with results from large-scale shell model calculations, assuming two different configuration spaces, with 40Ca and 56Ni as respective inert cores. In addition, calculations were performed for the heavy isotopes with an inert 48Ca core, which permits extending the model space for protons to the Ojr/z shell. Model space I (LSSM I) refers to a 40Ca core with the valence orbitals Ojr;z , lpa/r, lpllB and Ojr+ Due to the extremely large dimensionality involved, the configuration space was restricted to a maximum of 4p - 4h excitations for 62Z’n and 64Zn This restriction, however, has only a minor influence on the calculated quantities. For instance, in 66Zn the energy gain between the 4p - 4h space and the full space results is only 17 IceV. In all calculations KBSG was used as the effective interaction. Model space II (LSSM II) is based on a 56Ni core considering 0j5/a, lpala, lpi/r and Oge/z valence orbitals. In this case, the twobody matrix elements are from a G matrix calculation [108, 1091whose monopoles were modified to reproduce energy systematics in nickel isotopes and N = 50 isotones [llO, 1111. All calculations were performed with the computer code ANTOINE [71]. Moreover, a polarization effective charge of 0.5 was used for the E2 transition rates and effective g factors for the magnetic moments (see sect. 1). In view of the results and discussions for the Ni isotopes (see above), the jp shell valence space is expected to be relevant also for the Zn isotopes up to neutron number N = 36, whereas the inclusion of the ge/r orbital becomes extremely important for describing the heavier Zn isotopes. This feature was also emphasized in earlier calculations (see e.g. [97]).

‘.*c

-

1.6 L

Figure 31: Experimental g factors of the 2: states (large solid circles and star) are compared with results from shell model calculations. Small circles refer to a 40Ca core (LSSM I), squares to a 5sNi core (LSSM II) and triangles to a 4sCa core (LSSM III). The lines are drawn to guide the eye (see text and [106]).

The new g factor and B(E2) data are displayed together with the results of LSSM (I,II,III) calculations in Table 7 and Figs. 31 and 32. As expected, the LSSM I results are in rather good agreement with the values for “Zn through m.Zn. With respect to the g(2:) values, the calculations show almost no dependence on neutron number for these isotopes. The rise (not seen in the experimental data) for “Zn and more dramatically for “Zn, simply reflects subshell closures in the limited fp shell valence

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

0

t

I

"Zll

I

I

I

I

@Zll 66Zn @Zn "Zn

I

129

1

72Zn

Figure 32: Experimental B(E2)‘s of (2: --f 0:) in Weisskopf units (large solid circles and star) are compared with results from shell model calculations. Small circles refer to a %a core (LSSM I), squares to a “Ni core (LSSM II) and triangles to a 48Ca core (LSSM III). The lines are drawn to guide the eye (see text and [106]).

space. The striking deviation from the experimental data is therefore a strong indication of the need for a larger valence space that includes the Ogs/r orbital. The importance of the additional Ogs/r orbital for describing the heavy Zn isotopes originates from the strong T = 1 attraction between the 0f5iz,

62Zn

%n %n

“Zn

‘Ozn

Figure 33: Experimental B(E2)‘s of (4: + 2:) in Weisskopf units (large open triangles and star [106, 1071) are compared with results from shell model calculations. Small circles refer to a %‘a core (LSSM I), squares to a 56Ni core (LSSM II). The lines are drawn to guide the eye.

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

130

lp3i2, 1p~1~ and the 0gsj2 orbitals: at the top end of the fp shell, filling the neutrons brings down the Ogslz shell. Its spectacular effect on the g(2:) factor of ‘OZn is shown by the LSSM II result which explicitely includes this orbital. Evidently, for the lighter isotopes particle excitations from the Ofr/z shell significantly improve the agreement with the data (see Figs. 31 and 32). The same tendency was seen for the Ni isotopes, smoothing the g factor dependence on neutron number (sect. 4.2.4). On the other hand, the insignificantly small difference between the LSSM I and II results for the g factors of 62Zn and 64Zn shows that excitation of particles from the Ofrlz shell and population of the @Q/2 shell play only a minor role for the light Zn nuclei. This feature is, however, not seen in the B(E2) values (Fig. 32). From these results it becomes clear, that large-scale model calculations in the fpg configuration space would certainly be the optimum approach. However, this is not currently feasible due to computer limitations. An almost equivalent approach in this direction are calculations considering a 48Ca core (LSSM III): the configuration space includes for protons the Ofrlr, 1~312, lpi/r and Ofs,r orbitals and Of5/2,lps/z, lpl/p and @Q/2 orbitals for neutrons. In the residual interaction strong fs/z - ggp T = 0 attraction has been included. The results for the g factors and the B(E2)‘s of mZn and “Zn are displayed in Table 7 and Figs. 31 and 32. Evidently, there is very good agreement with the experimental g factors of both ssZn and “Zn, whereas the B(E2) predictions only reproduce well the “Zn value. The same striking improvement over the LSSM I and II results is not obtained for the B(E2) of 6sZn, where the collectivity is overestimated. However, one should bear in mind that the calculations which involve (6p - 6h) excitations have not yet fully converged. It is further noteworthy that the same calculations, when applied to the heavy nickel isotopes 62-74Ni, yield equally good results a.s former calculations baaed on a 40Ca core. Another interesting result is that the rather accurate B(E2) value of the (4: --t 2:) transition of 62Zn agrees very well with both the LSSM I and II calculations (Fig. 33 and Table 7). More precise measurements are highly desirable, in order to test the calculations, which predict a decrease of the B(E2; 4: --t 2:) with increasing neutron number, in contrast to previous data. In conclusion, the g factor and B(E2) results for the Zn isotopes are rather well described by largescale shell model calculations assuming a 40Ca core. That 40Ca is a more suitable core than 5sNi was already an essential feature in explaining the Ni data. On the other hand, the inclusion of the OgQ12 orbital is very important for describing the heavy Zn isotopes. With this enlargement of the valence space, calculations are restricted to a 48Ca core which, however, seem to be a reasonable compromise, in view of the presently-unfeasible 40Ca core calculations. The selected results using 48Ca as core demonstrate the well known fact that the N = 28 shell closure is a rather stable configuration. Moreover, the calculated g factors of 4: states exhibit rather large variations compared to the 2: values, a pattern which should definitely be investigated in future experiments. However, these require higher beam energies in order to enhance the Coulomb excitation cross sections.

4.3

Magnetic moments shell closure

of Selenium and Krypton

isotopes:

the N = 50

The stable even-A 74-82Se (Z = 34) and 78-86Kr (Z = 36) isotopes are transitional nuclei displaying rather particular features. Beside a varying collectivity, reflected by the B(E2) values, they also exhibit pronounced single particle and shell effects. Both series of isotopes approach or reach the N = 50 shell closure, where proton excitations should dominate the nuclear wave functions. In an early work by Kaup et al. [112] systematic calculations were performed in the framework of the interacting boson IBA-II model, to determine the extent to which single particle excitations compete with collective degrees of freedom. 4.3.1

g factors

of excited

states of even - A Selenium

isotopes

Until recently, the magnetic moments of the first 2+ states of all even-A Se isotopes were only known to rather low accuracy [86] permitting no more than crude interpretations of the underlying structure.

131

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

In fact, the g(2:) values turned out to be equal for all isotopes, as consistent with the prediction for vibrational nuclei, g = Z/A N 0.43. Within the large uncertainties of the experimental data no structure dependence of the wave functions on neutron number was seen.

Figure 34: Low-lying states of “-szSe

In the kinematics rays, but technique 2: states.

isotopes with relevant y transitions.

new measurements [24], Coulomb excitation of isotopically-pure selenium beams in inverse was applied, to increase the statistical accuracy via the enhanced detection efficiency of y also to improve the reliability of the data by using the same target for all isotopes. This allowed to measure not only g factors of the 2: states but also of the higher excited 4: and Fig. 34 displays the main decay features of low-lying states of the isotopes in question.

1

1w

*‘*‘~‘*‘~“‘~‘~‘~’ 300

500

700

900

1100

1300

15M

1700

1m

Energy(keV)

Figure 35: Spectra of 76Se y rays detected in coincidence with Si ions in a NaI(T1) scintillator and a Ge detector at forward angles. The relevant y lines are assigned according to the known level scheme (Fig. 34, [24])

Beams with energies between 230 and 262 MeV were urovided bv the tandem accelerator at Yale. All five isotopes 74+& (0.9 %), 76Se (9.0 o/o), “Se (23.5-%/o), @‘Se “(49.6 %) and ‘*Se (9.4 %) were produced according to their natural abundancies (in brackets) in the ion source of the accelerator

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

132

Table 8: Comparison of new experimental g factors with Z/A and results from IBA-II calculations with N = 50 shell closure. The B(E2) values in Weisskopf units refer to (2: + O:)-, (4: + 2:)- and (2: + 2:) transitions [24].

r(J) Nucleus

I”

74Se

‘?3e

‘Y3e

T3e

%e

r

[PSI

I

13(E2; 1i 3 1,) 4

I

1

Calc.

[W.U.]

new [24]

talc. -

Z/A - t

new [24]

Il.459

1.000

1.000

0.447

0.942(36)

1.021

0.436

0.898(35)

1.070

0.425

1.017(35)

1.123

0.415

1.159(39)

1.133

2+ 1

10.2(l)

41.3(5)

+0.428(27)

0.383

4:

<1.68(12)

79(4)

+0.50(10)

0.368

2:

5.8(16)

47( 13)

+0.55(9)

0.360

2:

17.7(3)

43.5(7)

+0.403(23)

0.391

4:

2.19(7)

70(2)

+0.64(9)

0.381

2:

4.9(3)

42(3)

+0.35(6)

0.373

2:

14.0(4)

33.3(9)

+0.384(25)

0.410

4:

1.51(7)

48(2)

+0.39(12)

0.408

2:

6.1(4)

22(l)

+0.33(11)

0.408

2+ 1

12.4(2)

24.2(4)

+0.435(27)

0.430

‘

g(2:;A Se)/g(2f;74 Se)

g(I)

4+ 1

0.95(3)

35(l)

+0.68(25)

0.447

2:

2.81(10)

18.5(7)

+0.35(10)

0.465

2:

18.9(3)

16.7(3)

+0.496(29)O

0.434

4:

1.38(22)

19(3)

+0.57(38)

0.553

2:

1.36( 16)

4.5(5)

-

0.508

with adequate intensities on the target. The target was a composite of several materials as in former similar measurements. A 0.95 mg/n2 layer of natural silicon was vacuum-deposited on a 4.4 mg/n2 Gd layer, which had itself been evaporated on a 1.0 mg/cm’ tantalum foil backed by 1.35 mg/cm’ of aluminium. The Se nuclei were excited in the silicon layer and experienced spin precession while traversing the subsequent magnetized Gd layer. Finally, the nuclei were stopped in the Al backing, a hyperfine-interaction-free environment. The forward scattered Si ions were energetic enough to pass through the entire target and an additional copper foil beam stop, to be detected in a 0” Si detector. This set-up permitted to measure the de-excitation y rays in coincidence with the Si ions, as in former measurements in this geometry. The +yrays were detected with NaI(T1) scintillators, whose energy resolution was sufficient to separate all relevant y lines. The absence of disturbing background y lines in the energy region of interest was confirmed by the spectrum of a Ge detector taken at forward angles. Typical y-coincidence spectra are shown in Fig. 35. Contrary to the above-mentioned measurements on Ti, 0, Ni and Zn nuclei, the analysis of the measured precessions of the 2: states of the Se isotopes involved significant corrections due to feeding from higher lying states. This procedure requires the precise knowledge of the excitation cross sections of the feeding states, which have been reliably calculated for the specific experimental conditions. The

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

133

feeding affects the precession in two ways: (i) the slope of the measured (2: --t 0:) angular correlation is the composite of contributions from the feeding states, and (ii) the precessions of the 4: and the 2; states themselves (the 0: state does not precess) must be taken into account explicitly. Both these effects are characterized by the population strength of the feeding states and by the angular correlations, in which the feeding transitions to the 2: states are not observed. Details of the analysis are described in [24].

1.3 ( IBA-II calculation (N=50 shell closure) ../

,__.._.._.... -.--.---i

Figure 36: Ratio of measured g(ASe(2:)) to g(‘“Se(2:)) in comparison with the IBA-II calculations. The dashed curve shows the results for the N = 50 shell closure; the continuous curve assumes an additional subshell closure at N = 38 (see [24]).

For the determination of the g factors from the measured precessions the transient field strength must be known. In order to obtain results independent of existing parametrizations, the field was calibrated with the known g factor of s*Se(2:), g = +0.496(29) [87]. This procedure also accounts for ion-beam-induced attenuations of the TF strength, which was determined independently and amounts to G=0.66(6), consistent with the expectations for the specific conditions of probe and beam ions (see Fig. 3). The derived absolute values of g factors of the individual excited states are summarized in Table 8. In Fig. 36 the results are displayed as ratios of g(ASe(2:)) to g(74Se(2:)), the g factor of the lightest isotope in the series. For these ratios small differences in ion velocities were corrected using TF parametrizations. The Rutgers as well as the linear parametrizations (eqs. (10) and (11)) gave the same results, showing that these ratios are insensitive to the inherent differences in these parametrizations. The errors on the g factor ratios reflect the experimental uncertainties of the measured precessions. It is evident from the data that, while the experimental g factors are of the order of the Z/A collective values, they also exhibit significant trends with neutron number. The results were discussed in the framework of the IBA-II model, which should be applicable to the Se nuclei due to their transitional character. The atomic number 2 = 34 is far from the closed shell 2 = 28. The neutron number varies from N = 48 for **Se, near shell closure at N = 50, to N = 40 for 74Se, significantly far from N = 50, but at a possible subshell closure at N = 40, as observed for Zr (Z = 40) nuclei. Extensive IBA-II calculations have been carried out in the past to explain several properties of these nuclei in a consistent manner [112, 1131, but g factors of the first 2+ states were only calculated by the authors of [113]. Their results, a constant g = +0.40 for 74-7SSe, and a slightly larger g = +0.43 for 8oSe,

134

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154

are in disagreement

with the present data which show a distinct decrease from 74Se to ‘*Se. In addition, the new g factors of the 2: and 4: states have not yet been discussed within IBA-II. Therefore, new calculations based on slightly different parameters were required in order to obtain better agreement with the magnetic properties of these nuclei and simultaneously reproduce the experimental electric quadrupole moments and transition probabilities. Details of these calculations based on the work by Otsuka et al. [114] are described in [24] and presented in Table 8 and Fig. 36. As can also be seen in Fig. 36, the calculations suggest a monotonic increase with neutron number and do not explain the observed drop at N = 42,44. However, if one assumes an additional subshell closure at N = 38, the model predicts g factor ratios in fairly good agreement with the experimental values (Fig. 36). At present, however, it is not clear that the assumption of this subshell closure is indeed physical. More data, including results on neighbouring nuclei such as krypton, are needed to test this intriguing feature. 4.3.2

g factors

of excited

states

of even - A Krypton

isotopes

3 IoooT0 4:

2:

h P 5w-I 0 - 0:

‘kr

72Kr Tln=ll.5m

r,n = 14.8 h

Figure 37: Low-lying states with relevant y transitions for even-A radioactive and stable Kr isotopes [119]. The new lifetime values are supplemented by those of ref. (1191.

The krypton nuclei represent a rich laboratory for the study of nuclear structure effects. Partial energy level schemes of the isotopes in question are shown in Fig. 37. The radioactive isotopes 72Kr,

K.-H. Speidel et al. / Prog. Part. Nucl.

Phys. 49 (2002) 91-154

135

74Kr, and 76Kr belong to the class of nuclei with N N Z, where both neutrons and protons occupy the same single particle orbital5 and are characterized by strong neutron-proton pairing correlations in comparison with the better established like-nucleon (nn and pp) pairing modes [115]. In like-nucleon pairing, the nucleons can only form I = 0, T = 1 isovector pairs, whereas for np pairing also the coupling to isoscalar pairs with I = 1, T = 0 is allowed. The existence and properties of the two pairing modes in N N Z nuclei has recently become an issue of major theoretical interest. These nuclei, together with the heavier stable isotopes “Kr, 80Kr and ‘*Kr, exhibit dominant collective structures with coexistent prolate and oblate shapes [116] (see also Fig. 37). AS the N = 50 shell closure is approached for 84Kr and 86Kr, single particle degrees of freedom of both protons and neutrons play the dominant role in the nuclear structure. In particular, particle-hole states in the fp shell (l~~,~, 0f5,2r and lp& as well as the 0gsj2 orbital for protons and Ogs/r hole states for neutrons, are expected to be the principal components of the nuclear wave functions. Moreover, the proximity to the Se isotopes, for which the g factors of the 2: states were well described by IBA-II calculations with an assumed closed neutron shell at N = 38, raises the intriguing question as to whether this subshell closure might also play a role in the Kr isotopes. A systematic study of the magnetic moments for the Kr isotopes as function of neutron number could shed light on the many facets of the structure of isotopes in this mass region. The very first measurements of g factors for the excited 2: and 4: states of all stable even Kr isotopes were performed only recently, employing the technique of projectile Coulomb excitation in inverse kinematics [117]. Only distinguishing characteristics pertaining to this specific experiment, which has many aspects in common with former measurements, will be presented here. Isotopically pure beams of 78-86Kr were accelerated at the Berkeley 88 Inch Cyclotron to energies between 220 and 261 Mel/. For all isotopes the same multilayered target was used. It consisted of 0.9 mg/cm* of enriched 26Mg evaporated onto a 4.0 mg/cm’ gadolinium layer, itself deposited on a 1.1 mg/cm2 tantalum foil, backed by a 3.9 mg/cm2 copper layer. 26Mg was selected for Coulomb excitation of the Kr isotopes, as the cross sections were large and the (2: + 0:) 7 line from target excitation was of no disturbance in the y spectra. The recoiling Mg ions were detected in a Si solar cell detector placed at 0” with respect to the beam direction. The y rays emitted from the excited states were measured in coincidence with the forward scattered Mg ions, either with NaI(T1) scintillators or Ge detectors, depending on the y energies and the energy resolution required. A typical spectrum is shown for ‘*Kr in Fig. 38. Together with the precessions, the lifetimes of most excited states of interest were also measured simultaneously, employing the DSAM technique.

lo’-

I 200

I

I 600

1000

1400 1800 Energy [keV]

Figure 38: Typical y coincidence spectrum of ‘sKr obtained with a Ge detector. The broad high energy peak refers to (2: -+ 0:) y rays emitted in flight from Coulomb excited “Mg target nuclei [117].

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 6’@2) 91-1-W

136

Table 9: New experimental

g factors and lifetimes with deduced E(E2) values in Weisskopf units [lli’]. The B(E2) values refer to (2: + OF)-, (4: --t 2:)- and (2: + a:)-transitions.

**Kr

s4Kr

s6Kr

2:

6.42(26)

6.7(2)

20.4(6)

+0.402(15)

4:

0.97(36)

-

32( 12)

+0.29(20)

2:

6.28(26)

5.8(2)

12.0(4)

+0.267(13)

4:

0.649(+&)

0.95(18)

15(3)

2:

0.433(?:;‘)

0.35(7)

13.0(2)

2:

0.318(26)

0.44(3)

8.7(5)

4;

4473(865)

-

0.05(l)

-

+1.12(14)

The g factors and B(E2) values deduced are displayed in Table 9 and Fig. 39. The data clearly indicate that the Kr isotopes evolve through a variety of structures. The light 78-8zKr nuclei have large B(E2) values, implying a predominantly collective behaviour, albeit decreasing as the neutron occupation of the ge,r shell further increases at 84Kr. The g(2:) values were calculated within the IBA-II model, using the parameters already successfully applied to the Se isotopes, but replacing the number of proton bosons of N,, = 3 for Se by N, = 4 for Kr. One has to bear in mind that the neutron boson number NV is connected to N,, by the constraint N%Q,, + N”E~” = 0 (see e.g. [118]), Ed* and s&, being the single-particle energies of the proton and neutron bosons, respectively. In this context, it is especially interesting to find out whether a neutron subshell closure at N = 38, as postulated for the Se isotopes, is also applicable to the Kr isotopes. Fits obtained with the appropriate parameters and shell closures reproduce the g factors reasonably well but fail completely for the B(E2)‘s. The best simultaneous fits of both g factors and B(E2) ‘s were obtained assuming a shell closure at N = 28 (Fig. 39). At the top end of the shell, the large g factor of the 2: state of 86Kr, g = +1.12(14), with its closed neutron shell at N = 50, clearly reflects dominant proton configurations in the wave function. This feature is also reflected in the small B(E2) value. The same feature is seen for the neighbouring isotones “Sr, “Zr and ‘*MO (see further discussion below). It is noteworthy that the measured g factors of these nuclei, ssSr : g(2:) = +1.15(17) WZr : g(2:) = +1.25(21) ‘*MO : g(2:)

= +1.15(14)

[120] [121] [122]

137

K.-H. Speidel et al. / Prog. Part. Nucl, Phys. 49 (2002) 91-154 N_

0.25

P “2

0.20

lexperiment - -

-

28cN60 !,B
; :

O 0.15 t +-

3 0.10 % 0.05 0.00

I

8

1.25

t

1.00 -

_ +Sc?

0.75

m

0.50

-

74 38

76 40

70 42

80 44

02 46

84 48

86 50

Kr mass and neutron numbers

Figure 39: Experimental g(2:) factors and B(E2) ‘s in comparison with results from IBA-II calculations (see text and [117]).

Table 10: Comparison for Sr [120] isotopes.

N &e 3&r 3&

of newly-measured

g(2:) factors for Se [24] and Kr [117] with those

40

42

44

46

48

50

+0.428(27)

+0.403(23)

+0.384(25)

+0.435(27)

+0.496(29)

-

-

+0.432(27)

+0.378(47)

+0.402(15)

+0.267(13)

+1.12(14)

-

-

+0.419(47)

+0.273(50)

+1.15(17)

are within errors the same and consistent with the value for “Kr. A large shell model calculation [123], in which 10 protons and 22 neutrons can occupy the Of+, lps/r, lpi/r, Oga/z and 2d5,2 orbits, yields g(2:) = +1.34 and B(E2; 2: + 0:) = 5.8 W.U. in very good agreement with the experimental values. For 84Kr, in which two neutrons are removed from the ga/r shell, the g factor shows a striking drop similar to that observed for ssSr. This large reduction is caused by contributions of the gs,r neutron configuration with its negative Schmidt value, g(vga,r) = -0.425. Moreover, the measured g factor of

138

K.-H. Speia’elet al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

“jSr, g(2:) = +0.273(50) [120], 1sremarkably close to that of 84Kr, suggesting very similar neutron and proton configurations in the wave functions of these nuclei. In fact, the g(2:) factor of s6Sr ww well described by means of a configuration consisting of equal weights of two neutron holes in the gg/z, pi/r, f5/2and ~313 orbits coupled to the ground and first excited states of *sSr [124]. The close similarity of the nuclear wave functions of the Kr isotopes to neighbouring Se and Sr isotopes is well demonstrated by the experimental g(2:) factors, which are all very close in magnitude (see Table 10). This comparison strongly suggests that supplementary data are highly desirable for the lighter isotopes 72Se, 74,‘6Kr and 76-82Sr, which are all unstable. These nuclei, however, could be studied, if produced as radioactive beams, a project presently being pursued at various laboratories. In conclusion, the g factor and B(E2) data clearly show a transition from single particle configurations at and near the N = 50 shell closure, where neutrons and protons compete in the wave functions, to more collective structures arising from the removal of neutrons from the gal2 shell. The clear signature of ga/r neutrons is exhibited in the magnitude of the measured g factors. While the collective g factors and B(E2) values are well described in the framework of the IBA-II model, microscopic shell model calculations are certainly needed for explaining the structure of the two heavier isotopes 84@Kr.

4.4

Magnetic moments of Zirconium nuclei and the N = 50 shell closure

The Zr (2 = 40) nuclei are, due to shell closures in both protons and neutrons, of fundamental importance for the understanding of the structure of nuclei near Z = 40 and N = 50 closed shells. The four stable even-even 90~g2*g4~g6Zr isotopes, in which the neutrons fill the l&/a orbit, have almost identical and small B(E2; 2: + 0:) values of 4 - 6 W.U. Partial level schemes are displayed in Fig. 40. In a naive shell model picture, the low-lying states of g2Zr and g4Zr should be dominated by the two neutrons beyond the magic N = 50 shell. For the same reason, in “Zr only proton excitations are expected. On these simple grounds, the g factors of the 2: and 4: states of g2*g4Zr should be negative, since the Schmidt value for a d5,2 neutron is g(Vdg,2) = -0.76. If the quenched spin g value, g;/f = 0.7. g6’““, is used, an effective g factor, geff(vd512) = -0.54, is obtained which agrees very well with the measured g factors of the 5/2+ ground states of the odd isotopes ‘lZr [86] and g5Zr [125]. In case of “Zr the proton gal2 configuration should be the main component in the wave functions, implying a large positive g factor, represented by the Schmidt value, g(rgs,2) = +1.323, which is calculated with a quenched gS value. This prediction was indeed confirmed in measurements on isomeric states in g”Zr [126, 1271: g(8:) = +1.356(7)

4.4.1

g factors

of the 2: and 3; states

and

g(5;) = +1.25(3).

of semi-magic

9oZr

The magnetic moment of the first 2+ state of “Z’r was never previously measured, because of its large excitation energy of 2.186 MeV and its extremely short lifetime of r = 0.126(3) ps (Fig. 40). These experimental difficulties have been successfully overcome with the new technique of projectile Coulomb excitation on light target nuclei in inverse kinematics, which provides the necessary high detection efficiency of coincident y rays. In this experiment [121], a beam of isotopically pure “Z’r was accelerated to energies of 270 and 300 Mel/ at the Yale tandem accelerator and Coulomb excited by scattering from thin layers of highly enriched 24Mg or “Mg of two different multilayered targets. The Coulomb excitation was restricted to only two levels in “Zr: to the very short-lived 2: state at 2.186 MeV, and in addition, to the 3; state at 2.747 Mel/ with a considerably longer lifetime of r = 202(72) ps (see Fig. 40). The Mg layers had different thicknesses because of the different nuclear lifetimes. A thin layer (0.24 mg/crn2) was required for the short-lived 2: state, in order to minimize the decay fraction in the Mg layer; for the long-lived 3; state a thick layer (0.9 mg/cm2) could be tolerated. The Mg layers were evaporated on Gd layers which were backed by a thin tantalum layer and a thicker copper layer in which the excited Zr ions were stopped. The beam itself was stopped (as usual) in a thick copper

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Figure 40: Low-lying states of Zr isotopes with relevant y transitions. also displayed the low-level scheme of radioactive ssZr.

For comparison

is

IO’

Id

kc?

B

3

IO

10

10

IC MO

Iwo

I503

2ocm

2ml

3ca

Energy [keV]

Figure 41: Spectra of y rays emitted from Coulomb excited “Zr projectiles in coincidence with (a) 24Mg ions at a beam energy of 270 MeV and (b) “jA4g ions at 300 MeV. The symbols S,U refer to Doppler-shifted and un-shifted components of the (2: + 0:) transition, respectively [121].

140

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

foil placed behind the target. The forward scattered Mg ions were registered in a Si solar cell detector. The de-excitation 7 rays were detected in four Ge Clover detectors. An additional Ge detector was placed at 0” to the beam direction to redetermine the lifetime of the 2: state by the DSAM technique. Fig. 41 shows -(-coincidence spectra of “Zr from Clover detectors placed at backward angles with respect to the beam direction. At 300 MeV the long-lived 3; state is preferentially populated by nuclear interactions while Coulomb excitation of the short-lived 2: state predominates at 270 Mel/. The g factor of the 3; state was determined from two 7 transitions: (i) the (3; + 2:) 561 IceV and (ii) the (2: + 0:) 2.186 Mel/ y-ray lines fed by the 561 keV transition. This was possible, as the precession of the 3; state is fully transferred to the 2: state due to the lifetime of the 3; state, which is much longer than the transit time of the ions through the Gd layer. The clear signature of these events is associated with the Doppler-unshifted component of the (2: + 0:) -y line (Fig. 41). In contrast, the r-ray line from the directly populated 2: state is fully Doppler-shifted due to the very short lifetime. The great merit of this indirect observation of the precession of the 3; state lies in an enhanced sensitivity to the precession due to the larger anisotropy of the angular correlation. This technique clearly has further applications to other nuclei. The g factors derived are g(2;) = +1.25(21)

and

g(3;) = +0.986(56).

These large and positive values provide clear evidence of rather pure proton configurations resulting from excitations from the fp shell into the gs,r orbit. In fact, the experimental g(2:) agrees very well with the Schmidt value for a pure (ng&)s+ configuration: g = +1.323. The g factor of the 3; state is mainly determined by the configuration (ng9,2p,;‘,)3- . Applying the additivity relation to the two orbitals one obtains, g(3;) = +1.08, in good agreement with the experimental value. The inclusion of other configurations such as (ags,&$ does not cause a significant change in the g factor. In conclusion, the g factors of the 2: and 3; states imply rather pure wave functions involving predominantly proton configurations. These findings are consistent with the results on the isomeric 5; and 8: states at even higher excitation energies [126, 1271. 4.4.2

g factors of the 2: and 4: states of g2Zr and g4Zr

Contrary to WZr, the g factors of the 2: states of the g2~g4Zrisotopes had been previously measured, by means of standard Coulomb excitation with backscattering of sulphur beams from isotopically enriched Zr targets [128, 1291. The deduced g values were surprisingly small and could not be explained by a dominant d5/2 neutron configuration in the wave functions (see also sect. 4.4). In fact, considerable admixtures from other neutron configurations as well as proton configurations in the pi/a and gsla orbits had to be assumed. These rather unexpected and not really understood results were the main motivation for a new set of measurements under improved experimental conditions aiming at higher data precision and reliability and also extending the measurements to the 4: states (see Fig. 40). Like in the WZr measurements, the technique of projectile excitation in inverse kinematics was applied. Again, isotopically pure beams of g2,g4Zr were accelerated to an energy of 300 MeV. The Zr nuclei were Coulomb excited by a natural silicon layer of a multilayered target, whereby the excited nuclei were stopped in an aluminium backing. The de-excitation y rays were measured with Ge Clover detectors in coincidence with the forward scattered Si ions. As shown in the y-coincidence spectra (Fig. 42) the y-ray lines emitted from the 2: and 4: states were well resolved. In this case the nuclear lifetimes were taken from the literature. More experimental details are reported in (1301. The g factors deduced are summarized in Table 11. The values of the 2: and 4: states in both isotopes are definitely negative, consistent with the expectation that d5,2 neutrons are indeed the dominant components in the wave functions of these states. In addition, it is quite clear that g(4:) is more negative than g(2:). This behaviour bears strong similarity to the oxygen isotopes 180 and “0 where the two or four neutrons beyond the magic N = 8 shell occupy the OdSlz orbit. As seen in Table 12, the g factors of the 2: and 4: states of 18~20 0 [86] compare very well in sign and magnitude with those of g2,g4Zr. In both cases the g factors of the 2: states are smaller than the g factor of a pure (d5,2)2 configuration suggesting admixtures from other neutron configurations as well as proton

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

141

Figure 42: Ge spectra of y rays emitted from Coulomb excited g2vg42r projectiles in coincidence with forward scattered Si ions. The peaks relevant to precessions are denoted by dots.

configurations which, however, seem to be less important in the higher spin states, The g factors of the 4: states are indeed significantly larger than those of the 2: states emphasizing the purity of the d5,z neutron component in the wave functions. This result for the Zr isotopes has been predicted in the early work by Talmi [131] who pointed out, that the 2 = 40 protons are in a closed subshell and, therefore, neutron excitations require less energy. Talmi also mentioned that, due to the different spatial separation of the valence neutrons in

Table 11: Comparison of new experimental results from shell model calculations [132].

g factors of g2,g42r(2:,4:)

Nucleus

I”

new [130]

92Zr

2:

-0.180(10)

-0.08

4:

-0.50(11)

-0.32

2:

-0.329( 15)

-0.17

4+ 1

-0.8(4)

-0.30

=Zr

talc. [132]

states

[130] with

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

142

Table 12: Comparison of experimental in 18~200(861 Nucleus

g factors in

g2,g4Z~

with those of corresponding

9(2?)

9(4:)

1&/s)’

-0.180(10)

-0.50(U)

94ZT(ld5,*)-"

-0.329(15)

-0.8(4)

180(Od5/2)2

-0.287(15)

-0.62(4)

*so(o&,*)-

-0.352(15)

-

'*2T(

states

isO and ‘*ZT, the wave functions of the 2: and 4: states of ‘*ZT should have a larger d5/2 neutron component than the corresponding wave functions in 180. This, however, does not seem to be confirmed by the present data. In the same context, another interesting feature is evident from the similar energy level spacings of the two ZT isotopes (see Fig. 40), which clearly reflects a well-established particle-hole symmetry with correspondingly similar wave functions. If g2Z~ is the (1dsj2)* particle configuration, then g4Z~ represents the (lds,s)-* complementary hole configuration. Finally we wish to emphasize that in recent shell model calculations by Werner et al. [132] considering a “Sr core, the predominant d5/2 neutron nature of the 2: and 4: states of the g2,94Zr isotopes is confirmed and the calculated g factors agree fairly well with the experimental values (see Table 11).

4.5

Magnetic moments of even. - A Xenon and Neodymium the role of the N = 82 shell closure

isotopes and

Like in the Se and KT isotopes, where the N = 50 shell closure gave rise to pronounced single particle effects in the structure of these nuclei due to neutron hole configurations in the relevant gs,s orbital, the same feature is expected for Xe and Nd isotopes, which lie in the vicinity of the N = 82 shell closure. In this case, hole states in the hills orbit and particle states in the f,,2 orbit are the expected dominant single particle neutron configurations for the heavy Xe and the light Nd isotopes, respectively. Again, since the magnetic moments are sensitive to specific neutrons and protons in the wave functions, one should observe their relative contributions in particular for nuclei with neutron numbers close to the magic N = 82 shell. Naturally, these single particle effects will be masked by collective effects, which become stronger upon removal of neutrons from the closed shell. Therefore, only a systematic study of the magnetic moments for a set of isotopes, varying the neutron nzlmber and spin, will reveal the effects of single neutron configurations. 4.5.1

g factors

of 2: and 4: states

of 130-136Xe isotopes

with evidence

for proton

excitations

The stable 124-13sXe isotopes belong to the same class of nuclei as the isotopes of Ru, Pd, Cd, Te and Ba, which are characterized by configurations lying between spherical and deformed r-soft vibrators. This interpretation arises primarily from examinations of energies of excited states as well as transition probabilities. The energy ratios E(4:)/E(2:) remain fairly stable between 2.0 and 2.5 for 62 < N < 80, and the B(E2)‘s decrease from about 80 to 20 Weisskopf units over the same interval. Even within the large statistical errors of early measurements (861, the g factors of the 2: states conflict with the traditional interpretation, lying well below the Z/A value expected for a collective vibrational state. This general trend is usually attributed to differences in pairing strengths for protons and neutrons [133]. However, in the case of the Xe isotopes and other nominally-vibrational nuclei in the transitional region of closed shells, the effect of pairing’might be less important and single particles could play the key role, in particular for g factors.

143

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

4: 2:

5’m cI 0.52

;

0+7 ‘:46Xes2 Figure 43: Low-energy level schemes of Xe isotopes with relevant y transitions

The interacting boson model in its IBA-II version, in which neutrons and protons are treated separately, has been very successful in describing the g factors of 2: states of even-A nuclei from Ru to Te [86, 1341. The B(E2; 2: + 0:)‘s calculated with constant effective charge for the entire isotopic chain were in good agreement with the experimental values [135]. A recent calculation of the g factors of the 2: states of Xe and Ba isotopes [136] with the wave functions of [135] and boson g values , gr = 1 and gy becoming more negative for larger A, yielded results slightly smaller than the lowest-order IBA-II mediction. namelv

N, g=g”‘N,+~v

N” +g”’ N, + N,

(24)

where gn = 1 and gV = 0 and the proton and neutron boson numbers NT+, refer to a 132Sn (2 = 50, N = 82) core. Moreover, the model also predicted the g factors of the 4: states to be equal to those of the 2: states. With respect to the Z/A rule of collective g factors where the protons of the entire body of the nucleus contribute to the magnetic moment, in the IBA-II model only valence nucleons, as bosons in pairs of protons and neutrons, determine the magnetic moment of a nuclear state via its hole or particle character. Hence, g factor measurements play an indispensable role. The stable even-A 124-136Xe isotopes, with four protons above the Z = 50 shell and spanning the range from zero to 12 neutron holes below the closed shell at N = 82 provide fertile ground to probe the development of collective structures as the number of neutron holes increases. Single particle effects are expected to dominate near the N = 82 shell closure and may lead to variations in the g factors of low-energy states. As the nuclei become collective, g factor variations may still occur but they are harder to explain in collective models, which, to first order, predict that all low-lying states have the same g factors. In the IBA-II model F-spin mixing can account for g(2.$)/g(2:) variations of the order of N 10 % [137, 1381 but differences between g(2:) and g(4:) require the invoking of mechanisms beyond this model. With respect to the experimental situation, g factors of the 2: states were known from measurements employing the PAC technique combined with internal static or transient magnetic fields in ferromagnets [86, 1391 and also external fields, where sufficiently long nuclear lifetimes were involved. The latter refers only to the 13sXe(4:) state [139]. First determinations of the g factors of the 2: states of 134Xe and 136Xe were obtained in measurements employing conventional Coulomb excitation in combination with TF in iron [140]. In these experiments, the xenon nuclei were excited by backscattering of sulphur beams from isotopically pure Xe targets obtained by implantation with the Bonn isotope separator. These measurements yielded a dramatic increase of the g factors going from 132Xe to ‘%Xe, consistent with expectations due to the filling of the hills neutron orbit and the increasing role of protons above the Z = 50 shell. In the new experiments, g factors were measured for the four heaviest stable isotopes 130*132*134,136Xe, not only for the 2: but also for the 4: and 2: states, employing projectile excitation in inverse kinemat-

K.-H. Speidelet al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154

144

its, aiming at high precision and reliability of the data [141]. Fig. 43 displays the low-energy levels of the four isotopes together with the relevant y transitions contributing to the y spectra. Contrary to the earlier TF experiments, where for each isotope a separate target was used, in the new measurements a single target was sufficient to excite the different Xe nuclei furnished as isotopically pure beams. 1.5

1

,

,

! l

i+

134 80

136 82

1.4 1.3 10.2; 1.2 -

n

1.1 1.0 0.9 -

A ~

4: 2; IBA-II

5

!

0.6 I

0.5 3

0.4 -0.8 0.7 0.3 0.2 -

;A /

o/-0-.

a

i

c-----

0.1 0.0 -0.1 122

124 70

I

I

126 72

120 74

/ 130 76

132 70

/ A N

Figure 44: Comparison of g factors from new TF measurements (solid symbols) with earlier data for the lighter Xe isotopes (open symbols). The 136Xe(4:) g factor is the average with the more precise previous value [139]. The solid line represents the IBA-II model prediction (see text and [141]).

The Xe projectiles were accelerated by the Berkeley cyclotron to energies between 485 and 508 MeV and excited by scattering from a “BtTi target layer deposited on a Gd layer and backed by a copper layer in which the excited nuclei were stopped. An additional copper foil was placed behind the target to stop the beam. The recoiling Ti ions passed through the target and beam stop and were detected in a solar cell detector. In coincidence with these ions de-excitation y rays were measured with four Ge detectors. A fifth Ge detector, placed at 0” to the beam axis, was used for simultaneous measurements of Doppler-broadened shapes of y lines, in order to determine the lifetimes of the excited states. The experimental procedure and subsequent analysis of the spectra followed closely the work on the KT isotopes (sect. 4.3.2 and [117]). The g factors were derived from the measured precession angles taking into account contributions from feeding states and using the Rutgers parametrization as well as the linear parametrization with the appropriate attenuation factor (see Fig. 3) to evaluate the transient field strength in the experimental conditions. The precession of the magnetic moment of the long-lived 4: state of ‘36Xe was corrected for precession in the external field. The values obtained are displayed in Fig. 44 and compared with former results in Table 13. There is generally good agreement, the only exception being the 134Xe(2:) value of the earlier TF measurement 11401,which turned out to be larger. This discrepancy is not understood and highly surprising, as excellent agreement is found for the other isotopes 13’Xe and 13sXe. The figure also includes the data for the lighter isotopes [86]. The g factors of the 2: and the 4: states in ‘%Xe are large, indicative of proton excitations. As a first approximation, both states are expected to be described by a gr/z proton configuration. A recent calculation within the framework of the cranked Hartree-Fock-Bogoliubov formalism [142] predicts for ‘36Xe, g(2:) = +1.002 which is somewhat larger than the experimental value, and g(4:) = f0.823, in good agreement with the data. Shell model calculations in a model space consisting of gr,z, d5/z, d312, .s,/z and hll,l configurations outside of a closed i3’Sn core, yielded, g(2:) = +0.90 and g(4:) = +0.84,

145

K.-H. Speiakl et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Table 13: New g factor results from measurements nique [141] compared with earlier data [139, 1401.

Nucleus

I”

new [141]

issXe

2:

+0.334(11)

4:

+0.42(5)

22’

+0.45(9)

2:

+0.314(12)

4:

+0.61(11)

2:

+0.354(7)

4:

+0.80(15)

2:

+0.766(45)

4:

+1.08(43)

13’Xe

‘%Xe

lssXe

employing

the transient

field tech-

I

earlier

a [140], b [139]

in good agreement with the experimental

results [139].

The g(2:) values of the lighter isotopes show a steady decrease as the number of neutron holes in the N = 82 shell is increased, as expected, since the hills neutron configuration with its negative Schmidt value, should play an increasing role. On the other hand, the behaviour of g(4:) differs from that of g(2:). It is probably just a coincidence that g(4:) follows fairly well the prediction of the IBA-II model with gn = 1 and gV = 0, since these g factors are more realistically interpreted as evidence for proton excitations near neutron shell closure (see Fig. 44). It is noteworthy that for 13’Xe all states being studied have the same g factor, consistent with collective behaviour. A new set of shell model calculations has been performed for the Xe isotopes, where the valence protons occupy the 2 = 50-82 shell and the neutron (holes) are in the N = 50-82 shell. In other words, valence protons and neutrons refer to a ‘@‘Sn core. It follows that the valence proton excitations will involve predominantly lgr/s configurations with some 2d s/s admixtures and much weaker contributions from 2d3j2, lhll/l and 3Si/s. In contrast, the close proximity of the relevant neutron hole orbits near the Fermi surface, implies that the wave functions of the low-lying states have equally strong components from 2d3,2r Ihills, and 351/s neutron configurations. Results in a truncated model space with closed N = Z = 64 subshells reproduce nicely the experimental observations for 132J34Xe, namely, that the 4: states have consistently larger g factors than the 2: states (see Fig. 44 and [141]). In conclusion, high precision g factor measurements have yielded new relevant insights into the microscopic structure of Xe nuclei. The experimental method applied is particularly powerful in providing reliable and meaningful data for the comparison of g factors of different isotopes and nuclear states. This information is practically independent of uncertainties in the parametrization of the transient field strength.

146

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Figure 45: Low-energy level schemes of Nd isotopes with relevant 7 transitions

4.5.2

g factors

of Neodymium

isotopes

in the transition

from deformed

to spherical

nuclei

The stable 142-150Nd isotopes are usually labelled as transitional nuclei, as they exhibit collective structures of rotational degrees of freedom at lsoNd and clear signatures of single particle structure at and near the N = 82 shell closure for 142Nd and 144Nd, respectively. For the isotopes between these two extremes the nuclear structure will be determined by the interplay between single particle configurations and collective excitations. Hence, these isotopes provide a particularly good laboratory for studying subtle changes in structure noticeable in the excitation energies, transition probabilities and most sensitively in the nuclear moments, magnetic dipole and electric quadrupole moments. The change in collectivity is well represented by the properties of the first 2+ states: the excitation energies show a dramatic increase going from rotational *“Nd to spherical 142Nd (see Fig. 45). The same behaviour is seen in the static quadrupole moments [143] as well as the B(E2)‘s of the (2: -+ 0:) [144] transitions which both increase with increasing atomic number as a manifestation of an increase of collectivity when departing from shell closure. The experimental g factors of the 2: states exhibit a significant steady decrease from the nominal value of gn = Z/A = $0.4 in lsoNd to g N +0.2 in 144Nd [145, 1461; for semi-magic 142Nd a large g factor was observed, g = +0.844(73), a clear indication of dominating proton excitations [147]. To explain the evolution of the g factors from 144Nd to “‘Nd IBA-II calculations were performed that, however, failed completely in predicting both the magnitude and the dependence on neutron number. Single particle effects were suggested as cause for this spectacular failure of the model but the experimental evidence was missing. The new experiments were designed to clarify the structure by measuring the g factors with high t opes, not only of the 2: states but also for the higher excited 4: and precision for the 144~146,148,150 Nd ISO 6: states. As we know from the measurements on the Zr isotopes (see sect. 4.3.2) the spin dependence of the g factor is a particularly useful tool for unveiling the nature of single particle components in the wave functions. Like in the Kr and Xe measurements, the technique of projectile Coulomb excitation in inverse kinematics was applied to isotopically pure Nd beams, which were accelerated at the Berkeley cyclotron to energies between 584 and 608 MeV. Coulomb excitation occurred in a 1 mglcm2 layer of natural Ni of a target which further consisted of Gd for precession, a Ta substrate for deposition of the Gd, and a copper backing in which the excited nuclei were stopped. The de-excitation y rays were measured with Ge detectors in coincidence with forward scattered Ni ions registered in a Si detector. Other details of the experimental procedure and the data analysis, which are similar to the Kr and Xe measurements are described in [148, 1491. The g(Z) factors derived from the measured precessions are summarized in Table 14 and are plotted in Fig. 46. The Rutgers parametrization was used for determining the transient field strength. The very precise g(2:) values are in excellent agreement with previous results [145, 1461.

141

K.-H. SpeiaW et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

Table 14: Newly-measured g factors [148, 1491, measured quadrupole moments [143] and lifetimes 11441with deduced B(E2) values in Weisskopf units of (Zi + Zf) E2 transitions. The g(2$ values are the average of the new data withresults of [i45, 1461 Nucleus

I”

r IPI

B(E2) J [W.U.]

Q PI

‘44Nd

2:

4.50(25)

24.6( 13)

-0.07(15)

+0.201(6)

4:

10.7( 13)

18.8(22)

_

+0.131(36)

_

-0.56(22)

29.3(17)

-0.78(9)

+0.291(16) +0.193(26)

6: 146Nd

14sNd

=‘Nd

2:

31.2( 19)

9V)

4:

5.8( 14)

43(11)

-

2:

112.5(17)

59.3(9)

-1.46(13)

+0.357(8)

-

+0.360(25)

4:

10.1(32)

6:

-

2:

2153(22)

4+1

91(4

92(3)

+0.266(51) 115(l)

-2.00(51)

+0.42(2)

173(8)

_

+0.44(3) +0.35(7) +0.56(13)

6:

18(l)

209(12)

-

8+ 1

6.3(5)

-

-

10:

_

_

+0.14(19)

For “‘Nd the g factors of all the states in the rotational band are found to be equal near g = Z/A = O.;, as expected for a classical rotational nucleus. As neutrons are removed and thk neutron number approaches shell closure at N = 82, the g factors decrease continuously, consistent with the B(E2; 2: -+ 0:)‘s and the static electric quadrupole moments, that show the same trends. The exciting discovery in the new measurements is the pronounced spin dependence of the g factors, exhibited in particular for 144Nd, for which g(6:) is even negative. The observation shows clearly that the low-excitation structure of this nucleus is dominated by 2fr/r neutron configurations with its effective Schmidt value, g(vfr,2) = -0.383, using g:‘f = 0.7. gp. The trend of g with neutron number reflects the decreasing role of this configuration as the nuclei become more collective. The same effect is seen in the odd isotopes 14=Nd and 14’Nd for which the measured g factors of the 7/2- ground states are g(7/2-) = -0.304(l) and -0.187(l), respectively [86]. From these results it is evident that pure collective models cannot describe the data. Calculations involving the coupling of valence nucleons to a collective core have been carried out for lUNd [150, 1511. They predict reasonably well the energies and E( E2)‘s by assuming considerable (2fr,r)* neutron strength, which increases with spin as required by the observed g factors. Large-scale shell model calculations are certainly needed to describe the change in structure in the transition from deformed “‘Nd to more spherical ‘44Nd. Such calculations, which include dynamical many-body correlations that are extremely important in the transitional region, have recentlv been nerformed with great success for the transitional Ba isotopes, using Monte Carlo ‘techniques td describe energy levels, B(E2)‘s and B(Ml)‘s [35, 1521.

K.-H. Speidel et al. / Prog. Part. Nucl. Phys. 49 (2002) 91-154

148

-0.6

Figure 46: Summary of the measured g(1) factors as a function of the mass or neutron number of even-A Nd isotopes [148, 1491. Also displayed is the g(2:) value of 14*Nd [147].

5

Conclusions and perspectives

In the present work we have attempted to review the state-of-the-art in g-factor measurements for shortlived nuclear states, in terms of the experimental methodology, the scope and quality of the nezu body of data accumulated, as well as the highlights of the novel theoretical implications for the underlying nuclear structure. Specifically, the new methodology pertains to projectile Coulomb-excitation in inverse kinematics, combined with transient field precession measurements on high-velocity ions traversing ferromagnetic media. With respect to the ne2u body of data accumulated, the potential and versatility of the technique is demonstrated by the high-precision and reliability of the data obtained for a broad range of nuclei. The latter has engendered novel theoretical implications, in the sense of permitting critical testing of effective NN interactions and the associated relevant configuration space of valence nucleons for entire series of isotopes and levels not previously interrogated by experiment. In the present review we have focussed almost exclusively on experimental results, obtained by our own group or in collaboration with other groups, in order to highlight the power of the new method. In so doing, we have omitted the description of many other interesting and relevant measurements with important theoretical implications. Further to the theoretical aspects, Monte Carlo techniques are found to be highly promising in cases where the Hamiltonian matrix becomes too large to be diagonalized, namely, for describing heavy nuclei and/or large open shells. However, for the nuclei reviewed here, the potential of this approach for g-factor predictions has not yet been exploited. With regard to future experimental implications, the advent of radioactive beams should permit the

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154

149

reviewed methodology to be extended, among others, to nuclei far from stability. The high sensitivity of the technique should ensure adequate accuracy of the data (for significant theoretical statements), even for the low intensity beams available. Whereas many of these measurements are likely to be carried out with beam energies below the Coulomb barrier, at which the TF parameters are well known, there are also good prospects for g factors to be determined via Coulomb excitation of projectiles at relativistic energies. Since fragmentation is the most effective and versatile reaction for producing exotic nuclei at relatively high velocities, the ionic environments required for sensitive g-factor measurements appear to be readily accessible. Whether, in such reactions, the spin alignment of the residual nuclei is high enough and the TF adequately intense remain, at present, open questions to be studied in future experiments. However, in the latter respect, favourable indications have already been obtained that TF remain operative at the high velocities associated with single-electron ions and are therefore expected to be high.

Acknowledgement The authors are indebted to M.B. Goldberg for many stimulating discussions and valuable comments. This work was supported in part by the BMBF and the Deutsche Forschungsgemeinschaft.

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (200.2) 91-13

150

References [l] L.W. Alvarez and F. Bloch, Phys. Rev. 57 (1940) 111 [2] F. Bloch, W.W. Hansen, and M. Packard, Phys. Rev. 69 (1946) 127 and Phys. Rev. 70 (1946) 474 [3] N. Bloemherger,

E.M. Purcell, and R.V. Pound, Phys. Rev. 73 (1948) 679

[4] D.E. Groom et al. (Particle Data Group), Eur. Pfiys. J. C 15 (2000) 1 and 2001 partial update for edition 2002 [5] R.S. Dyck, P.B. Schwinberg, and H.G. Dehmelt, Phys. Rev. Lett. 59 (1987) 26 [6] H.N. Brown et al., Muon (g-2) collaboration, [7] A. Poves, J. Sanchez-Solano,

Phys. Rev. Lett. 86 (2001) 2227

E. Caurier, and F. Nowacki, Nucl. Phys. A 694 (2001) 157

[8] A. Arima, J. Phys. Sot. Jp. 34 (1973) 205 (91 I.S. Towner and F.C. Khanna,

Nuc1. Phys. A 399 (1983) 334

[lo] T. Yamazaki, Hyp. Int. 21 (1985) 85 [II] Y. Yamauchi, A. Buchmann,

and A. Faessler, Nucl. Phys. A 526 (1991) 495

[12] G. Goldring: Hyperfine Interactions Amsterdam (North-Holland, 1980)

in Isolated Ions, R. Bock (ed.), Vol. 3

[13] W.L. Randolph, N. Ayres de Campos, J.R. Beene, J. Burde, M.A. Grace, D.F.H. Start, and R.E. Warner, Phys. Lett. B 44 (1973) 36 [14] S. Kremeyer, K.-H. Speidel, 0. Jessensky, H. Busch, U. Grabowy, A. Gohla, J. Cub, G. Jakob, P. Maier-Komor, J. Gerber, A. Meens, F. Hagelberg, and T.P. Das, 2. Phys. A 348 (1994) 49 [15] M. Forterre, J. Gerber, J.P. Vivien, M.B. Goldberg, K.-H. Speidel, and P.N. Tandon, Phys. Rev. C 11 (1975) 1976 [16] M.B. Goldberg, G.J. Kumbartzki, Hyp. Int. 1 (1976) 429 [17] J. Lindhard

and A. Winther,

[18] N. Benczer-Koller,

K.-H. Speidel, M. Forterre, and J. Gerber,

Nucl. Phys. A 166 (1971) 413

M. Hass, and J. Sak, Annu. Rev. Nucl. Part. Sci. 30 (1980) 53

[19] F. Hagelberg, T.P. Das, and K.-H. Speidel, Phys. Rev. C 48 (1993) 2230 [20] F. Hagelberg,

T.P. Das, and K.-H. Speidel, Hyp. Int. Cl (1996) 151

[21] E. Lubkiewicz, H. Emling, H. Grein, R. Kulessa, R.S. Simon, H.J. Wollersheim, Ch. Ender, J. Gerl, D. Habs, and D. Schwalm, 2. Phys. A 335 (1990) 369 [22] U. Birkental, A.P. Byrne, S. Heppner, H. Hiibel, W. Schmitz, P. Fallon, P.D. Forsyth, J.W. Roberts, H. Kluge, E. Lubkiewicz, and G. Goldring, Nzlcl. Phys. A 555 (1993) 643 [23] A. Jungclaus, C. Teich, V. Fischer, D. Kast, K.P. Lieb, C. Lingk, C. Ender, T. H;irtlein, F. K&k, D. Schwalm, J. Billowes, J. Eberth, and H.G. Thomas, Phys. Rev. Lett. 80 (1998) 2793 and references therein [24] K.-H. Speidel, N. Benczer-Koller, G. Kumbartzki, C. Barton, A. Gelberg, J. Holden, G. Jakob, N. Matt, R.H. Mayer, M. Satteson, R. Tanczyn, and L. Weissman, Phys. Rev. C 57 (1998) 2181 [25] R. Ernst, K.-H. Speidel, 0. Kenn, U. Nachum, J. Gerber, P. Maier-Komor, N. Benczer-Koller, Jakob, G. Kumbartzki, L. Zamick, and F. Nowacki, Phys. Rev. Lett. 84 (2000) 416

G.

[26] R. Ernst, K.-H. Speidel, 0. Kenn, A. Gohla, U. Nachum, J. Gerber, P. Maier-Komor, N. BenczerKoller, G. Kumbartzki, G. Jakob, L. Zamick, and F. Nowacki, Phys. Rev. C 62 (2000) 024305 [27] 0. Kenn, K.-H. Speidel, R. Ernst, J. Gerber, P. Maier-Komor, N. Benczer-Koller, and F. Becker, Nucl. Ins&. Meth. in Phys. Res. B 171 (2000) 589 [28] T.T.S. Kuo and G.E. Brown, NucZ. Phys. 85 (1966) 40 [29] T.T.S. Kuo and G.E. Brown, Nucl. Phys. A 114 (1968) 241 [30] H. Nakada, T. Sebe, and T. Otsuka, Nucl. Phys. A 571 (1994) 467

G. Kumbartzki,

151

K.-H. Speidd et al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154

[31] S.E. Koonin, D.J. Dean, and K. Langanke, Phys. Rep. 278 (1997) 2 [32] S.E. Koonin, D.J. Dean, and K. Langanke, Annu. Rev. Nucl. Part. sci. 4’7 (1997) 463 [33] M. Honma, T. Mizusaki, ‘and T. Otsuka, Pfiys. Rev. Lett. 77 (1996) 3315

[34] T. Otsuka, M. Honma, and T. Mizusaki, Phys. Rev. Lett. 81 (1998) 1588 [35] T. Ctsuka, M. Honma, T. Mizusaki, and Y. Utsuno, Prog. Part. Nucl. Phys. 47 (2001) 319 [36] K. Hara, Y. Sun, and T. Mizusaki, Phys. Rev. Lett. 83 (1999) 1922 [37] J. Terssaki, R. Wyss, and P.H. Heenen, Phys. Lett. B 437 (1998) 1 [38] G.J. Kumbartzki, K. Hagemeyer, Speidel, Hyp. Int. 7 (1979) 253

W. Knauer,

G. Krosing, R. Kuhnen,

V. Mertens,

and K.-H.

[39] N. Rud, and K. Dybdal, Physics Scripta 34 (198G) 561 [40] J.L. Eberhardt, (1977) 195

R.E. Horstman,

P.C. Zalm, H.A. Doubt, and G. van Middelkoop,

Hyp. Int. 3

[41] K.-H. Speidel, 0. Kenn, J. Leske, G. Miiller, S.K. Mandal, H.J. Wollersheim, J. Gerl, P. Mayet, F. de Oliveira Santos, I. Matea, F. Becker, W. Korten, J. Gerber, and P. Maier-Komor, Proposal for GANIL, fiance, June 25 (2001) [42] H.J. Wollersheim and S.K. Mandal, GSI (2001), private communication [43] N.K.B. Shu, D. Melnik, J.M. Brennan, W. Semmler, and N. Benczer-Koller, 1828

Phys. Rev. C 21 (1980)

[44] H.R. Andrews, 0. Hlusser, D. Ward, P. Taras, R. Nicole, J. Keinonen, P. Skensved, and B. Haas, Nucl. Phys. A 383 (1982) 509 [45] P. Maier-Komor,

K.-H. Speidel, and A. Stolarz, Nucl. In&.

[46] M.B. Goldberg, W. Knauer, G.J. Kumbartzki, 4 (1978) 262

Meth. in Phys. Res. A 334 (1993) 191

K.-H. Speidel, J.C. Adloff, and J. Gerber, Hyp. Int.

[47] K.-H. Speidel, M. Knopp, W. Karle, M.-L. Dong, J. Cub, U. Reuter, H.-J. Simonis, P.N. Tandon, and J. Gerber, Phys. Lett. B 227 (1989) 16 [48] K.-H. Speidel, Hyp. Int. 61 (1990) 1295 [49] K.-H. Speidel, J. Cub, W. Karle, F. Passek, U. Reuter, M. Weidinger, and J. Gerber, 2. Phys. A 340 (1991) 333 (501 K.-H. Speidel, U. Reuter, J. Cub, W. Karle, F. Passek, H. Busch, S. Kremeyer, and J. Gerber, Z. Phys. D 22 (1991) 371 [51] W. Brandt in: Atomic collisions in solids, ed. S. Datz, B.R. Appleton, New York, London, Plenum Press 1975 [52] W. Brandt, R. Laubert,

M. Morino, and A. Schwarzschild,

C.D. Moak, Vol. 1, p. 261,

Phys. Rev. Lett. 30 (1973) 358

[53] H. Busch, G. Jakob, K.-H. Speidel, J. Cub, S. Kremeyer, U. Grabowy, A. Gohla, J. Gerber, A. M&ens, P. Maier-Komor, C. Rezny, M. Schulz, and P. Padberg, Z. Phys. A 355 (1996) 9 [54] K.-H. Speidel, L. Kleinen, A. Gohla, R. Ernst, H. Busch, G. Jakob, U. Grabowy, V. Roth, and J. Gerber, Z. Phys. A 358 (1997) 389 [55] C.A. Bertulani and G. Baur, Phys. Rep. 163 (1988) 299 [56] U. Grabowy, K.-H. Speidel, J. Cub, H. Busch, H.-J. Wollersheim, and M. Loewe, Z. Phys. A 359 (1997) 377

G. Jakob, A. Gphla, J. Gerber,

[57] E.M. Szanto, A. Szanto de Toledo, H.V. Klapdor, M. Diebel, J. Fleckner, and U. Mosel, Phys. Rev. Lett. 42 (1979) 622 [58] P.C. Zalm, J.F.A. van Hienen, and P.W.M. Glaudemans,

Z. Phys. A 287 (1978) 255

’

[59] A. Arima, T. Cheon, and K. Shimizu, Hyp. Int. 21 (1985) 79 [60] B.A. Brown, J. Phys. G: Nucl. Phys. 8 (1982) 679 (611 K.-H. Speidel, G.J. Kumbartzki, W. Knauer, V. Mertens, P.N. Tandon, N. Ayres de Campos, J. Gerber, and M.B. Goldberg, Phys. Lett. B 92 (1980) 289

K.-H. Speidelet al. / Prog. Part.

152

NW!.

Phys. 49 (2002) 91-154

[62] U.

Reuter, F. Hagelberg, S. Kremeyer, H.-J. Simonis, K.-H. Speidel, M. Knapp, W. KarIe, P.N. Tandon, and J. Gerber, Phys. Lett. B 230 (1989) 16

[63] K.-H. Speidel, P.N. Tandon, V. Mertens, W. Trijlenberg, G.J. Kumbartzki, and M.B. Goldberg, Nucl. Phys. A 378 (1982) 130

J. Cub,

N. Ayres de CamPos,

[64] W. T%lenberg, F. Hagelberg, H.J. Simonis, P.N. Tandon, K.-H. Speidel, M. Knopp, and J. Gerber, Nucl. Phys. A 458 (1986) 95 [65] T. Bright, D. Ballon, R.J. Saxena, Y. Niv, and N. Benczer-Koller,

Whys. Rev. C 30 (1984) 696

[66] J. Leske, K.-H. Speidel, 0. Kenn, G. Miiller, S. Schielke, J. Gerber, N. Benczer-Koller, bartzki, and P. Maier-Komor, to be published

G. Kum-

[67] A. P. Zuker, in Proceedings of the INPC98 Conference, Nucl. Phys. A 654 (1999) 747~ [68] A. Paves and A.P. Zuker, Phys. Rep. 70 (1981) 235 [69] W.A. Richter, M.J. Van der Merwe, R.E. Julies, and B.A. Brown, NucZ. Phys. A 523 (1991) 32 [70] J.C. Wells and N.R. Johnson, program LINESHAPE, (711 E. Caurier, code ANTOINE,

Strasbourg,

Feb. 1994, ORNL

1989

[72] L.K. Peter, Nucl. Data Sheets 68 (1993) 271; T.W. Burrows, ibid. 68 (1993) 1; Huo Junde, ibid. 71 (1994) 659; F. Brandolini et al., Nucl. Phys. A 642 (1998) 387 [73] E. Caurier, K. Langanke, G. Martinez-Pinedo, 240

F. Nowacki, and P. Vogel, Phys. Lett. B 522 (2001)

(741 S. Wagner, K.-H. Speidel, 0. Kenn, R. Ernst, S. Schielke, J. Gerber, P. Maier-Komor, N. BenczerKoller, G. Kumbartzki, Y.Y. Sharon, and F. Nowacki, Phys. Rev. C 64 (2001) 034320 [75] A. Pakou, R. Tanczyn, D. Turner, W. Jan, G. Kumbartzki, Liu, and L. Zamick, Phys. Rev. C 36 (1987) 2088

N. Benczer-Koller,

Xiao-Li Li, Huan

[76] A.A. Pakou, J. Billowes, A.W. Mountford, and D.D. Warner, Phys. Rev. C 50 (1994) 2608 [77] G. Martinez-Pinedo, A. Poves, L.M. Robledo, E. Caurier, F. Nowacki, J. Retamosa, Phys. Rev. C 54 (1996) R2150

and A. Zuker,

[78] A. Arima and H. Horie, Prog. Theor. Phys. 11 (1954) 509 179) L. Zamick and DC. Zheng, Phys. Rev. C 54 (1996) 956 (801 E. Caurier, A.P. Zuker, A. Poves, and G. Martinez-Pinedo, [81] E. Caurier, G. Martinez-Pinedo, C 59 (1999) 2033

Phys. Rev. C 50 (1994) 225

F. Nowacki, A. Poves, J. Retamosa,

and A.P. Zuker, Phys. Rev.

1821 A. Pakou, Phys. Rev. C 64 (2001) 069801 [83] T. Yamazaki, T. Nomura, S. Nagamiya, and T. Katou, Phys. Rev. C 25 (1970) 547 [84] T. Callaghan, M. Kaplan, and N.J. Stone, Nucl. Phys. A 201 (1973) 561 [85] H. Noya, A. Arima, and H. Horie, Suppl. Prog. Theor. Phys. 8 (1958) 33 186) P. Raghavan, At. Data and Nucl. Data Tables 42 (1989) 189 [87] M. Brennan, N. Benczer-Koller,

M. Hass, and H.T. King, Hyp Int.

[SS] K.-H. Speidel, 3. Cub, U. Reuter-Knopp, Hagelberg, 2. Phys. A 342 (1992) 17

4 (1978) 268

W. Karle, H. Busch, S. Kremeyer,

[89] K.-H. Speidel, R. Ernst, 0. Kenn, J. Gerber, P. Maier-Komor, N. Benczer-Keller, L. Zamick, M.S. Fayache, and Y.Y. Sharon, Phys. Rev. C 62 (2000) 031301(R)

J. Gerber, and F. G. Kumbartzki,

(901 0. Kenn, K.-H. Speidel, R. Ernst, J. Gerber, N. Benczer-Koller, and F. Nowacki, Phys. Rev. C 63 (2000) 021302(R)

G. Kumbartzki,

[91] 0. Kenn, K.-H. Speidel, R. Ernst, J. Gerber, P. Maier-Komor, (2000) 064306

and F. Nowacki, Phys. Rew. C 63

[92] R.B.M. Mooy and P.W.M. Glaudemans,

P. Maier-Komor,

2. Phys. A 312 (1983) 59

[93] T. Mizusaki, T. Otsuka, Y. Utsuno, M. Honma, and T. Sebe, Phys. Rev. C 59 (1999) R1846

K.-H. Speidel et al. /Prog. Part. Nucl. Phys. 49 (2002) 91-154 [94] M. Has,

N. Benczer-Koller,

1.53

J.M. Brennan, H.T. King, and P. Goode, Phvs. Rev. G 17 (1978) 997

[g5] R. Bhat, Nuc~. Data Sheets 80 (1997) 789; M.M. King, ibid. 69 (1993) 1 and 60 (1990) 337; BaIraj Singh, ibid. 78 (1996) 395 (961 0. Sorlin et al., Phys. Rev. Lett. 88 (2002) 092501 (971 J.F.A. van Hienen, W. Chung, and B.H. Wildenthal,

Nucl. Phys. A 269 (1976) 159

[98] V. Lopac and V. Paar, Nucl. Phys. A 297 (1978) 471 [99] H.F. de Fries and P.J. Brussard,

Z. Phys. A 286 (1978) 1

[loo] Y. Sun, Jing-ye Zhang, M. Guidry, J. Meng, and S. Im, Phys. Rev. C 62 (2000) 021601(R) [loll C.E. Svenson et al., Phys. Rev. Lett. 79 (1997) 1233 [102] R.K. Sheline, I. Ragnarsson,

and S.G. Nilsson, Phys. Lett. B 41 (1972) 115

[103] J. Dudek, W. Nazarewicz, Z. Szymanski, and G.A. Leander, Phys. Rev. Lett. 59 (1987) 1405 [104] N. Benczer-Koller, [105] C. Fahlander,

M. Hass, T.M. Brennan, and H.T. King, J. Phys. Sac. Jpn. 44 (1978) 341

K. Johansson,

and G. Possnert,

Z. Phys. A 291 (1979) 93

[106] 0. Kenn, K.-H. Speidel, R. Ernst, S. Schielke, S. Wagner, J. Gerber, P. Maier-Komor, Nowacki, Phys. Rev. C 65 (2002) 034308

and F.

[107] Huo Junde and B. Singh, Nucl. Data Sheets 91 (2000) 395; B. Singh, ibid. 78 (1996) 462; M.R. Bath, ibid. 8S (1998) 83; 76 (1995) 349; 68 (1998) 124

[108] M. Hjorth-Jensen,

(private communication)

[109] M. Hjorth-Jensen,

T.T.S. Kuo, and E. Osnes, Phys. Rep. 261 (1995) 125

[llO] F. Nowacki, Ph.D. thesis, IReS, Strasbourg, [ill]

1996

E. Caurier, F. Nowacki, A. Poves, and J. Retamosa,

[112] U. Kaup, C. Monkemeyer,

and P. von Brentano,

Phys. Rev. Lett. 77 (1996) 1954

Z. Phys. A 310 (1983) 129

[113] F.S. Radhi and N.M. Stewart, Z. Phys. A 356 (1996) 145 [114] T. Otsuka, A. Arima, and F. Iachello, Nucl. Phys. A 309 (1978) 1 [115] A.L. Goodman,

Phys. Rev. C 58 (1998) R3051 and Adv. Nucl. Phys. 11 (1979) 263

[116] S. Fraundorf and J.A. Sheik, Nucl. Phys. A 645 (1999) 509 [117] T.J. Mertzimekis, N. Benczer-Koller, J. Holden, G. Jakob, G. Kumbartzki, K.-H. Speidel, R. Ernst, A. Macchiavelli, M. McMahan, L. Phair, P. Maier-Komor, A. Pakou, S. Vincent, and W. Korten, Phys. Rev. C 64 (2001) 024314 [118] A. Wolf, 0. Scholten,

and R.F. Casten, Phys. Lett. B 312 (1993) 372

[119] S. Rab , Nucl. Data Sheets 76 (1995) 285; B. Singh, ibid. 66 (1992) 623; M.M. King and W.-T. Chou, ibid. 76 (1995) 285; J.K. Tuli, ibid. 81, (1997) 331; M.M. King, ibid. 80 (1997) 567, [120] A.I. Kucharska, J. Billowes, and M.A. Grace, J. Phys. G 14 (1988) 65 [121] G. Jakob, N. Benczer-KoIIer, J. Holden, G. Kumbartzki, T.J. Mertzimekis, K.-H. Speidel, R. Ernst, P. Maier-Komor, C.W. Beausang, and R. Kriicken, Phys. Lett. B 494 (2000) 187 [122] P.F. Mantica, A.E. Stuchbery, D.E. Groh, J.I. Prisciandaro,

and M.P. Robinson, Phys. Rev. C 63

(2001) 034312 [123] E.K. Warburton,

J.W. Olness, C.J. Lister, J.A. Becker, and SD. Bloom, J. Phys. G: Nucl. Phys.

12 (1986) 1017 [124] C.A. Fields, F.W.N. de Boer, and J. Sau, Nucl. Phys. A 398 (1983) 512 [125] N. Stone, Table af New Nuclear Moments, Oxford University, 1997 [126] 0. Hausser, LS. Towner, T. Faestermann, Broude, Nucl. Phys. A 293 (1977) 248

H.R. Andrews, J.R. Beene, D. Horn, D. Ward, and C.

[127] R. Eder, E. Hagn, and E. Zech, Nucl. Phys. A 468 (1987) 348 [128] M. Hass, C. Broude, Y. Niv, and A. Zemel, Phys. Rev. C 22 (1980) 1065

K.-H.

154

Speia!d etal./Prog.

Part. Nucl.

Phys. 49 (2002) 91-154

[129] J. Gerber, M.B. Goldberg, W. Knauer, G.J. Kumbartzki, and K.-H. Speidel, HYP. Int. 4 (1978) 257 [130] G. Jakob, N. Benczer-Koller, J. Holden, G. Kumbartzki, T.J. Mertzimekis, K.-H. Speidel, C.W. Beausang, and R. K&ken, Phys. Lett. B 468 (1999) 13 [131] I. Talmi, Phys. Rev. 6 (1962) 2116 [132] V. Werner, D. Belie, P. von Brentano, C. Fransen, A. Gade, H. von Garrel, J. Jolie, U. Kneissl, C. Kohstall, A. Linnemann, A.F. Lisetskiy, N. Pietralla, H.H. Pitz, M. Scheck, K.-H. Speidel, F. Stedile, and S.W. Yates, submitted to Phys. Lett. J3 [133] W. Greiner, Nucl. Phys. 80 (1966) 417 [134] N.K.B. Shu, R. Levy, N. Tsoupas, Phys. Rev. C 24 (1981) 954

A. Lopez-Garcia,

W. Andrejscheff,

and N. Benczer-Koller,

(1351 G. Puddu, 0. Scholten, and T. Otsuka, Nucl. Phys. A 348 (1980) 109 [136] T. Otsuka, Hyp. Id.

78 (1993) 19

[137] A.E. Stuchbery, I. Morrison, L.D. Wood, R.A. Bark, H. Yamada, and H.H. Bolotin, Nucl. Phys. A 435 (1985) 635 [138] S. Kuyucak and A.E. Stuchbery, Phys. Lett. B 348 (1995) 315 [139] Z. Berant, A. Wolf, J.C. Hill, F.K. Wohn, R.L. Gill, H. Mach, M. Rafailovich, H. Kruse, B.H. Wildenthal, G. Peaslee, A. Aprahamian, J. Goulden, and C. Chung, Phys. Rev. C 31 (1985) 570 [140] K.-H. Speidel, H. Busch, S. Kremeyer, U. Knopp, J. Cub, M. Bussas, W. Karle, K. Freitag, U. Grabowy, and J. Gerber, Nucl. Phys. A 552 (1993) 140 [141] G. Jakob, N. Benczer-Koller, G. Kumbartzki, J. Holden, T.J. Mertzimekis, K.-H. Speidel, R. Ernst, A.E. Stuchbery, A. Pakou, P. Maier-Komor, A. Macchiavelli, M. McMahan, L. Phair, and I.Y. Lee, Phys. Rev. C 65 (2002) 024316 [142] Ramendra

Majumdar,

[143] H.S. Gertzmann,

Phys. Rev. C 55 (1997) 745

D. Cline, H.E. Gove, and P.M.S. Lesser, &cl.

Phys. A 151 (1971) 282

[144] J.K. Tuli, Nucl. Data Sheets 56 (1989) 607; L.K. Peker, ibid. 59 (1990) 393; L.K. Peker, ibid. 60 (1990) 953; E. derMateosian, and J.K. Tub, ibid. 75 (1995) 827 [145] N. Benczer-Koller,

D.J. Ballon, and A. Pakou, Hyp. Znt. 33 (1987) 37

[146] A.E. Stuchbery, G.J. Lampard, and H.H. Bolotin, Nucl. Phys. A 516 (1990) 119 [147] D. Bazacco, F. .Brandolini, K.L. Liiwenich, P. Pavan, C. Rossi-Alvarez, and A.M.I. Haque, Nz~cl. Phys. A 533 (1991) 541

E. Maglione, M. de Poli,

[I481 J. Holden, N. Benczer-Koller, G. Jakob, G. Kumbartzki, T.J. Mertzimekis, K.-H. Speidel, A. Macchiavelli, M. McMahan, L. Phair, P. Maier-Komor, A.E. Stuchbery, W.F. Rogers, and A.D. Davies, Phys. Lett. B 493 (2000) 7 [149] J. Holden, N. Benczer-Koller, G. Jakob, G. Kumbartzki, T.J. Mertzimekis, K.-H. Speidel, C.W. Beausang, R. Kriicken, A. Macchiavelli, M. McMahan, L. Phair, A.E. Stuchbery, P. Maier-Komor, W. Rogers, and A.D. Davies, Phys. Rev. C 63 (2001) 024315 [150] S.J. Robinson,B.

Faircloth, P. MioiiinoviE , and A.S. Altgilbers,

Phys. Rev. C 62 (2000) 044306

[I511 J. Copnell, S.J. Robinson, J. Jolie, and K. Heyde, Phys. Rev. C 46 (1992) 1301 [152] N. Shimizu, T. Otsuka, T. Mizusaki, and M. Honma, Pfiys. Rev. Lett. 86 (2001) 1171