Experiments on multiphonon states with proton–neutron mixed symmetry in vibrational nuclei

Progress in Particle and Nuclear Physics 60 (2008) 225–282 www.elsevier.com/locate/ppnp

Review

Experiments on multiphonon states with proton–neutron mixed symmetry in vibrational nuclei N. Pietralla a,b,∗ , P. von Brentano b , A.F. Lisetskiy c a Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany b Institut f¨ur Kernphysik, Universit¨at zu K¨oln, 50937 K¨oln, Germany c Department of Physics, University of Arizona, Tucson AZ 85721, USA

Abstract Considerable progress has been achieved in the experimental investigation and theoretical understanding of isovector valence shell structures in heavy nuclei that are frequently called mixed-symmetry states. This article attempts a review of the experimental aspects and data on mixed-symmetry states of vibrational and weakly deformed transitional nuclei. After a brief introduction the main experimental tools are described that have lately been used for the investigation of mixed-symmetry states in vibrational and transitional nuclei. The main body reviews the available data where the mixed-symmetry assignments have been made on the basis of large magnetic dipole transition rates. c 2007 Elsevier B.V. All rights reserved.

Keywords: Nuclear structure; Spectroscopy; Electromagnetic transitions; Algebraic models; Collective states; Mixedsymmetry states

Contents 1. 2.

Introduction......................................................................................................................226 Mixed-symmetry states in the proton–neutron interacting boson model ................................... 228 2.1. Definition of MSSs.................................................................................................. 228 2.2. F-spin ...................................................................................................................229 2.3. Decay properties of MSSs ........................................................................................ 232 2.4. Q-phonon scheme................................................................................................... 234 2.5. d-parity.................................................................................................................. 236 2.6. Signatures for MSSs in near-vibrational nuclei ........................................................... 236

∗ Corresponding author at: Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany.

E-mail address: [email protected] (N. Pietralla). c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2007.08.002

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3.

4.

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Experimental techniques for identification of mixed symmetry ............................................... 238 3.1. Electron scattering .................................................................................................. 239 3.2. Nuclear resonance fluorescence ................................................................................ 241 3.3. Coulomb excitation .................................................................................................243 3.4. β-decay ................................................................................................................. 246 3.5. Inelastic neutron scattering.......................................................................................248 3.6. Light-ion fusion ...................................................................................................... 249 3.7. Other techniques ..................................................................................................... 252 Data on vibrational nuclear excitations with mixed symmetry................................................. 252 4.1. 94 Mo as an example ................................................................................................253 4.2. Other N = 52 isotones ............................................................................................ 259 4.2.1. 96 Ru ......................................................................................................... 259 4.2.2. 92 Zr ..........................................................................................................260 4.3. Other nuclei in the [28 < Z < 50, 50 < N < 82]-major shell ...................................... 265 4.3.1. 96 Mo ........................................................................................................ 265 4.3.2. 112 Cd........................................................................................................ 265 4.3.3. 114 Cd........................................................................................................ 266 4.4. Nuclei in the [50 < Z < 82, 50 < N < 82]-major shell .............................................. 266 4.4.1. 128 Xe........................................................................................................ 266 4.4.2. 126 Xe........................................................................................................ 267 4.4.3. 134 Ba ........................................................................................................ 267 4.4.4. 136 Ba ........................................................................................................ 268 4.4.5. 138 Ce ........................................................................................................ 268 4.5. Nuclei in the [50 < Z < 82, 82 < N < 126]-major shell ............................................ 270 4.5.1. 142 Ce ........................................................................................................ 270 4.5.2. 144 Nd........................................................................................................ 270 4.5.3. 148 Sm .......................................................................................................271 4.6. A = 60 mass region ................................................................................................272 4.6.1. 54 Cr..........................................................................................................273 4.6.2. 56 Fe..........................................................................................................273 4.6.3. 66 Zn ......................................................................................................... 273 4.7. F-vector E1 transitions ........................................................................................... 274 Summary and outlook........................................................................................................ 276 Acknowledgments.............................................................................................................278 References ....................................................................................................................... 278

1. Introduction Understanding the basic mechanisms of how the interactions between protons and neutrons lead to the formation of complex nuclear structures is a prime interest of modern nuclear structure physics. Information can be obtained from studies of nuclear states that are particularly sensitive to the proton–neutron degree of freedom. This article reports on the recent progress in the experimental identification and investigation of multiphonon proton–neutron mixedsymmetry states (MSSs) of vibrational nuclei based on measurements of energies and of absolute electromagnetic transition matrix elements. MSSs appear in nuclear structure models of the nuclear valence shell that have proton and neutron degrees of freedom. The first experimental observation of a mixed-symmetry state was made by A. Richter’s group in Darmstadt. They used the 156 Gd(e, e0 ) reaction at the DALINAC facility and observed [1] magnetic dipole excitations with a total strength of a few W.u. The experiment followed theoretical calculations that suggested [2–9] the existence of isovector

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valence shell excitations. Such theory was already indicated in an early paper by Faessler [2] which framed the concept of proton–neutron quadrupole surface vibrations in the collective model. MSSs of nuclei with various forms of quadrupole collectivity appeared in a natural way in the proton–neutron version of the Interacting Boson Model IBM-2 [4] of Arima, Otsuka, Iachello and Talmi. They were initially discussed in early papers by Iachello and Otsuka [3,5,9, 10]. Isovector magnetic dipole excitations of deformed nuclei were suggested from schematic RPA calculations [6] and geometrically modeled in the two-rotor model by Lo Iudice and Palumbo [8]. A related geometrical model has been suggested by Hilton [11]. Subsequent consideration of shell structure and density dependence of restoring forces in the early model of quadrupole surface vibrations resulted [12] in better agreement with data as compared to that for the initial formulation of the model. Heyde and Sau [13] identified the appearance of mixedsymmetry states as a general feature of two-component systems from schematic microscopic considerations. The first experimentally observed case of a mixed-symmetry state was the JKπ = 1+ 1 scissors mode in deformed nuclei [1,14]. The Darmstadt experiments triggered a tremendous amount of experimental and theoretical research on this phenomenon. The scissors mode in heavy nuclei typically lies at an energy of about 3 MeV and in most cases it does not appear as an isolated state but is fragmented over several 1+ states that carry the total M1 strength. We will not review here the scissors mode of strongly deformed nuclei, because this subject has been exhaustively discussed and reviewed already. Comprehensive reviews on different aspects of the scissors mode can be found in Refs. [14–18]. Increased interest in the mixed-symmetry scissors mode was accompanied by the search for MSSs in weakly deformed, vibrational and transitional nuclei. Examples for the vibrational 2+ 1,ms one-quadrupole phonon state were first suggested by Hamilton et al. [19] in the N = 84 isotones from the analysis of E2/M1 multipole mixing ratios. Similar arguments were subsequently used by several authors, e.g., in Refs. [20–24] and others, for the assignment of mixed-symmetry character to off-yrast low-spin states of vibrational and transitional nuclei in different mass regions. Other approaches made use of the isospin sensitivity of hadronic probes for the assignment of MSSs [25–27]. In this review we focus on one-phonon and twophonon MSSs of vibrational or transitional nuclei that were identified, mainly recently, from measurements of absolute M1 matrix elements. This overview is meant to summarize the recent data on these states. From an experimental point of view the investigation of MSSs in vibrational nuclei is rather different from the experiments done on the scissors mode in deformed nuclei. The identifying signature for the MS phonon modes (strong M1 transitions to proton–neutron + symmetrical phonon excitations with the same phonon number, e.g., 2+ 1,ms → 21 ) differs in this + + case from the excitation mechanism, e.g., 01 → 21,ms . This is in contrast to the situation for the scissors mode where the identifying strong M1 excitation strength from the ground state + (0+ 1 → 1sc ) was used for population of the scissors mode in electron and photon scattering reactions. Consequently, experiments of different types have to be performed or even to be combined in order to identify and study MSSs in vibrational and transitional nuclei. In view of this fact we have decided to restrict our review to the experimental techniques and available data on MSSs in vibrational and transitional nuclei. This is even more appropriate as there are already numerous and excellent review articles and seminal papers on the scissors mode and on theoretical aspects of MSSs [14–18,28–30]. Absolute electromagnetic decay strengths provide a solid basis for the identification of MSSs. Identifications of the 2+ 1,ms one-phonon MSS from measurements of absolute electromagnetic

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decay strengths were reported in the late 1980s and the 1990s by Lieb et al. [31], Vermeer et al. [32], Hartung et al. [33], Takamatsu et al. [34], Fazekas et al. [35], Vanhoy et al. [36], Garrett et al. [37], and Wiedenh¨over et al. [38]. There exists also evidence for a 2+ ms state in a deformed nucleus [39]. Recently, more comprehensive sets of data became available from the combined approach of various experiments that has enabled us to study vibrational MSSs and their multiphonon structure in much greater detail. We will mainly discuss data on nuclei for which besides the one-phonon state also two-phonon states have been identified. The next section briefly summarizes the definition of MSSs and discusses their electromagnetic decay properties that serve us as prominent signatures of MSSs in vibrational and transitional nuclei. A further section will outline the experimental techniques that have been used recently for the investigation of MSSs of several vibrational and transitional nuclei. The following main section summarizes most of the recent experimental progress on our topic and discusses some of the recent data in detail. A summary and an outlook conclude this review. 2. Mixed-symmetry states in the proton–neutron interacting boson model 2.1. Definition of MSSs Mixed-symmetry states (MSSs) can be defined in the framework of the IBM-2 [4], the proton–neutron version of the Interacting Boson Model [40,41]. We will use this definition throughout this article. Comprehensive review articles, e.g., [17,28–30,42–47], and textbooks [41,48–54] deal with several aspects of the IBM, its application to nuclear structure, and its geometrical interpretation. In order to avoid repetition we will sketch here as briefly as possible only the most relevant ideas for the definition and for sufficient theoretical understanding of experimental signatures for MSSs. For an in-depth study of the theoretical formulation of MSSs the reader is referred to the literature cited above. The IBM-2 represents an algebraic model for the structure of heavy nuclei in terms of proton bosons and neutron bosons—in its simplest form of monopole (s) and quadrupole (d) type, only. These bosons are considered as effective valence proton or neutron pairs and, hence, the IBM Hamiltonian conserves separately the total number of proton bosons Nπ and of neutron bosons Nν . A general Hamiltonian of the IBM-2 is given by X X H = E0 + α bα+ bα + vαβγ δ bα+ bβ+ bγ bδ + O(b+ b+ b+ bbb). (1) α

αβγ δ

Here the operators b and b+ denote boson annihilation and creation operators and the indices α, β, γ , δ codify additional quantum numbers for angular momentum, isospin, and magnetic sub-states. The Hamiltonian of Eq. (1) contains one-body boson energy terms and two-body boson–boson interactions. Terms of higher order than two-body are typically neglected in applications of the IBM to nuclear data. In the following we will do the same. Transition operators are usually restricted to one-body operators X T = t0 + tαβ bα+ bβ . (2) αβ

The parameters E 0 , α , vαβγ δ , t0 , and tαβ are subject to further constraints due to hermiticity, spherical tensor rank, or other symmetries that are relevant. Once these constraints are fulfilled the remaining parameters are treated as a set of effective model parameters, {q}, that must be calculated from microscopic considerations or fixed phenomenologically by comparison to data.

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The eigenvalue problem of the IBM Hamiltonian H (q)Ψq = E Ψq (q)Ψq

(3)

is analytically solvable for special classes of parameter sets, {qds }, in imposing dynamical symmetries on the model [42–44]. One particularly interesting symmetry of the IBM-2 is the F-spin symmetry [17] which results in the existence of special classes of IBM-2 eigenstates, characterized by their symmetry under the pairwise exchange of an arbitrarily chosen pair of proton and neutron bosons. States that are totally symmetric under this exchange are called full symmetry states (FSSs) and they correspond to the complete Hilbert space of the IBM-1 model [40] with the same total boson number, where no distinction is made between proton bosons and neutron bosons. Besides the FSSs the F-spin symmetry of the IBM-2 also predicts [4] the existence of mixed-symmetry states (MSSs) that contain at least one antisymmetric pair of proton and neutron bosons. The properties of MSSs are particularly sensitive to the effective proton–neutron interaction. This sensitivity makes the MSSs highly appealing objects for experimental studies. 2.2. F-spin The proton–neutron boson symmetry of the IBM-2 wave functions can be quantified by the F-spin quantum number. The formalism of isospin for a system of protons and neutrons [55] can be straightforwardly applied to a system of proton and neutron bosons. In such a system, the bosons are formally considered as “elementary” particles that form an isospin doublet with projection +1/2 (proton boson) and −1/2 (neutron boson). Boson type

bπ

bν

F Fz

1/2 +1/2

1/2 −1/2

In order to avoid confusion with the ordinary isospin for nucleons the “boson isospin” is called F-spin [4,7]. Looking at the above table we see that in the simplest proton–neutron version of the IBM (namely the IBM-2) there is no deuteron (pn) boson. Therefore, F-spin in the IBM-2 is different from the total isospin of corresponding valence nucleon pairs. However, formally, isospin and F-spin are analogous. The former applies to “elementary” nucleons, the latter to “elementary” bosons. The treatment of the bosons as an F-spin doublet imposes an SU (2) group structure on the transformations of the proton–neutron boson basis states. For a system of s- and d-bosons only, the generators of this group are Fˆ+ = dπ+ · d˜ν + sπ+ sν

(4)

Fˆ− = dν+ · d˜π + sν+ sπ i 1h + ˜ Fˆz = dπ · dπ + sπ+ sπ − dν+ · d˜ν − sν+ sν 2  1ˆ = Nπ − Nˆ ν 2

(5) (6) (7)

with the total proton (neutron) boson number operator Nˆ π(ν) . Here, sρ+ and dρ+ denote s-boson and d-boson creation operators for proton and neutron bosons and “·” denotes the scalar product

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of two spherical tensors of equal rank λ. The operators sρ and dρ µ denote s-boson annihilation and d-boson annihilation in the tensor component µ of isospin ρ, respectively. Requiring the µ-components of the creation operators to be spherical tensors, results in the operators sρ and d˜ρ µ ≡ (−)µ dρ −µ being spherical tensors, too. The generators (4)–(6) satisfy the same commutation relations h i Fˆz , Fˆ± = ± Fˆ± (8) h i Fˆ+ , Fˆ− = 2 Fˆz (9) as the familiar angular momentum operators in a spherical basis or as the isospin operators. Consequently, boson basis states can be formed that are simultaneous eigenstates of the quadratic F-spin operator Fˆ 2 = Fˆ− Fˆ+ + Fˆz ( Fˆz + 1)

(10)

and the z-component Fˆz of the F-spin operator with eigenvalues F(F + 1) and Fz , respectively, just like eigenstates of angular momentum or isospin. For a given nucleus with fixed numbers of proton and neutron bosons, Nπ and Nν , the z-component of the F-spin is a good quantum number. Its value is Fz = Nπ (+1/2) + Nν (−1/2) = (Nπ − Nν )/2. Multiboson wave functions with good total F-spin quantum number can be formed. For a given total boson number, N = Nπ + Nν , the maximum Fz value formed by N F-spin-1/2 particles is Fmax = N /2. The total F-spin quantum number can take values between Fz = |Nπ − Nν |/2 and Fmax = (Nπ + Nν )/2. The F-spin quantum number quantifies the symmetry of an IBM-2 wave function with respect to the pairwise exchange of proton and neutron boson labels as can be seen from Eqs. (4)–(10). Basis states that are characterized by a maximum F-spin quantum number F = Fmax can be transformed by the successive action of the F-spin raising operator Fˆ+ into a state that consists of proton bosons only. Such a state stays obviously unchanged under a pairwise exchange of proton and neutron labels because it does not contain any neutron bosons. Therefore, IBM2 states with maximum F-spin quantum number are called Full Symmetry States (FSSs) with F(FSS) = Fmax . All other basis states with good F-spin quantum numbers F < Fmax are called MSSs. We will discuss only MSSs with quantum numbers F = Fmax − 1 in this article because no examples of states with F < Fmax − 1 have been identified so far on the basis of absolute transition matrix elements. It is easy to see that IBM-1 wave functions with N bosons correspond to IBM-2 wave functions formed by Nπ + Nν = N bosons with maximum F-spin quantum number F = Fmax : Any IBM-1 wave function of N bosons is identical to a corresponding IBM-2 wave function formed by Nπ = N proton bosons and no neutron boson. This is obtained by simply adding a proton boson label “π” to each IBM-1 boson. Applying Eq. (10) to such an IBM-2 wave function Ψπ containing only proton bosons yields Fˆ 2 Ψπ = Fmax (Fmax + 1)Ψπ .

(11)

This result is obtained by using Eq. (10) and the relations Fˆ+ Ψπ = 0 and Fˆz Ψπ = Nπ /2Ψπ = N /2Ψπ = Fmax Ψπ . The Nν -times-repeated use of the F-spin ladder operator Fˆ− on the pure proton boson (IBM-1 type) wave function Ψπ with F = Fmax then yields an IBM-2 wave function formed by Nπ = N − Nν proton bosons and Nν neutron bosons. It has a total Fspin quantum number F = Fmax because the lowering operator Fˆ− does not change the total

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F-spin. Thus, all IBM-1 states correspond to the FSSs of the IBM-2. Eigenstates of IBM-2 Hamiltonians can have good F-spin if H commutes with the F-spin operator, i.e., [H, Fˆi ] = 0. Charge independence of the IBM-2 Hamiltonian is a sufficient condition for all eigenstates to have a good total F-spin quantum number. If the Hamiltonian parameters remain constant (as a function of proton boson and neutron boson numbers) then the eigenstates of boson systems with a given total boson number N form degenerate F-spin multiplets with 2F + 1 members. Approximately degenerate F-spin multiplets have been observed over wide regions of the nuclear chart [56–58,28,59]. These observations support F-spin being an approximate quantum number for low-lying collective states of heavy nuclei. The lowest states in a given nucleus are those formed by the fully symmetric states with the quantum number F = Fmax . As mentioned before, the (approximate) validity of the F-spin limit is supported by the sheer fact that the IBM-1 successfully describes many properties of the lowest-lying states of heavy open-shell nuclei and that M1 transitions between such states are observed to be small on a single-particle scale. The latter property of FSSs originates in the predominantly isovector character of the M1 transition operator and will be discussed in more detail below. In fact, M1 transitions between states with maximum F-spin are exactly forbidden in the IBM-2, and the IBM-1 states correspond exactly to the F-spin symmetric states of the IBM-2. Both facts, i.e., successful description of low-energy nuclear spectra by the IBM-1 and absence of strong M1 transitions at low energies, seem to point to a dominant F = Fmax component in the wave functions of the low-lying states in typical realistic descriptions of open-shell nuclei in the IBM-2. The new qualitative feature emerging in going from the IBM-1 to the IBM-2 is the prediction [3–5,9,10] of entire new classes of collective states with F-spin quantum numbers F < Fmax , namely the MSSs. The experimental discovery of such a state (the 1+ scissor mode) by A. Richter and his group at the DALINAC at TU Darmstadt [1,14] was, thus, a spectacular success which triggered corresponding experimental activities. Fig. 1 shows the spectrum of the schematic IBM-2 Hamiltonian H = (n dπ + n dν ) + λ Mˆ for the boson numbers Nπ = Nν = 1. The quantity n dπ (n dν ) denotes the number of proton (neutron) d-bosons, and Mˆ = [Fmax (Fmax + 1) − Fˆ2 ]/2 is a simple Majorana operator (here in Casimir form) which has non-zero eigenvalues for MSSs, only. The Hamiltonian shown has U (5) symmetry and possesses a vibrational spectrum with a symmetric J π = 2+ one-quadrupole phonon state and a symmetric two-phonon triplet with J π = 4+ , 2+ , 0+ states. The boson wave functions for this example are displayed in Fig. 1, too. Besides the 2+ 1 one-quadrupole phonon FSS there exists also an antisymmetric (with respect to proton–neutron labels) linear combination of the two configurations with n d = 1, forming the mixed-symmetry one-quadrupole phonon 2+ 1,ms state with F-spin quantum number F = Fmax − 1. Antisymmetric angular momentum coupling of two (non-identical) d-bosons leads to mixed-symmetry two-phonon states with odd spin quantum numbers, 3+ and 1+ . A larger boson number N allows for a greater variety of FSSs and MSSs as given in the simple example from Fig. 1. A comprehensive review of the F-spin symmetry of the IBM-2 has been given by Van Isacker et al. [17]; an experimental review has been given by Lipas et al. [28]. Good F-spin implies the applicability of the F-spin tensor formalism in the F-spin space for quantifying the proton–neutron symmetry of boson wave functions or for evaluating the proton–neutron contributions to matrix elements of operators. One major result of this approach is given by the proton–neutron contribution to the matrix elements of any one-body operator between FSSs [17] + hFmax , α k bρ,β bρ,β 0 k Fmax , α 0 i = Nρ cαα 0 ββ 0

(12)

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Fig. 1. Spectrum of a schematic IBM-2 Hamiltonian with U (5) symmetry with boson numbers Nπ = Nν = 1. The boson wave functions are displayed, too. |0i denotes the boson vacuum and []L denotes tensor couplino to rank L.

where α, α 0 , β, β 0 denote additional quantum numbers, Nρ is the number of bosons with isospin character ρ = π or ν, and cαα 0 ββ 0 is independent of ρ. The importance of the F-spin formalism lies in its broad range of applicability to situations outside of the fully analytically solvable dynamical symmetries of the IBM as long as the Hamiltonian is an F-scalar or even to situations where the Hamiltonian is not an F-scalar but F-spin is approximately restored in the wave functions of the eigenstates by a large enough Majorana interaction that sufficiently separates components of the wave functions with the largest values of the F-spin quantum number for the lowest-energy states. 2.3. Decay properties of MSSs The most distinct and measurable feature of MSSs (those with F = Fmax − 1) is their electromagnetic decay by allowed F-vector (1F = 1) M1 transitions to symmetric states. This is important because M1 transitions between FSSs are forbidden and M1 transitions can thus serve as an outstanding signature for MSSs. The standard M1 transition operator in the IBM-2 is given by r 3 T (M1) = (13) [gπ L π + gν L ν ] µ N 4π r   3 Nπ gπ + Nν gν Nπ Nν = L tot + (gπ − gν ) (L π /Nπ − L ν /Nν ) µ N (14) 4π N N √ where L ρ = 10[dρ+ × d˜ρ ](1) with ρ ∈ {π, ν} denotes the angular momentum operator of isospin character ρ and gπ (ν) is the effective proton (neutron) boson g-factor. Here, the notation for the magnetic sub-states is suppressed while d˜ is defined to be a spherical tensor with substates d˜µ = (−1)µ d−µ , and [. . .](L) denotes tensor coupling to rank L. L tot = L π + L ν is the total angular momentum operator of the IBM-2. It is diagonal in IBM-2 by construction and, hence, cannot induce transitions between different states. Using Eq. (12) it is obvious that for transitions between FSSs also the second term in Eq. (14) yields vanishing matrix elements. Thus M1 transitions between FSSs states are forbidden. This prediction of the F-spin symmetry of the IBM is in qualitative agreement with the vast majority of observations on M1 transitions

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Table 1 Some analytical expressions for reduced electromagnetic transition strengths and electromagnetic moments in the U (5) dynamical symmetry of the IBM-2 + B(M1; 2+ 1,ms → 21 ) = + B(M1; 1+ 1,ms → 02 ) =

3 4π (gπ 3 4π (gπ + + 3 B(M1; 11,ms → 22 ) = 4π (gπ + 3 B(M1; 2+ 2,ms → 22 ) = 4π (gπ + 3 B(M1; 3+ 1,ms → 22 ) = 4π (gπ + + 3 (g B(M1; 31,ms → 41 ) = 4π π + µ(21,ms ) = (gπ Nν + gν Nπ ) N2 µ(1+ 1,ms ) = (gπ + gν )/2

− gν )2 62 Nπ Nν N

4 − gν )2 (N −1)N Nπ Nν 7 − gν )2 (N −1)N Nπ Nν

− gν )2 3(N −2)2 Nπ Nν (N −1)N

− gν )2 7(N24 −1)N Nπ Nν − gν )2 7(N18 −1)N Nπ Nν

+ 2 1 B(E2; 2+ 1,ms → 01 ) = (eπ − eν ) N Nπ Nν + 2 2 B(E2; 2+ 2 Nπ Nν 1,ms → 02 ) = (eπ − eν ) 5(N −1)N

+ 2 1 B(E2; 1+ 1,ms → 21 ) = (eπ − eν ) N Nπ Nν + 2 1 B(E2; 1+ 1,ms → 21,ms ) = (eπ Nπ + eν Nν ) N + 2 1 B(E2; 3+ 1,ms → 21 ) = (eπ − eν ) N Nπ Nν + + B(E2; 31,ms → 21,ms ) = (eπ Nπ + eν Nν )2 N1

From Refs. [17,61].

between low-energy collective states, that in most cases have a strength of the order of about 0.01 µ2N or less. In contrast to that, M1 transitions between MSSs and FSSs are F-spin allowed. Since the relevant difference of boson g-factors gπ − gν is of the order of one nuclear magneton, one can expect allowed M1 matrix elements to be roughly of order 1 µ N . For the dynamical symmetries the matrix elements of transition operators can be calculated analytically. This task has been carried out [60] for many matrix elements relevant to the interpretation of data and has been reviewed in Ref. [17]. Further results were published in Ref. [61,62]. Tables 1 and 2 summarize analytical formulae for transition rates and electromagnetic moments relevant to the discussion of MSSs in the U (5) and O(6)F-spin symmetries of the IBM-2, respectively. The M1 transition operator is given in Eq. (13). The E2 transition operator is defined by T (E2) = eπ Q χππ + eν Q χν ν

(15)

with χ

Q ρ ρ = sρ+ d˜ρ + dρ+ sρ + χρ [dρ+ d˜ρ ](2)

(16)

where eπ (ν) is the effective electric quadrupole proton (neutron) boson charge. The M3 transition operator is defined by r i 35 h T (M3) = Ωπ [dπ+ d˜π ](3) + Ων [dν+ d˜ν ](3) (17) 8π with the effective magnetic octupole boson charges Ωρ . In the F-spin symmetry all M1 transitions between FSSs are forbidden, M1 transitions between MSSs and FSSs are proportional to the quantity (gπ − gν )2 Nπ Nν . E2 transitions between FSSs are proportional to (eπ Nπ + eν Nν )2 , and E2 transitions between MSSs and

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Table 2 Some analytical expressions for reduced electromagnetic transition strengths in the O(6) dynamical symmetry of the IBM-2 + 3 2 3(N +2)(N +4) N N B(M1; 2+ π ν 1,ms → 21 ) = 4π (gπ − gν ) 4N 2 (N +1) + 3 2 1 B(M1; 1+ 1,ms → 01 ) = 4π (gπ − gν ) N +1 Nπ Nν + 3 2 (N +4)(N +5) B(M1; 1+ 1,ms → 22 ) = 4π (gπ − gν ) 2(N −1)N (N +1) Nπ Nν 3 2 12(N +4)(N +5) B(M1; 3+ → 2+ 1,msP 2 ) = 4π (gπ − gν ) 49(N −1)N (N +1) Nπ Nν

µ(2+ 1,ms ) =

ρ=π,ν [N (3N +8)−2(N +4)Nρ ]gρ

2N (N +2) + µ(11,ms ) = (gπ + gν )/2

+ 2 2(N +2) B(E2; 2+ 1,ms → 01 ) = (eπ − eν ) 5N (N +1) Nπ Nν + 2 9(N +2)(N +4)(N +5) B(E2; 2+ 1,ms → 41 ) = (eπ − eν ) 140(N −1)N 2 (N +1) Nπ Nν + 2 N +4 B(E2; 1+ 1,ms → 21 ) = (eπ − eν ) 2N (N +1) Nπ Nν P { ρ=π,ν [N −(N +4)Nρ ]eρ }2 4N (N +2) + + 2 B(E2; 31,ms → 21 ) = (eπ − eν ) 2NN(N+4 +1) Nπ Nν P { ρ=π,ν [N −(N +4)Nρ ]eρ }2 + + B(E2; 31,ms → 21,ms ) = 4N (N +2) + B(E2; 1+ 1,ms → 21,ms ) =

+ 35 7 2 B(M3; 3+ 1,ms → 01 ) = 8π (Ωπ − Ων ) 10(N +1) Nπ Nν

From Refs. [17] except for the M1 and E2 formulae involving the 3+ 1,ms state.

FSSs are proportional to the expression (eπ − eν )2 Nπ Nν . The proportionality factors depend on the structures of the wave functions involved in a given transition and can be analytically calculated for the dynamical symmetries. The M1 operators conserve the U(5) quantum numbers. Therefore M1 transition between configuration with different numbers of d-Bosons do not occur. However, most nuclei exhibit spectra that correspond to more or less broken symmetries and are frequently described best with IBM Hamiltonians outside of the dynamical symmetries [46]. The Q-phonon scheme [63–66] provides approximations to the wave functions that often help to semi-quantitatively understand the decay properties of boson states in regions other than those for analytically solvable cases. 2.4. Q-phonon scheme The collective properties of heavy nuclei are often well described with IBM Hamiltonians outside of the dynamical symmetries. For numerical studies one frequently uses the following IBM-2 Hamiltonian: H = π nˆ dπ + ν nˆ dν + κππ Q χππ · Q χππ + 2κπν Q χππ · Q χν ν ˆ 1 , ξ2 , ξ3 ) + κνν Q χν ν · Q χν ν + M(ξ

(18)

which contains single d-boson energies, mutual proton–neutron quadrupole–quadrupole interactions and a symmetry energy term. This boson Hamiltonian effectively mimics the nucleonic pairing-plus-quadrupole Hamiltonian. The nˆ dπ and nˆ dν are the d-boson number

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235

χ

operators; Q ππ and Q ν ν are the quadrupole operators defined previously. The most general threeparameter Majorana interaction Mˆ can be defined as ˆ 1 , ξ2 , ξ3 ) = 1 ξ2 (sπ+ dν+ − dπ+ sν+ ) · (sπ d˜ν − d˜π sν ) M(ξ 2  i(K )  X (K ) h  − ξ K dπ+ dν+ · d˜π d˜ν .

(19)

K =1,3

The Hamiltonian of Eqs. (18) and (19) is not in general an F-scalar and thus allows for F-spin breaking. However, F-spin remains an approximate quantum number in most practical cases, in particular for the lowest states. A special form of Eq. (18) is the IBM-2 Hamiltonian  
2
(20) H =  nˆ dπ + nˆ dν + κ Q χπ + Q χν + λ Fmax (Fmax + 1) − Fˆ2 for which F-spin is an exact quantum number. We will focus on nuclei that are described with structural parameters χπ ≈ χν ≈ 0. For χ = 0 the Hamiltonian has O(5) symmetry. Particularly in this case, and for good F-spin, the approximative Q-phonon scheme provides a useful framework for the semi-quantitative discussion of FSSs and MSSs in the IBM even outside of the exact dynamical symmetries. The Q-phonon scheme is, therefore, a convenient tool for understanding the main structural mechanism in the formation of collectivity and it is a particularly useful formalism for the design of new experiments. The Q-phonon scheme was introduced by Otsuka et al. [64] and was developed by the Cologne–Tokyo collaboration [63–69]. In the Q-phonon scheme for mixed-symmetry states [70] the unnormalized wave functions of the lowest-lying fully symmetric and mixed-symmetry states can be written in a compact way as Q-phonon excitations of the strongly correlated ground state |0+ 1 i, e.g., + |2+ 1 i = Q s |01 i

(21)

+ |2+ ms i = Q m |01 i

(22)

(4) + |4+ 1 i = [Q s Q s ] |01 i (3) + |3+ ms i = [Q s Q m ] |01 i

|1+ ms i =

[Q s Q m ](1) |0+ 1 i.

(23) (24) (25)

Here Q s = Q π + Q ν and Q m = [Q π N /Nπ − Q ν N /Nν ] denote the F-scalar and the mixed-symmetric quadrupole operators, respectively, with Q ρ from Eq. (16). The normalization factors of the wave functions were suppressed in Eqs. (21)–(25) for simplicity. These Q-phonon expressions represent a good approximation to the unnormalized wave functions for large ranges of values for the parameters of realistic IBM-2 Hamiltonians even in regions outside of the dynamical symmetries [70]. The validity of the Q-phonon scheme is easy to understand in the U (5) and O(6) dynamical symmetry limits, which is the type of nuclear structure that we focus on here, using the property of matrix elements of one-body operators between FSSs from Eq. (12). This relation ensures that the configuration Q m |0+ 1 , Fmax i does not have any overlap with FSSs. Since Q m is a onebody operator it can change the F-spin quantum number at most by one unit and, therefore, the configuration on the r.h.s. of Eq. (22) is completely exhausted by the space of states with quantum number F = Fmax − 1. The d-parity quantum number which will be discussed below ensures in conjunction with τ selection rules of the quadrupole operator the validity of the Eqs.

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(21)–(25) for eigenstates of IBM-2 Hamiltonians with O(5) symmetry. Note that the action of the F-spin scalar Q s on a wave function cannot change the F-spin. Thus, acting with Q s on the wave function of the 2+ 1,ms state with F = Fmax − 1 yields two-phonon states with mixed symmetry. The Q-phonon scheme makes the importance of the 2+ 1,ms state particularly apparent: it is the fundamental mode or building block of the quadrupole-collective mixed-symmetry structures. Therefore, it is a particularly appealing object to study. Moreover, the identification of mixedsymmetry structures built on top of the 2+ 1,ms state is of importance because the existence of a whole class of mixed-symmetry states is a parameter-free prediction of the IBM-2. Eq. (22) shows that the 2+ 1,ms state is a one-phonon excitation which has proton and neutron E2 excitation matrix elements of about the same size as those of the 2+ 1 state. Due to the destructive interference between the proton and neutron E2 matrix elements, the total E2 strength of excitation to the + 2+ 1,ms is reduced in comparison to that of the 21 state. However, weakly collective E2 strengths of the order of 1 W.u. are expected since an accidental exact cancellation of proton and neutron parts is in general unlikely. Although MSSs are defined in the framework of the IBM-2, the Q-phonon scheme also allows for an interpretation of MS structures in other nuclear models that are capable of describing proton and neutron quadrupole matrix elements. The Q-phonon scheme can, for instance, be applied to the results of shell model calculations in order to interpret the microscopic wave functions in terms of symmetric and mixed-symmetry parts. 2.5. d-parity For the discussion of electromagnetic decays in near-vibrational nuclei the concept of d-parity is useful. [38,71–73]. The Hamiltonian (18) has d-parity symmetry for the parameter values χπ = χν = 0. In this case it is invariant under the inversion of the sign of the d-boson operators. This d-parity operation [38,71,73] Pd : dρ → −dρ , sρ → sρ (ρ = π, ν) does not change the Hamiltonian for χπ = χν = 0 since it contains only terms with an even number of dboson operators. Due to this d-parity symmetry of the special Hamiltonian with χπ = χν = 0 its eigenstates have a good d-parity πd . Their boson wave functions contain either only even numbers or only odd numbers of d-bosons. This corresponds to positive or negative d-parity. The M1 operator does not change the number of d-bosons and, hence, for Hamiltonians with d-parity symmetry it connects only states with the same d-parity. M1 transitions that change the d-parity quantum number are exactly forbidden, also outside of the analytical dynamical symmetry limits. On the other hand, the E2 transition operator with d-parity symmetry, i.e. with χπ = χν = 0, changes the number of d-bosons by one unit and therefore connects only states with different d-parity. The d-parity selection rules [73] are important signatures for MSSs of β-soft or γ -soft vibrational nuclei. Fig. 2 illustrates the d-parity selection rules for one-phonon and two-phonon MSSs for Hamiltonians with d-parity symmetry. The d-parity quantum number πd survives, at least approximately, in cases where the values χπ and χν differ from zero but are small, in which case the d-parity selection rules remain approximately valid [73]. 2.6. Signatures for MSSs in near-vibrational nuclei On the basis of the arguments given above we expect the following signatures for mixedsymmetry one-phonon and two-phonon excitations for vibrational and transitional nuclei with,

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Fig. 2. F-vector M1 (left) and E2 (right) transitions of the lowest symmetric and mixed-symmetry 2+ states in the O(6)F-spin limit. Solid arrows indicate allowed decays while transitions that are forbidden due to the d-parity selection rules are plotted with dashed arrows. The values of the d-parity quantum number πd are shown in the middle. From Ref. [73].

at least approximate, O(5) symmetry, i.e., for which d-parity is a good or approximate quantum number: • The one-phonon 2+ 1,ms state is the lowest-lying MSS. + • The 21,ms state decays to the 2+ 1 by a strong M1 transition with an absolute matrix element of order 1 µ N . • Since the 2+ 1,ms state is a one-phonon excitation it should have sizeable collective E2 matrix elements for a transition to the ground state for both protons and neutrons. The matrix elements for protons and neutrons have however opposite signs, which might lead to partial + cancellation in the total h0+ 1 k E2 k 21,ms i matrix element. We thus expect a weakly collective + E2 transition strength of a few W.u. for the 2+ 1,ms → 01 transition. • Two-phonon MSSs decay by a collective E2 transition to the one-phonon 2+ 1,ms state with an + E2 strength comparable to that of the 2+ → 0 transition. 1 1 • Two-phonon MSSs decay by a strong M1 transition to the symmetric two-phonon states with + an M1 strength comparable to that of the 2+ 1,ms → 21 transition, i.e., with a matrix element of about 1 µ N . • Two-phonon MSSs should not decay by a strong M1 transition to the symmetric 2+ 1 state because of the d-parity selection rule. • Two-phonon MSSs should instead decay to the symmetric 2+ 1 state by a weakly collective E2 + transition comparable in strength to the 2+ → 0 transition. 1,ms 1 • All MSSs are expected to be very short lived, typically a few hundred femtoseconds or less, because of the strong M1 matrix elements for transition to particular FSSs and typical transition energies of about 1 MeV in vibrational nuclei. At this point we like to stress that the observation of small E2/M1 multipole mixing ratios δ for specific transitions to low-energy FSSs is not a sufficient signature for the assignment of a mixed-symmetry character to a nuclear state. Even an M1 transition strength too small for a safe MS assignment can generate a small E2/M1 multipole mixing ratio if the E2 transition is not collective, in particular, and for relatively small γ -ray energies. For example, a 0.5 MeV 2+ → 2+ transition in an A = 100 nucleus with a small B(M1) value of 0.02 µ2N and a typical

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non-collective E2 transition strength of half a Weisskopf unit would have an E2/M1 multipole mixing ratio |δ| = 0.15, although the B(M1) value would be almost an order of magnitude too small for being in agreement with the predictions for MSSs. Instead, a firm assignment of mixed symmetry should be based on the measurement of large absolute B(M1) values. In this review we consider only those nuclei where absolute B(M1) transition values have been determined. The measurement of a complete set of B(M1) values for all states with a given spin in a certain energy region is preferred not only for identifying the corresponding MSS but also for judging its possible fragmentation into several neighboring states. 3. Experimental techniques for identification of mixed symmetry The signatures for mixed-symmetry states are their quadrupole collectivity and the simultaneous existence of specific enhanced M1 transitions to FSSs. As has been discussed above, a firm identification of MSSs requires the measurement of the absolute B(M1) values for the expected enhanced M1 transitions. Preferably, we should have rather complete information on the fragmentation of the MSS under investigation. This requires typically the measurements of: • • • •

spin and parity quantum numbers J π , lifetimes τ , γ -decay branching ratios Γi /Γ , and E2/M1 multipole mixing ratios δ 2 = Γi,E2 /Γi,M1 for a complete set of states with J π in a given energy range.

In order to gather the desired information, one typically has to combine a variety of experimental techniques which have different sensitivities to different observables. The experimental approach used recently for the identification of multiphonon MSSs in vibrational nuclei gained its power from a new combination of classic techniques of nuclear γ -ray spectroscopy. Nuclear low-spin states of vibrational nuclei have been investigated by means of electron scattering and by γ -ray spectroscopy using the electromagnetic interaction, the strong interaction, and the weak interaction in the population mechanisms. In particular various reactions such as photon scattering, Coulomb excitation, proton- and helium-induced light-ion fusion evaporation, inelastic neutron scattering, and β-decay were used in the γ -ray spectroscopy studies. All of these different experimental approaches have specific capabilities and special advantages and drawbacks. Combining the complementary information from the various reactions one can obtain a new richness of nuclear structure data which enables one to identify MS multiphonon structures from absolute M1 and E2 transition strengths [74–77] and even study their fragmentation. These techniques are standard methods in low-spin nuclear spectroscopy and we will only briefly describe them below along with examples of their application to the investigation of MSSs. The experimental techniques might be classified into two categories: those that provide high-precision information on the relative values of electromagnetic decay matrix elements through high-statistics experiments and those that yield data on absolute values for electromagnetic decay matrix elements through the measurements of cross sections or level lifetimes. Ideally, information from the two approaches can be combined. In this subsection we give a brief overview of the experimental techniques: nuclear spectroscopy by electron scattering and the use of γ -ray spectroscopy in photon scattering, Coulomb excitation, β-decay, inelastic neutron scattering, and light-ion fusion evaporation reactions for the identification of MSSs.

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3.1. Electron scattering In electron scattering reactions electrons with incident beam energy E e interact with the charge, current and magnetization density distributions of the target nuclei by virtual photon exchange and are scattered at an angle θ where their final energy E e0 (θ ) is measured with an electron spectrometer. Excited nuclear states at excitation energies E x show up as peaks in the energy loss spectrum E e − E e0 (θ ). The ratio of the experimentally observable cross section (dσ/dΩ ) E x (E e , θ ) for a level at energy E x to the Mott cross section 

dσ dΩ



(E e , θ ) = Mott



Z e2 2E e

2

cos2 θ/2 sin4 θ/2

defines the square of the form factor FE x (E e , θ ) according to     dσ dσ = |FE x (E e , θ )|2 . dΩ E x dΩ Mott

(26)

(27)

The form factor contains the information on the charge density, current and magnetization density distributions. The full formalism of electron scattering can be found, e.g., in Refs. [78,79]. In the typical approach, the scattering process, on a given density distribution hypothesis, is modeled in the distorted wave Born approximation (DWBA) and compared to the data. For not too high nuclear charge (Z α  1) and low momentum transfer the scattering process can be analyzed with sufficient accuracy in the plane wave Born approximation (PWBA). The form factor can then be written as a function of the 3-momentum transfer (neglecting recoil correction at this point) q q = 2E e (E e − E x )(1 − cos θ ) + E x2 /h¯ c (28) as a decomposition in electric and magnetic components of multipolarity λ:      X  dσ  dσ dσ = + . dΩ PWBA dΩ Eλ dΩ Mλ λ

(29)

Let us discuss here the simple example of excitation of a 2+ state from a 0+ ground state, e.g., for a typical situation in even–even nuclei on which we focus in this article. We further restrict ourselves to the limit of low momentum transfer of a few fm−1 at most. The excitation proceeds then entirely due to the E2 cross section. It can be separated into longitudinal and transverse parts according to 

dσ dΩ



 = E2

Z e2 Ee

2

[VL |F(C2, q)|2 + VT |F(E2, q)|2 ]

(30)

where VL =

1 + cos θ 2(y − cos θ )2

(31)

denotes the longitudinal kinematical factor and VT =

2y + 1 − cos θ 4(y − cos θ )2 (1 − cos θ )

(32)

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is the transverse kinematical factor with y =1+

E x2 2E e (E e − E x )

(33)

in the relativistic limit E e  m e c2 . Within this approximation, the form factors from Eq. (30) are related to the nuclear transition charge density ρi→ f (r ) according to qλ ˆ Cλ (q) k ψi i hψ f k M 2Ji + 1 · (2λ + 1)!! Z (2λ + 1)!! Cλ ˆ Jλ (qr )Yλµ ρi→ f (E r )d3r hψ f |Mµ (q)|ψi i = qλ F(Cλ, q) = √

(34) (35)

and to the transition current density Eji→ f (E r ) according to √

λ + 1 · qλ ˆ Eλ (q) k ψi i hψ f k M F(Eλ, q) = √ (2Ji + 1)λ · (2λ + 1)!! Z   ˆ µEλ (q)|ψi i = (2λ + 1)!! E × Jλ (qr )YEλ±1λµ Eji→ f (E hψ f |M ∇ r )d3r. λ+1 q

(36) (37)

The (typically small) contribution of the nuclear magnetization density to the expression in Eq. (37) has been neglected for simplicity. Except for extreme scattering angles θ ≈ nπ the longitudinal part of the cross section typically dominates over the transverse part that moreover can be separated by a Rosenbluth separation. At the photon point q = E x /h¯ c the momentumtransfer-dependent reduced transition strength + 2 + + ˆ C2 B(E2, q; 0+ 1 → 2 ) = |h2 k M (q) k 01 i|

(38)

equals the reduced E2 strength for γ -radiation. This can be seen from expansion of the Bessel function Jλ in Eq. (35) for small q and keeping the first term only which then involves the E2 operator r 2 Y2 . However, electron scattering not only offers the measurement of data that are equivalent to γ excitation by extrapolation of the form factor to the photon point, but, and more importantly, also is sensitive to the radial densities of charges and currents. This fact originates in the decoupling of energy and momentum transfer in electron scattering. It thereby allows for unique tests of nuclear models in a way which is beyond what can be inferred from γ -decay data alone. Consequently, electron scattering has developed into a major tool for nuclear spectroscopy which has been documented in several textbooks and review articles, e.g. [78–81,14]. The experiment that pioneered electron scattering for research on MSSs of vibrational nuclei was done by Hartung et al. [33] on the nucleus 56 Fe at the DALINAC facility at Darmstadt and at NIKHEF at Amsterdam. Fig. 3 shows two corresponding parts of the energy loss spectra 56 Fe suggested from that experiment in the energy region of the one-phonon 2+ 1,ms state of + previously [82,31] to be located around 2.8 MeV. Here the 22 at 2.66 MeV and the 2+ 3 at 2.96 MeV share symmetrical two-phonon and mixed-symmetry one-phonon character [82,31,33]. The agreement of the measured form factors [33] with the model predictions corroborates the MS assignments independently from the large M1 transition strength observed in γ -ray spectroscopy to the 2+ 1 state.

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Fig. 3. Parts of energy loss spectra from electron scattering experiments on 56 Fe performed [33] at the DALINAC facility 56 at Darmstadt (top) and at the NIKHEF facility at Amsterdam (bottom) for the energy range of the 2+ 1,ms state of Fe [82, 31]. From Ref. [33].

3.2. Nuclear resonance fluorescence The resonant photon scattering reaction or nuclear resonance fluorescence (NRF) is a resonance reaction and thus a two-step process. First, a photon is absorbed and lifts the nucleus to an excited state, which decays subsequently by the emission of a second photon to lowerlying states. The emitted photon can be detected with highly efficient large-volume germanium detectors which enables studies with high energy resolution. The reaction mechanism is purely electromagnetic and offers a variety of observables, including the natural level width and spin and parity quantum numbers, as has been extensively discussed in previous review articles, e.g., Refs. [15,83,84]. NRF is sensitive to the partial ground state transition width Γ0 and is hence well suited for investigating very short-lived nuclear excitations that possess a large matrix element for the electromagnetic transition to the ground state. The energy-integrated cross section Is,0 for the elastic resonant photon scattering reaction is given by the relation Γ02 (39) Γ with the reduced wavelength λ = h¯ c/E x and the statistical spin factor g = (2J + 1)/(2J0 + 1), where J and J0 are the spin quantum numbers of the excited state and of the ground state, Is,0 = π 2 λ2 g

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Fig. 4. Part of a γ -ray spectrum scattering off a 1 g isotopically enriched sample of 136 BaCO3 using an incident bremsstrahlung beam with energies E γ ≤ 2.8 MeV at the IfS of Stuttgart University. The angular distribution of the resonant γ -ray scattering intensity at 2129 keV indicates the excitation of a J = 2 state. The peak area at 2129 keV is proportional to the electromagnetic excitation cross section and allows one to determine the lifetime of this level. From Ref. [85].

respectively. Γ = h¯ /τ is the total level width of the excited state and Γ0 is the partial width of the electromagnetic decay to the ground state. From measurements of the energy-integrated photon scattering cross sections one can determine the level lifetime and transition matrix elements in a model-independent way once the spin quantum numbers J and the decay branching ratios Γi /Γ0 are known. The former can be extracted from photon scattering experiments by an analysis of angular intensity distributions. Due to small angular distribution anisotropies in cases of halfinteger ground state spins J0 > 1/2, spin assignments in NRF are, however, practically restricted to even–even nuclei in most cases. Branching ratios for the decay to excited states E i are obtained from the observation of γ -ray intensities at corresponding decay energies E γ 0 = E x − E i in conjunction with the Ritz combination principle. Due to typically high background at low energies in NRF spectra and due to possible accidental matches of the Ritz energy combination it is preferable to obtain relative decay data from γ -ray coincidence methods. In the late 1990s the NRF technique was first applied to the lifetime measurement of the 2+ 1,ms one-quadrupole phonon MSS [85]. Of course, the NRF technique has been a workhorse in the investigation of the scissors mode [14,15,84, and references therein]. Its application to the study of 1+ MSSs was an obvious choice. The Q-phonon formalism emphasizes the one-phonon character of the 2+ 1,ms and suggests, due to the weakly collective E2 coupling to the ground state, accessibility in NRF experiments, too. Fig. 4 shows the data from the first investigation of a

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243

+ 2+ 1,ms state with the NRF method which resulted in the observation of the 21,ms state and the two+ 136 phonon 1 mixed-symmetry states in the nucleus Ba. Application of the NRF technique to the study of MSSs, in particular to two-phonon states with J π = 1+ , has recently been reviewed in Ref. [84]. The NRF method is however not very sensitive to small γ -decay branching ratios and typically not useful for the measurement of γ -transition multipole mixing ratios. The NRF method furthermore requires the handling of considerable amounts (e.g. 1 g) of isotopically enriched target material, which can be very expensive and thus limits the method in practice to investigating more abundant stable isotopes.

3.3. Coulomb excitation In Coulomb excitation (COULEX) [86–88] the nucleus is excited by the Coulomb field of impinging charged projectiles passing the nucleus with velocity v at a short distance without penetrating into the range of nuclear forces. COULEX can be considered as an absorption process for virtual photons. COULEX is sensitive to electromagnetic matrix elements for excitation from the ground state and has become a major tool for γ -ray spectroscopy which has been documented in several review articles, e.g. [88–92]. In general, the excitation cross section contains interfering terms from the sum over different excitation paths that correspond to multiple COULEX processes. The situation is considerably simplified in the one-step case where one excitation path predominates. This is the case, for instance, for the COULEX of one-quadrupole phonon states that are excited from the ground state in one-step E2 excitations. The total cross section for such a one-step E2 COULEX process is given in the semi-classical approximation by [89]   Z 1 e 2 −2 σ E2 = a B(E2; 2+ → 0+ (40) 1 ) f E2 (ξ ) h¯ v with a=

Z 1 Z 2 e2 µv 2

(41)

and where Z 1 , Z 2 are the charge numbers of the projectile and the target, µ is the reduced mass of the scattering problem, and v is the velocity of the projectile with respect to the target. The kinematical function f E2 (ξ ) is monotonically decreasing with increasing ξ , where ξ is given by ξ=

a1E h¯ v

(42)

where 1E is the excitation energy of the Coulomb excited state. Eq. (40) shows that the one-step COULEX cross section is proportional to the reduced E2 transition strength. The cross section furthermore increases with decreasing excitation energy for a given transition matrix element. This is in contrast to the case for NRF where the cross section increases with the transition energy taken to the power of 2λ − 1 for multipole radiation of rank λ and a fixed transition matrix element hJ f k Mˆ λ k gsi. The Coulomb excitation to a MSS was first observed by Vermeer et al. [32] who demonstrated 142 Ce at 2004 keV excitation energy from the MS character of the 2+ 3 state of the N = 84 nucleus the measurement of its E2 ground state excitation matrix element. This information together with the previously known branching ratio for γ -decay to the ground state and to the 2+ 1 state and the E2/M1 multipole mixing ratio δ of the latter made it possible to deduce the large

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+ + B(M1; 2+ 3 → 21 ) value which served as the main signature of the 21,ms state. This assignment was later confirmed by Vanhoy et al. [36] in a measurement using the Doppler Shift Attenuation Method in Inelastic Neutron Scattering (DSAM-INS; see below). Depending on the experimental details the COULEX analysis can be partly model dependent through the estimate of the impact of two-step excitation processes due to incomplete knowledge about quadrupole moments of nuclear states. This model dependence might limit the method to an accuracy of about 10%–20%. This accuracy is however sufficient for making reliable MS

Fig. 5. Gamma-ray spectra from the bombardment of a 1 mg/cm2 carbon foil with 280 MeV 96 Ru nuclei at the WNSL, Yale University. The beam energy ensured dominant Coulomb excitation (“safe COULEX”). Panel (a) displays the raw γ -ray in-beam singles spectrum observed at an observation angle of 0◦ with respect to the beam axis. Panels (b)–(d) show the in-beam spectra at observation angles of 0◦ , 42◦ , and 138◦ after subtraction of the room background. Panel + 96 (e) displays the coincidence spectrum gated with the 832.6 keV 2+ 1 → 01 transition in Ru. The measured Coulomb 96 Ru helped to unambiguously identify this state as the 2+ excitation cross section for the 2+ state of 3 1,ms one-phonon excitation. From Ref. [93].

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Fig. 6. Top: Angular distribution functions for γ -ray transitions observed in projectile Coulomb excitation of a 480 MeV + beam of 138 Ce ions with the Gammasphere array for the 2237 keV 2+ 4 → 01 E2 transition (left), for the 1448 keV + + + 24 → 21 predominantly M1 transition (middle), and for the 1354 keV 23 → 2+ 1 mixed E2/M1 transition (right). The solid lines show fits to Legendre polynomials. Bottom: Resulting A2 /A0 vs. A4 /A0 distribution coefficients for the γ -ray lines at 1354 and 1448 keV compared to the theoretical ellipse for a 2 → 2 transition plotted as a parametric function of the E2/M1 multipole mixing ratio δ whose local values are indicated along the ellipse. From Ref. [94].

assignments. Multistep processes can be further reduced by using reactions with a low-Z reaction partner such as carbon or helium particles which leads however to a smaller total cross section. One important advantage of COULEX experiments over both NRF and DSAM-INS experiments is that they do not require large quantities of enriched target material and that they might even be performed with beams of β-unstable nuclei in inverse kinematics if sufficient beam intensity is available at radioactive-ion beam facilities. The method of studying MSSs in COULEX in inverse kinematics has been demonstrated [93] + 96 by the identification of the 2+ 3 state of Ru at an excitation energy of 2284 keV as the 21,ms state in a projectile Coulomb excitation experiment with a beam of stable 96 Ru ions impinging with an energy of 280 MeV on a carbon foil at the ESTU-TANDEM accelerator of the WNSL at Yale + University. Fig. 5 shows some γ -ray spectra from that experiment. Doppler-shifted 2+ 3 → 21 transitions are clearly visible, indicating the excitation of the 2+ 3 state in that reaction. The corresponding excitation cross section, and thus the E2 excitation strength of about two singleparticle units in this case, can be obtained from a comparison to the cross section for the 2+ 1 state at 832.6 keV with a known E2 excitation strength. Systematic errors due to multiple COULEX processes are small due to the small charge of the carbon target nuclei and can be estimated quantitatively from the simultaneous observation of the weak two-step COULEX of the 4+ 1 state + visible in the coincidence spectrum gated on the 2+ → 0 transition [93]. 1 1 More recently the technique of projectile Coulomb excitation has been developed into a self-sufficient method for the investigation of the 2+ 1,ms one-Q-phonon MSS. The full set of

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signatures for the one-Q-phonon MSS has been obtained in Ref. [94] from one-step projectile Coulomb excitation of 138 Ce ions on a carbon target. Gamma rays have been observed with the Gammasphere Ge detector array at ANL, consisting of 98 HPGe detectors surrounding the target in rings at 17 different polar angles relative to the beam axis. Besides relative COULEX cross sections, γ -ray branching ratios and lifetimes, the data yielded independent information on the multipolarities of the γ -ray transitions from the analysis of their angular intensity distributions. Fig. 6 shows an example of the data [94]. This made the derivation of complete information + + on B(E2; 2+ → 0+ 1 ) and B(M1; 2 → 21 ) strength distributions possible up to an excitation energy of about 2.7 MeV. These distributions not only allow for the identification of the 2+ 1,ms one-phonon MSS but also yield information on its fragmentation. 3.4. β-decay Gamma-ray spectroscopy following β-decay [95] of a low-spin state of an odd–odd nucleus can be very sensitive to γ -decay branching ratios and multipole mixing ratios, for gamma transitions between highly excited low-spin states of the even–even daughter nucleus. The population mechanism is spin selective (proportional to the β-decay matrix elements) and has the disadvantage that it cannot be controlled by the experimentalist. However, in well-chosen cases β-decay can yield very high population rates of highly excited low-spin states of interest as can be seen from the middle panel of Fig. 7 which was obtained from the β + -decay of the 52 min (2)+ isomer 94 Tcm . The off-beam γ -spectra can be extremely clean, in particular when the technique of Comptonescape suppression is used. Gamma-decay branching ratios can then be measured precisely either from γ -ray single spectra without the necessity for corrections for angular distribution effects or from γ γ -coincidences. Very small γ -decay branching ratios (<1%) can be measured with high accuracy for those states that are strongly populated. Precise multipole mixing ratios can be obtained from the analysis of γ γ -coincidence intensities as a function of relative observation angle θ (angular correlation from unoriented states [96]) or from the measurement of linear polarizations of coincident γ -rays with Compton polarimeters [97,98]. For coincident γ -radiation from an initially unoriented source, the angular correlation function can be expanded in terms of Legendre polynomials in cos θ: W (θ ) = A0 + A2 P2 (cos θ ) + A4 P4 (cos θ ) + · · · = A0 [1 + a2 P2 (cos θ ) + a4 P4 (cos θ ) + · · ·]

(43) (44)

where A0 denotes the total coincidence intensity. The relative expansion coefficient an can be calculated by the quantum theory provided that the spin quantum numbers and γ -ray multipolarities are known. L 1 ,L 0

L 2 ,L 0

For a typical coincidence cascade Ji (γ1 1 )J (γ2 2 )J f with leading γ -multipolarities L 1 and L 2 and with next-to-leading multipolarities L i0 = L i + 1 the expansion coefficients separate into two parts corresponding to each γ -emission: an = Bn (γ1 )An (γ2 )

(45)

with Bn (γ1 ) =

i 1 h 0 2 0 0 F (L , L , J , J ) − 2δ F (L , L , J , J ) + δ F (L , L , J , J ) (46) n i n i n i 1 1 1 1 1 1 1 1 1 + δ12

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and An (γ2 ) =

i 1 h 0 2 0 0 F (L , L , J , J ) + 2δ F (L , L , J , J ) + δ F (L , L , J , J ) . n f n f n f 2 2 2 2 2 2 2 2 1 + δ22 (47)

The F-coefficients are geometrical factors and are tabulated [96]. The δi denote the multipole mixing ratios. The E2/M1 mixing ratios δ are defined here in the phase convention of Ref. [96] as √ hΨ f k Ej N AE2E k Ψi i 3 E γ hΨ f k E2 k Ψi iBM . (48) δ= · = M 10 h¯ c hΨ f k M1 k Ψi iBM hΨ f k Ej N AE1 k Ψi i

Fig. 7. Top: Photon spectrum scattering off 94 Mo nuclei using energies of E γ ≤ 3.3 MeV for the incident bremsstrahlung photons at the DYNAMITRON accelerator in Stuttgart. Middle: Spectrum of γ -rays following the β + decay of the J π = (2+ ) low-spin isomer, 94 Tcm , observed at the Cologne cube spectrometer. Bottom: Gamma-ray spectrum of 94 Mo nuclei excited at the Lexington Van De Graaf accelerator by inelastic scattering of neutrons with kinetic energies of 3.6 MeV. Some low-spin levels of interest are populated in all three reactions. The combined analysis yields absolute electromagnetic transition matrix elements for a unique identification of J π = 1+ , 2+ , and 3+ MSSs. The measuring times needed for taking these γ -ray spectra are indicated in units of hours. From Ref. [77].

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The second equation in (48) gives the relation of the mixing ratio δ to the reduced matrix elements of the transition operators as defined by Bohr and Mottelson [99]. In a typical situation some of the spin quantum numbers and multipolarities are known, e.g. for + the very useful Ji → 2+ 1 → 01 coincidence cascade. The theoretical angular correlation function then contains two unknowns, Ji and δ1 , that can be unambiguously determined from a sufficiently precise measurement of the an ’s. For the most important case of dipole and quadrupole radiation Eq. (44) terminates at a4 , and the measurement of the coefficients A0 , a2 , and a4 is possible from the observation of the coincidence intensity at three different angles, preferably at 90◦ , 180◦ , and 54.7◦ where P2 (cos θ ) has local extrema or a zero, respectively. Information on the absolute size of transition matrix elements cannot be obtained with this method for very short-lived nuclear states with lifetimes well below 1 ps which is typical for MSSs. 3.5. Inelastic neutron scattering Inelastic neutron scattering (INS) refers to the excitation of a nucleus in an (n, n 0 γ ) reaction. Quasi-monochromatic neutron beams can be produced, e.g., at the 7 MV electrostatic Van De Graaff accelerator at the University of Kentucky, by the 3 H( p, n)3 He reaction induced by a pulsed mono-energetic proton beam bombarding a tritium gas cell at a pressure of nearly 1 atm [100]. The kinetic energy of the neutrons can be controlled by adjusting the incident proton energies. Excited states of the secondary target nuclei can be populated up to the energy of the neutron beam and their decays can be studied by γ -ray spectroscopy [101]. The bottom part of Fig. 7 displays a γ -ray spectrum from a 36.6 g sample of metallic molybdenum powder enriched in the isotope 94 Mo to 91.6% [77]. The population of excited states can be controlled by selecting the incident neutron energy. Measurements of excitation functions, i.e., variations of the yield of a given γ -ray as a function of the incident neutron energy, allow the placement of the γ -rays in the level scheme of the nucleus under investigation. Neutron scattering reactions populate low-spin states below the beam energy non-selectively in a certain spin window and thus offer a tool for obtaining comprehensive spectroscopic information. A state of the target nucleus excited in the neutron scattering reaction acquires an alignment with respect to the beam axis. Angular distributions of γ -ray intensities show an anisotropy due to that alignment and can be analyzed according to the formalism of directional correlations of oriented states. Such an analysis yields information on γ -ray multipolarities and spin quantum numbers of excited states. √ The impinging neutron also transfers a linear momentum of magnitude p = 2m n E n to the excited nucleus which typically corresponds to recoil velocities of almost 10−3 of the speed of light. Subsequent γ -ray decays of short-lived states, that emit the γ -ray before the recoiling nucleus has been stopped to rest, are Doppler shifted as a function of the observation angle with respect to the beam axis according to   v E γ (θ ) = E γ0 1 + F(τ ) cos θ (49) c where F(τ ) is the Doppler shift attenuation factor. Precision measurements of F(τ ) yield information on the lifetimes of short-lived states through a comparison to calculated Fcalc. (τ ) values using a stopping theory. This technique of lifetime measurements using the Doppler Shift Attenuation Method following Inelastic Neutron Scattering (DSAM-INS) has been developed

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and extensively used by S.W. Yates and collaborators at the University of Kentucky. A more comprehensive review of this method can be found in Ref. [102]. The γ -ray spectroscopy after Inelastic Neutron Scattering (INS) is ideally suited for studying low-spin MSSs in stable vibrational nuclei [36,37,103,77,104]. However, similarly to the NRF technique it requires the availability of considerable amounts (e.g., 10 g) of isotopically enriched target material. 3.6. Light-ion fusion Light-ion fusion evaporation reactions populate low-spin states non-selectively in a certain spin and energy window and thus allow comprehensive in-beam γ -ray spectroscopy in the sensitive energy and spin region. Fig. 8 shows a comparison of γ -ray coincidences from the + 94 + 3+ 2 state of Mo populated in β -decay and (α, n) reactions. The 32 state has been identified as 94 the 3+ 1,ms state of Mo [75]. Gamma-ray spectroscopy in light-ion fusion evaporation reactions can provide all the observables that are necessary for the identification of MSSs. Gamma-ray multipole mixing ratios and spin quantum numbers can be obtained from the measurement of angular distributions of in-beam oriented states or from γ γ -coincidence angular correlation from in-beam oriented states [96]. For coincident γ -radiation from an initially aligned source in the state Ji to the intermediate state J and subsequently to the final state J f , the

Fig. 8. Relevant parts of the γ γ -coincidence spectra from 94 Mo populated in the β-decay of 94 Tcm (top: (a)–(c)) and + in the 91 Zr(α, n) reaction (bottom: (d)–(f)) gated with the 702.6 keV 4+ 1 → 21 transition ((a), (d)), with the 993.2 keV + + + 2+ → 2 transition ((b), (e)), and with the 1196.7 keV 2 → 2 transition ((c), (f)). The arrows mark the decays of 2 1 3 1 94 Mo at an excitation energy of 2965.4(2) keV. This information helped in the experimental discovery of the 3+ state of 2 the two-phonon 3+ 1,ms state. From Ref. [75].

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γ γ -coincidence angular correlation function can be expanded in terms of products of spherical harmonics Yλµ (Ω ) as a function of the polar angles θ1 , θ2 of the observational directions with respect to the alignment axis (beam axis for in-beam aligned sources) and of the polar angle φ between the reaction planes spanned by the alignment axis and the momentum vectors of the two coincident γ -rays [96] X Aλ1 (Ji )Aλλ2 λ1 (γ1 )Aλ2 (γ2 )Hλ1 λλ2 (θ1 θ2 φ) (50) W (θ1 , θ2 , φ) = λ1 λλ2

where Hλ1 λλ2 (θ1 θ2 φ) =

λ0 =min(λ,λ X 2) q=−λ0

4π hλ1 0λq|λ2 qiYλq (θ1 , 0)Yλ∗2 q (θ2 , φ). 2λ2 + 1

(51)

The directional distribution coefficients Aλλ2 λ1 (γ1 [Ji → J ]) for the first transition Ji → J and Aλ2 (γ2 [J → J f ]) for the subsequent transition J → J f are given by Aλλ2 λ1 (γ1 ) =

i 1 h λ2 λ1 λ2 λ1 0 2 λ2 λ1 0 0 F (L L J J ) + 2δ F (L L J J ) + δ F (L L J J ) (52) i i i 1 1 1 1 1 1 1 1 λ λ λ 1 + δ12

and Aλ (γ2 ) =

i 1 h 0 2 0 0 F (L L J J ) + 2δ F (L L J J ) + δ F (L L J J ) λ f λ f λ f 2 2 2 2 2 2 2 2 1 + δ22

(53)

if one considers two mixed multipoles L i and L i0 , only. The multipole mixing ratios δi are given according to Eq. (48). The generalized F-coefficients are tabulated in Ref. [96]. The orientation parameters Aλ1 (Ji ) follow from the relative population of magnetic sub-states P(m) according to X p Aλ1 (Ji ) = 2Ji + 1 (−1) Ji +m hJi − m Ji m|λ0iP(m) (54) m

94 Mo at an excitation energy of Fig. 9. Measurement of the spin quantum number J = 3 for the 3+ 1,ms state of 2965.4 keV from the γ γ angular correlation analysis of an (α, nγ ) reaction. The solid squares show the measured + + π relative intensities of the 703 keV–1391 keV γ γ -coincidence from the J2965 keV → 41 → 21 γ γ -cascade for the five geometrically independent coincidence groups of the spectrometer used in that work. The coincidence groups are labeled with three integers, which denote in units of π/4 the two angles between the observational directions and the beam and between the planes defined by the beam and by the γ -directions. The lines represent the least-squares fits for the spin + π hypotheses J = 3 and J = 2. The Gaussian width σ of the m-sub-state distribution and the J2965 keV → 41 multipole mixing ratio δ were treated as free parameters. From Ref. [75].

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where P(m) must be computed from the reaction theory. For in-beam orientation from fusion evaporation reactions the relative sub-state population is well approximated by a discrete Gaussian distribution e−m /2σ P(m) = +J Pi −m 2 /2σ 2 e 2

2

(55)

m=−Ji

with width σ [105]. For γ γ -coincidences from an unoriented source with σ = ∞ and P(m) = const., Eq. (50) reduces to Eqs. (43)–(47). In a typical in-beam angular correlation measurement γ γ -coincidences are taken for more than five angle combinations (θ1 θ2 φ). If the second half of the cascade is known, e.g., (J → J f ) = (4+ → 2+ ), one discrete (Ji ) and two continuous unknowns (σ, δ1 ) are fitted to the data points for a best match. Fig. 9 shows the measurement of the spin quantum number J = 3 for 94 the 3+ 1,ms state of Mo from the analysis of in-beam γ γ angular correlation measured with an (α, n) light-ion fusion evaporation reaction. Partial level lifetime information can be obtained for short-lived states from the observation of Doppler shifts of γ -transitions. The Doppler shifts are, however, small due to the low recoil velocity induced by the impinging light ion. The small Doppler shifts and the problem of unknown side-feeding times from the highly excited entry states limit the precision for lifetime measurements with this method for short-lived states such as MSSs. Fig. 10 shows 94 Mo that were observed in Doppler shifts for the γ -decays of the short-lived 3+ 1,ms state of the 91 Zr(α, n) 94 Mo reaction [75]. From this measurement a lifetime of τ (3+ 2 ) = 80(30) fs + was deduced [75] in agreement with the more accurate value τ (32 ) = 79(8) fs measured later [77] using the DSAM-INS technique. Beside the two-phonon 3+ 1,ms state the two-phonon

+ Fig. 10. Doppler shift of the 1101 keV 3+ 1,ms → 22 transition observed in coincidence with the (unshifted) 993 keV + + ◦ ◦ 22 → 21 transition at the angles of 45 and 135 compared to the unshifted transition observed at 90◦ . The upper spectra are vertically displaced by 1800 counts (135◦ ) and 2800 counts (45◦ ) for better visualization. The dashed line marks the unshifted position of the peak of interest. The curves show line shapes calculated using the observed 5% longlived discrete feeding, 95% side-feeding with a calibrated side-feeding time τSF = 80 fs, and a level lifetime τ (3+ ) = 80 fs. From Ref. [75].

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2+ 2,ms state was initially discovered [76] from in-beam γ -ray spectroscopy of light-ion-induced fusion evaporation reactions. 3.7. Other techniques Techniques in addition to those described in more detail above have occasionally been employed for assignment or characterization of MSSs of vibrational nuclei. Analysis of Gamma-Ray Induced Doppler broadening (GRID) [106,107] has been used [31] for lifetime measurements of low-spin states of 54 Cr. The experiments made use of the ultrahighresolution double-crystal spectrometer GAMS4 installed at the ILL Grenoble high-neutron-flux reactor. Low-spin states of 54 Cr were populated by strong primary γ -ray transitions following 53 Cr(n, γ ) capture reactions. The initial recoil energy due to the primary transition amounted to 0.5 keV and thus to a maximum recoil velocity of the order of 10−4 c. Measurements of the Doppler profiles for subsequent γ -radiation with typical resolution in the eV range have been shown to be sensitive to the short lifetimes of the issuing energy levels via simulations of the stopping process. Hadron scattering has occasionally been used [25–27] for identifying the one-phonon 2+ 1,ms state of heavy vibrational nuclei. While this technique is not directly sensitive to the identifying + 2+ 1,ms → 21 M1 transition strength, the isoscalar and isovector parts of the quadrupole transition + + matrix elements for the direct 0+ 1 → 2 excitation of low-lying 2 states are derived from the inelastic scattering of two probes with different isospin. Data from ( p, p 0 ) proton scattering reactions have been compared to data from (d, d 0 ) deuteron scattering reactions in pioneering experiments by Pignanelli et al. [25] and De Leo et al. [26,27]. The data analysis makes use of an appropriate choice of parameters of the optical model. Coupled-channel calculations serve for evaluating the ratio of direct to multistep contributions to the L = 2 excitations. Assuming, as is usually done at the beam energies used in the literature [25], an effective p–n interaction three times larger than the p– p interaction, one obtains that the ( p, p 0 ) and (d, d 0 ) cross sections are proportional to the square of M( p, p 0 ) = 1/4 M p +3/4 Mn and of M(d, d 0 ) = 1/2 M p +1/2 Mn , respectively. These proton and neutron matrix elements might then be combined [25] to form an isoscalar matrix element M S = 1/2 (M p + Mn ) = M(d, d 0 ) and an isovector matrix element MV = 1/2 (M p − Mn ) = 2[M(d, d 0 ) − M( p, p 0 )]. Alternatively, a more sophisticated scheme for the relation between the interaction strength M(x, x 0 ) of probes x and the individual matrix elements for protons and neutrons might be used [27]. An enhanced value of the ratio MV /M S is taken [26,27] as a signature of a mixed-symmetry state. In particular the recent advances in performing high-resolution inelastic hadron scattering experiments at medium energies in the range of 100 MeV per nucleon where direct reactions clearly dominate over multistep excitations, e.g., at RCNP, Osaka, Japan, or at iThembaLabs, Somerset West, South Africa, make this technique a very promising tool for the study of MSSs in the years to come. 4. Data on vibrational nuclear excitations with mixed symmetry This subsection summarizes the data for vibrational nuclear excitations with predominant one-phonon or two-phonon MS structure assigned on the basis of their large absolute M1 decay matrix elements. These data have mostly been obtained by at least one of the experimental techniques presented in the preceding subsection. We limit our overview to those cases shown in + Table 3 for which at least a large lower limit for the B(M1; 2+ 1,ms → 21 ) value which allows a MS assignment has been reported in the literature up to June 2006.

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Table 3 Overview over vibrational nuclei for which at least the one-phonon 2+ 1,ms MSS has been reported in the literature up to June 2006 based on measurement of large M1 decay strength Nucleus

E(2+ 1 ) (keV)

E(2+ 1,ms ) (keV)

56 Cr

96 Ru

1007 847 1039 934 871 778 833

112 Cd

618

114 Cd

558

126 Xe

389

128 Xe

443

134 Ba

605

136 Ba

819

138 Ce

789

142 Ce

641

144 Nd

697

148 Sm

747

2327 2658 2938 1847 2067 2096 2284 2156 2231 2219  2064 (2359) 2127 2029 2088 2129 2143 2237 2005 2365 2073 2369 2146

56 Fe 66 Zn 92 Zr 94 Mo 96 Mo

+ 2 B(M1; 2+ 1,ms → 21 ) µ N

>0.22 0.30(2) 0.21(2) 0.37(4) 0.56(5) 0.18 0.78(23)

References [31] [82,33] [108] [109,110] [74,77] [112] [93,111]

0.099(7)

[37]

0.089(9)

[104]

(>0.09)

[113]

0.07(2)

[38]

0.20(4)

[21,35]

0.26(3)

[85]

0.18(2)

[94]

0.59(5)

[36]

0.32(5)

[103]

>0.12

[114]

The brackets indicate the energy of the two dominant fragments.

The most significant amount of new comprehensive information has been obtained for the N = 52 isotone 94 Mo. We will begin with a detailed discussion of 94 Mo as an example. 4.1.

94 Mo

as an example

The low-spin level scheme of the near-spherical nucleus 94 Mo has recently been extensively investigated with γ -ray spectroscopy using photon scattering reactions [74,77], (α, nγ ) light-ioninduced fusion evaporation reactions [75–77], inelastic neutron scattering reactions [77], and βdecay reactions [74,77] as populating mechanisms. Photon scattering 94 Mo(γ , γ 0 ) experiments were performed [74] at the bremsstrahlung facility [15] at the Stuttgart DYNAMITRON accelerator of the Institut f¨ur Strahlenphysik, Universit¨at Stuttgart. Continuous-energy photon beams with maximum energies of 3.3 and 4.0 MeV were used for the measurement of the cross sections of the 94 Mo(γ , γ 0 )94 Mo∗ reaction. Clean off-beam γ γ -coincidence spectroscopy of γ transitions between excited states of 94 Mo populated in the β-decay of the (2+ ) low-spin isomer 94 Tcm was performed [74,77] with the Cologne OSIRIS cube spectrometer at the FN-TANDEM accelerator of the Institut f¨ur Kernphysik, Universit¨at zu K¨oln. The 94 Tcm nuclei were produced with the low-angular-momentum-generating 94 Mo( p, n) fusion reaction. The target was located in the center of the spectrometer; the measurements were carried out in a beam-pulsing mode. In-beam γ γ -coincidence spectroscopy experiments on 94 Mo were performed in Cologne with the OSIRIS cube spectrometer using the “cold” 91 Zr(α, n) 94 Mo fusion evaporation reaction at beam energies close to the Coulomb barrier in order to populate low-spin states with high cross

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Table 4 Comparison of analytical IBM-2 predictions for the U (5) and the O(6) symmetry with the core 100 Sn for allowed M1 strengths of low-lying MSSs with experimental data on 94 Mo Observable

U (5)

O(6)

Expt.

+ B(M1; 1+ ms → 01 ) + B(M1; 1ms → 2+ 2) + B(M1; 2+ 1,ms → 21 ) + B(M1; 2+ → 2 2,ms 2) + → 4 B(M1; 3+ ms 1) + B(M1; 3+ ms → 22 )

0 0.33 0.23 0.09 0.12 0.16

0.16 0.36 0.30 0.10 0.13 0.18

+0.011 0.160−0.010 0.44(3) 0.56(5) 0.27(3) 0.075(10) 0.24(3)

Ideal orbital values, gπ = 1 µ N and gν = 0 µ N , are used for the boson g-factors. All M1 strengths are given in µ2N . From Ref. [77].

section [75–77]. The 94 Mo(n, n 0 γ ) measurements were done at the neutron facility of the 7 MV electrostatic accelerator of the University of Kentucky. In total, a comprehensive level scheme of 57 low-spin states below an excitation energy of 4 MeV and 120 γ -ray transitions between them was established and lifetime information for 50 levels was obtained [77, Table I]. For 73 γ -ray transitions it has been possible to measure the multipole contributions which, together with the level lifetimes, lead to numerous absolute E1, M1, and E2 transition rates. Table 4 compares the data on the expected allowed M1 decays to the IBM-2 predictions. Lifetimes and decay transition rates for the eight lowest-energy 2+ states are available. The top part of Fig. 11 shows the observed distribution of B(M1; 2+ → 2+ 1 ) values. Five of them + are considerably smaller than 0.1 µ2N . Only the 2+ → 2 transition exhibits a large reduced M1 3 1

+ 94 Fig. 11. Identification of the 2+ 3 state of Mo as the 21,ms state from absolute E2 and M1 transition strengths. From Ref. [77].

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+ 2 transition strength of B(M1; 2+ 3 → 21 ) = 0.56(5) µ N corresponding to a reduced M1 matrix + + element of |h21 k M1 k 23 i| = 1.67(8) µ N . The bottom part of Fig. 11 displays the strength distribution for the E2 excitation from the + ground state on a logarithmic scale. This distribution is dominated by the B(E2; 0+ 1 → 21 ) value of 0.196(3) e2 b2 [115] which corresponds to a decay strength of 15.44(24) W.u. All of the other 2+ states, beside the 2+ 3 , decay to the ground state with E2 strengths that are about two orders + of magnitude smaller. Only the 2+ 3 → 01 transition exhibits a comparatively large reduced E2 + + transition strength of B(E2; 23 → 01 ) = 2.2(2) W.u. which corresponds to 14% of the E2 decay strength of the 2+ 1 state and it can be considered a weakly collective transition. Comparing these observations to the list of signatures for the one-phonon 2+ 1,ms state given at + 94 the end of Section 2 we find that the 23 state of Mo at an excitation energy of 2067.4(1) keV matches those signatures very closely. On the basis of this unique combination of absolute M1 + 94 and E2 transition rates, the 2+ 3 state of Mo is firmly assigned as the 21,ms state of that nucleus. However, the information displayed in Fig. 11 is even more comprehensive. From the strength distributions we see that there does not exist another 2+ state of 94 Mo below 3.5 MeV that would + 94 show similar properties. Consequently, the 2+ 1,ms assignment to the 23 state of Mo is unique. This situation differs from that for the scissors mode in other heavy deformed nuclei where the total M1 excitation strength is typically fragmented over several states. Fig. 11 suggests that at + most a small part of the MS character might be shared between the 2+ 3 and the 24 states due to + + + B(M1; 24 → 21 ) = 0.118(16) µ2N which is the second largest 2+ → 21 M1 transition strength + + + for this nucleus. The measured ratio B(M1; 2+ 4 → 21 )/B(M1; 23 → 21 ) = 0.21(3) indicates + that not much more than 20% of the 21,ms wave function might be spread to neighboring states. The significantly reduced fragmentation of the 2+ 1,ms state with respect to what has been found + for the 1 scissors mode in deformed nuclei is presumably due to the low excitation of about 2 MeV and to the low mass number A = 94. + From this identification of the 2+ 3 state as the 21,ms state one should expect from a simple harmonic phonon coupling scheme the existence of two-phonon MSSs close to the sum energies + of the constituent one-phonon states, i.e., approximately at excitation energies E x ([Q s Q m ] J ) ≈ + + E x (21,ms ) + E x (21 ) = 2.94 MeV.

Fig. 12 shows that there do indeed exist J π = 1+ , 2+ , and 3+ states that decay with a large + M1 transition strength to the 2+ 2 state as is expected for two-phonon MS states. The 12 state at + 3129 keV, the 2+ 5 state at 2870 keV, and the 32 state at 2965 keV dominate the distribution of + π B(M1; 22 → J ) values. All three states have very similar transition strengths of B(M1; 2+ 2 → J π ) ≈ 0.3 µ2N , each corresponding to reduced M1 matrix elements of about 1 µ N . These large M1 matrix elements suggest MS assignments for these three states. A partial level scheme for low-spin states of 94 Mo is displayed in Fig. 13. It emphasizes the assignment of one-phonon and two-phonon MSSs based on large M1 transition matrix elements. In addition, the measured matrix elements for M1 transitions from the two-phonon MSSs to the 2+ 1 state are one order of magnitude smaller than the d-parity allowed transitions shown in Fig. 13 [77]. This observation agrees with the signatures for MSSs from the end of Section 2, too. Moreover, the E2 transition + + + rates for these decays, B(E2, 1+ 2 → 21 ) = 0.7(3) W.u., B(E2, 25 → 21 ) = 0.4(2) W.u., + + B(E2, 32 → 21 ) = 0.9(5) W.u., are close to 1 W.u., and agree within a factor of 2–5 with the decay strength of the one-phonon 2+ 1,ms state to the ground state with a value of + B(E2, 2+ → 0 ) = 2.2(2) W.u. 3 1

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Fig. 12. Identification of two-phonon MS states of 94 Mo from the M1 strength distribution from the FS 2+ 2 two-phonon state. From Ref. [77].

Fig. 13. Partial low-spin level scheme of 94 Mo with FS and MS one-phonon and two-phonon states. The numbers at the level bars denote their excitation energies in keV. The arrows indicate transitions with large M1 matrix elements, |hJ f k M1 k Ji i|. Experimental values were taken from Refs. [74–77] and are shown in units of µ N . From Ref. [116].

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Fig. 14. First observation of a MS → MSγ -ray transition and thus direct evidence for MS states forming a separate + class of nuclear states [10]. A coincidence spectrum gated with the 2+ 3 → 21 transition is shown on the right which + + contains the 12 → 23 transition at 1061.1(5) keV. From off-beam γ γ -coincidence spectroscopy of γ -transitions in 94 Mo following β + -decay of 94 Tcm . Updated from Ref. [74]. + + A fourth supporting fact for the two-phonon MS assignment of the 1+ 2 , 25 , and 32 states of should come from the measurement of the expected collective E2 transition to the 2+ 3 one-phonon MS state. These transitions have indeed been observed recently [74] despite the fact that they have a small intensity due to their comparatively low transition energy and due to the competition with the fast M1 decay transitions with higher transition energy. Fig. 14 shows + 94 evidence for the 1+ 2 → 23 transition in Mo which represents the first observation of a twophonon to one-phonon transition between MSSs and thereby first supported directly [74] the view of MSSs as forming a whole class of collective nuclear states [10]. Decay transitions to the 94 one-phonon 2+ 1,ms state were subsequently observed for the other two-phonon MSSs of Mo as + + well. Parts (c) and (f) of Fig. 8 show the 3ms → 21,ms transition observed in γ -ray coincidence spectroscopy using two different population mechanisms. Unfortunately, the comparatively low intensities of these MS→MS 1J = 1 transitions did not then permit a measurement of their E2/M1 multipole mixing ratios with sufficient precision. This prevented up to now the accurate determination of the E2 strength for the MS→MS transitions although their transition rates are 94 Mo

Fig. 15. Experimental excitation energies of MSSs of 94 Mo compared to a schematic IBM-2 calculation. From Ref. [117].

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Table 5 + + + Comparison of measured M1 and E2 transition strengths of the 2+ 3 , 25 , 32 , and 12 states to predictions in O(6) + + + symmetry of the IBM-2 (third column) for the 2+ , 2 , 3 , and 1 states ms 1,ms 2,ms ms Observable B(M1; 2+ 3 B(M1; 1+ 2 B(M1; 1+ 2 B(M1; 1+ 2 B(M1; 1+ 2 B(M1; 2+ 5 B(M1; 2+ 5 B(M1; 2+ 5 B(M1; 3+ 2 B(M1; 3+ 2 B(M1; 3+ 2 B(M1; 3+ 2

Expt. → → → → → → → → → → → →

2+ 1) 0+ 1) + 21 ) 2+ 2) 2+ 3) 2+ 1) 2+ 2) 2+ 3) 2+ 1) 4+ 1) 2+ 2) 2+ 3)

IBM-2

SM

QPM

0.56(5)

0.30

0.51

0.20

+0.011 0.160−0.010

0.16

0.26

0.08

0.012(3)

0

0.002

0.003

0.44(3)

0.36

0.46

0.42

<0.05

0

0.08

0.003

0.0017+0.0010 −0.0012

0

0.004

0.008

0.27(3)

0.100

0.17

0.56

<0.16

0

0.06

+0.003 0.006−0.004

0

0.10

0.004

0.075(10)

0.13

0.058

0.12

0.24(3)

0.18

0.09

0.17

+0.016 0.021−0.012

0

0.001

0.006

0.09(2) B(E2; 2+ 1 B(E2; 2+ 3 B(E2; 2+ 5 B(E2; 2+ 3 B(E2; 1+ 2 B(E2; 1+ 2 B(E2; 1+ 2 B(E2; 2+ 5 B(E2; 2+ 5 B(E2; 2+ 5 B(E2; 3+ 2 B(E2; 3+ 2 B(E2; 3+ 2 B(E2; 3+ 2

→ → → → → → → → → → → → → →

0+ 1) 0+ 1) 0+ 1) 2+ 1) 2+ 1) 2+ 2) 2+ 3) 2+ 1) 2+ 2) 2+ 3) + 21 ) 4+ 1) 2+ 2) 2+ 3)

16.0(2)

18.4

2.2(2)

1.2

0.14(2)

0

0.29

0.34

1.0+0.6 −0.5

0

0.001

0.039

0.72+0.27 −0.24

1.9

0.51

2.99

0.99+0.92 −0.62

0

0.055

0.35

<27

21.9

16.5 1.65

8.98

0.40+0.19 −0.14

0.626

1.8

0.4+0.9 −0.4

0

1.2

16.9

5.5

<140 +0.5 0.9−0.4 0.14+0.30 −0.13 0.9+1.6 −0.8

58(14)

16.3 0.79

29.1 3.31 22.5

1.90

1.70

2.76

0

0.91

0.007

0

6.70

14.6

7.81

0.35 25.4

+12.5 9.5−8.3

Many transition strengths are reproduced on an absolute scale using only a single free parameter, the proton boson effective charge eπ = 9 e fm2 . M1 strengths are given in µ2N and E2 strengths in W.u. In addition, columns 4 and 5 give the results of microscopic calculations for these states in the spherical shell model (SM) (from [118]) and in the QPM (from [119]). For E2 multipolarity the factor for conversion between Weisskopf units for mass number A = 94 and the unit e2 fm4 is given by 1 W.u. = 25.39 e2 fm4 . From Ref. [77].

fairly well known and agree with the expected collective E2 transitions. At least for the 1+ ms → 2+ 1,ms transition it could be established that the E2 transition strength must be between 3 and 27 W.u. A schematic description of excitation energies of MSSs by the IBM-2 is given in Fig. 15. The MS structure in 94 Mo has been studied theoretically in various microscopic approaches [118–120]. Table 5 compares relevant data on M1 and E2 strengths to the IBM, the Shell Model (SM) and the Quasiparticle Phonon Model (QPM).

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4.2. Other N = 52 isotones 4.2.1. 96 Ru The nucleus 96 Ru is the next heavier even–even N = 52 isotone after 94 Mo. Off-beam and in-beam γ γ -coincidence spectroscopy of γ -transitions between excited states of 96 Ru populated in the β-decay of the (3+ ) low-spin isomer 96 Rhm and with the 95 Mo(3 He, 2n)96 Ru reaction was carried out [111] with the Cologne OSIRIS cube spectrometer and in the 12 C(96 Ru, 96 Ru∗ ) Coulomb excitation reaction with the S PEEDY array at the Wright Nuclear Structure Laboratory at Yale University [93]. The Coulomb excitation experiment was performed in order to gain absolute information 96 Ru. on the electromagnetic transition matrix elements for the one-phonon 2+ 1,ms state of The performance of photon or neutron scattering experiments was not possible because of the unavailability of large quantities of isotopically enriched material. Due to the lack of enriched 96 Ru material, 96 Ru ions were extracted from the ion source of the ESTU-TANDEM accelerator at Yale which was loaded with ruthenium in natural composition. The ruthenium ion beam was mass-separated to obtain a pure beam of 96 Ru ions. The 96 Ru beam was shot on a 1 mg/cm2 thick carbon foil of natural composition. Single γ -ray events and γ γ -coincidences were recorded. The + + 96 2+ 1 state, 41 state, and 23 state of Ru were Coulomb excited with cross sections large enough for observation. Fig. 5 displays some of the data. Gamma-ray spectroscopy of low-spin states of 96 Ru was carried out after population by βdecay of the (3+ ) low-spin state 96 Rhm . The 96 Rhm nuclei were produced with the 96 Ru( p, n) reaction in a beam-pulsing mode. Branching ratios and γ multipolarities were determined [111]. The in-beam study of 96 Ru in the reaction 95 Mo(3 He, 2n)96 Ru was performed at a 3 He beam energy of 13.5 MeV. The complementary angular correlations of the γ γ -coincidences from the in-beam oriented states helped in some cases with deciding on multipole mixing ratios which were ambiguous from the β-decay. Doppler shifts were observed for γ -transitions depopulating states of 96 Ru within less than 1 ps which made measurements of effective lifetimes possible in eight cases [111]. In this way short-lived states in the level scheme of 96 Ru could be identified. Absolute M1 and E2 matrix elements for 96 Ru were obtained from the combination of Coulomb excitation cross sections measured in the 96 Ru + nat C reaction in inverse kinematics

Fig. 16. Comparison of the low-lying low-spin level schemes of the even–even N = 52 isotones 94 Mo and 96 Ru. The numbers denote relative γ intensities, absolute M1 matrix elements and B(E2) values. The information on the large M1 + and E2 transition strengths was used to identify the 2+ 3 states of both nuclei as their mixed-symmetry one-phonon 21,ms state. From Ref. [93].

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Fig. 17. Example of the identification of a two-phonon MS state of 96 Ru in comparison to the decay scheme of the 2+ 1,ms 94 one-phonon and the 3+ ms two-phonon MSSs of the isotonic neighbor Mo. From Ref. [111].

at Yale University [93] with information on decay branching ratios and multipole mixing ratios from γ -ray spectroscopy performed in Cologne. The combined data from Cologne and Yale 96 yield enough information for the unambiguous identification of the 2+ 1,ms state in Ru [93,111] and convincing candidates for being mixed-symmetry two-phonon states [111]. In particular, the + + large matrix element |h2+ 1 k M1 k 23 i| = 2.0(3) µ N was measured [93] and establishes the 23 + state of 96 Ru as the 21,ms state. The properties of the identified one-phonon MSSs in 96 Ru and 94 Mo are compared in Figs. 16 and 17. Excitation energies and E2 and M1 decay properties of the 2+ 1,ms states are very similar for the two N = 52 isotones. 96 Ru performed in inverse The identification of the 2+ 1,ms state in Coulomb excitation of kinematics is interesting from an experimental point of view because it initiated a program to study the one-phonon MSS in projectile Coulomb excitation (see Ref. [94] for first results) and it furthermore demonstrates that MSSs are experimentally accessible with a beam of nuclei of interest. This fact supports the prospect of identifying and investigating MSSs in unstable nuclei with future high-intensity exotic beam facilities. Higher-lying dipole excitations of 96 Ru have recently been investigated [121] by means of photon scattering using the Stuttgart bremsstrahlung beam with end point energy of 4 MeV. The dipole excitation strength distribution is dominated by a state at 3154 keV which is suggested to 96 be the two-phonon 1+ ms state of Ru [84,121].

4.2.2. 92 Zr The nucleus 92 Zr is the lightest of the three stable even–even N = 52 isotones. In contrast to the case for 94 Mo and 96 Ru, discussed above, where the valence protons occupy the π(1g9/2 ) orbital, zirconium isotopes lie at the Z = 40 proton sub-shell closure at the upper end of the p f -shell. This proton sub-shell closure leads to pronounced shell effects on the formation of pre-collective nuclear states and, in particular, to a breaking of F-spin symmetry of the IBM-2. It is interesting to investigate to what extent properties of MSSs can survive or are modified in the presence of a sub-shell closure because such information will allow for a test of the robustness of MS structures. We therefore stress here that one should be aware of pronounced non-collective shell effects at Z = 40. For simplicity we prefer using the term “phonon” in the context of excitations of 92 Zr although severe modifications of the collective picture are apparent. This preference is solely motivated by a wish to simplify the discussion of the evolution of the structures along the N = 52 isotonic chain.

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261

+ + Fig. 18. Comparison of the 2+ → 2+ 1 M1 strength distribution (left) and 01 → 2 E2 strength distribution (right) in 94 Mo (top) and 92 Zr (bottom). From Ref. [110].

In order to study the evolution of MSSs in N = 52 isotones and to search for reminiscences of MSSs at the Z = 40 proton sub-shell closure the low-spin level scheme of 92 Zr was very recently investigated in extensive experiments with photon scattering [109,123] and in neutron scattering [110]. Comprehensive data on low-spin states of 92 Zr comparable to those on 94 Mo were obtained. Fig. 18 shows a comparison of the relevant data for the identification of the 94 92 one-phonon 2+ 1,ms state in Mo to the data obtained on Zr. The sub-shell closure affects the spectrum of 2+ states in several ways. The 2+ state which carries the largest part of the 2+ → 2+ 1 M1 strength lies in 92 Zr at even lower excitation energy than in 94 Mo, however with a smaller total B(M1; 2+ → 2+ 1 ) value. Both features might be understood [109,110,129,130], at least qualitatively, from a reduced proton–neutron interaction at the Z = 40 sub-shell closure which results in a different distribution of proton and neutron contributions to the predominantly onephonon 2+ states in 92 Zr. The lower-lying state has a more pronounced neutron character and the higher-lying state has a predominant proton contribution. The reduced proton–neutron interaction at the sub-shell closure results in a smaller repulsion between these proton and neutron onephonon configurations and thus in a smaller energy separation between the resulting one-phonon + states. In 92 Zr the 2+ state with the largest B(M1; 2+ → 2+ 1 ) value happens to be the 22 state. + 94 + Like for Mo it is exactly this state (which carries the largest B(M1; 2 → 21 ) strength) that decays to the ground state by the most collective E2 transition above the 2+ 1 state. In a sense, the symmetrical two-quadrupole phonon state and the one-phonon MSS cross when going from 94 Mo to 92 Zr; however this crossing is accompanied by a breaking of the F-spin symmetry and a decrease in collectivity of the 2+ 1 state. This qualitative picture is supported by the decrease of the total M1 strength between the one-quadrupole phonon states with predominant FSS or MSS character and by the increase of the strength of the E2 ground state excitation to the one-quadrupole phonon state with predominant MSS character in the N = 52 isotonic chain from 96 Ru to 92 Zr. The properties of these predominantly one-phonon states are summarized in Fig. 19. The quantitative amount of F-spin breaking in these predominantly one-quadrupole phonon states of 92 Zr is subject to much current discussion [109,110,129,130]. Table 6 shows the data on the most relevant observables for the assigned one-phonon MSSs in N = 52 isotones.

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Fig. 19. Systematics of some properties of the fundamental one-phonon 2+ states in N = 52 isotones as a function of the nuclear charge number. Part (a) shows the excitation energies, part (b) displays the B(E2; 2+ → 0+ 1 ) value in Weisskopf units (1 W.u. = 0.06 A4/3 e2 fm4 ) on a logarithmic scale and part (c) gives the strength of the M1 transition between these 2+ states in µ2N . From Ref. [122]. + 92 Note that we use here for brevity the label 2+ 1,ms state also for the 22 state of Zr although this state is not considered to be a pure MSS according to the previous discussion. Like for 94 Mo there exists information on J π = 1+ states in 92 Zr with properties close to those expected for the two-phonon 1+ MSS. Fig. 20 shows the comparison between the M1 excitation strength distributions in 94 Mo and 92 Zr. These strength distributions are quite similar. There is however an overall decrease of strength by about 20% and a slight increase of excitation energy by about 10% from 94 Mo to 92 Zr. Note that the parity quantum numbers of the 1+ states have been measured explicitly [123] in order to avoid a misinterpretation due to the presence of the quadrupole–octupole coupled 1− state that is present in both of these nuclei at similar excitation energies. Fig. 21 shows two examples of data on parity quantum numbers of dipole excitations of 92 Zr. These NRF spectra were obtained at the High Intensity γ -ray Source (HIγ S) at the Duke Free Electron Laser Laboratory [124,125]. The quasi-monochromatic completely polarized γ -ray beam has been tuned to the energies of the dipole excitations of interest. NRF intensity at a polar scattering angle of 90◦ is either observed in the polarization plane of the beam for the case of M1 radiation or perpendicular to the polarization plane for E1 radiation [126–128]. The radiation characters of the dipole excitations of 92 Zr studied at excitation energies of 3472 keV (1+ ) and 3638 keV (1− )

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Fig. 20. Comparison of the measured M1 excitation strength distributions in 94 Mo and 92 Zr. From Ref. [123].

Table 6 Some key observables for the predominant one-phonon 2+ 1,ms states of N = 52 isotones Observable

Unit

96 Ru

94 Mo

92 Zr

E(2+ 1)

keV

832.6(1)

871.09(10)

934.46(10)

τ (2+ 1)

ps

4.32(13)

4.15(6)

7.21(58)

+ B(E2; 2+ 1 → 01 ) + E(21,ms )

W.u.

18.1(5)

16.0(2)

6.4(6)

keV

2283.8(2)

2067.4(1)

1847.3(1)

+ Irel (2+ 1,ms → 21 ) + Irel (2+ 1,ms → 01 )

%

100(3)

100.0(7)

100.0(23)

%

7.5(10)

15.1(7)

44.6(23)

τ (2+ 1,ms ) + B(E2; 2+ 1,ms → 01 ) + + δ(21,ms → 21 ) + B(M1; 2+ 1,ms → 21 )

fs

22(7)

50.8(43)

138(14)

W.u.

1.6(3)

2.2(2)

3.4(4)

0.03(10)

0.15(4)

−0.04(2)

0.78(23)

0.56(5)

0.37(4)

µ2N

The data are taken from the most recent literature discussed in the text.

can immediately be concluded from the observation of NRF intensity in one of these directions. Assignment of the parity quantum numbers is obvious and unambiguous. The excitation energies of relevant quadrupole-collective states of the stable even-even N = 52 isotones 92 Zr, 94 Mo, and 96 Ru are summarized in Table 7.

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Fig. 21. Parity measurement of 1− and 1+ states of 92 Zr. The boxes show photon scattering spectra of 92 Zr taken at the High Intensity γ -ray Source (HIγ S) at the Duke Free Electron Laser Laboratory in the polarization plane of the incident polarized γ -ray beam (top row) and perpendicular to the polarization plane (bottom row). From Ref. [123].

Table 7 Excitation energy (in keV) of dominant fragments of basic symmetrical phonon modes and MSSs identified in stable even–even N = 52 isotones in previous literature 92 Zr

94 Mo

π Jτ,F

π Jexpt

π Jexpt

2+ 1,s

2+ 1

2+ 2,s 4+ 2,s 2+ 1,ms 1+ 2,ms 2+ 2,ms 3+ 2,ms

2+ 3

Ex

Ex

π Jexpt

Ex

934.5

2+ 1

871.1

2+ 1

832.6

2066.6

2+ 2 4+ 1 2+ 3 1+ 2 2+ 5 3+ 2

1864.3

2+ 2

1931.1

1573.7

4+ 1

1518.1

2067.4

2+ 3

2283.8

3128.6

11

3154.2

2870.0

2+ (5) 3+ (1)

2739.8

Inconclusive 2+ 2 1+ 2

96 Ru

1847.3 3472.1 Inconclusive Inconclusive

2965.3

(+)

2897.7

The left column specifies the state with O(5) quantum number τ = τ1 + τ2 in the F-spin limit with O(5) symmetry where the F-spin is indicated by “s” (“ms”) for F = Fmax (F = Fmax − 1). Due to strong F-spin breaking in 92 Zr considerable deviations between the observables and the simple F-spin limit occur in this nucleus (see text). The data are taken from the most recent literature discussed in the text.

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4.3. Other nuclei in the [28 < Z < 50, 50 < N < 82]-major shell 4.3.1. 96 Mo Low-spin states of 96 Mo have recently been studied using photon scattering [131,123,84] and using inelastic neutron scattering [112]. Yates et al. reported [112] the observation of a 2+ state at 2096 keV excitation energy that decays by a strong M1 transition with B(M1; 2+ 2096 keV → 2 . This reduced strength corresponds to an absolute M1 matrix element of 2+ ) = 0.18 µ N 1 + 0.95 µ N , the size expected for the 2+ 1,ms → 21 transition. Experimental uncertainties were not given. The decrease of M1 strength by about a factor of 3 from 94 Mo to its neighboring even–even isotope 96 Mo is not understood [112]. Photon scattering experiments at the Stuttgart bremsstrahlung facility yield information [131, 84] on the dipole excitation strength distributions up to 4 MeV. Five pronounced dipole excitations were observed between 2.8 and 3.9 MeV. Their parity quantum numbers have been measured at the HIγ S-facility at TUNL, Durham, NC, using intense, quasi-monochromatic, fully polarized γ -ray beams in the entrance channel for photon scattering experiments in the same way as was described above for the case of 92 Zr. Three 1+ states of 96 Mo P at excitation energies of + 2795, 3300, and 3425 keV [123] share a total M1 excitation strength B(M1; 0+ 1 → 1 ) = 2 0.63(8) µ N . The other two dipole excitations, observed at 3600 and 3895 keV, have negative parity. 4.3.2. 112 Cd 112 Cd at 1469 keV had been suggested [132] as the 2+ The 2+ 3 state of 1,ms one-phonon MSS + based on a small E2/M1 multipole mixing ratio for the 2+ → 2 transition. Later measurements 3 1 + 112 on Cd favored a much larger value of δ and showed that the 23 state of 112 Cd belonged to the intruder configuration [133,134]. Previously, Pignanelli et al. [25] had determined a large 112 Cd at 2231 keV based on ( p, p 0 ) and (d, d 0 ) scattering neutron amplitude for the 2+ 6 state of cross sections. This was not inconsistent with a mixed-symmetry assignment. Absolute B(M1) measurements were later contributed by Garrett et al. [37] from Doppler shift measurements in inelastic neutron scattering at Lexington. They observed almost pure + + + M1 radiation for the 2+ 5,6 → 21 transitions from the 25 state and from the 26 state of 112 Cd at 2156 and 2231 keV, with M1 strengths of B(M1; 2+ → 2+ ) = 0.044(5) µ2 and N 5 1 + 2 B(M1; 2+ 6 → 21 ) = 0.055(5) µ N , respectively. These M1 strengths are an order of magnitude larger than those for the transitions from lower-lying 2+ states to the 2+ 1 state. This experiment was considered the first observation of a MSS in a good U (5) nucleus [37]. Like the situation for vibrational nuclei in the A ≈ 90 mass region the 2+ 1,ms state of 112 Cd occurs at an excitation energy of a little bit above 2 MeV. However, it is observed to be fragmented into at least two states and carries a total M1 strength which is smaller than the values observed in the 94 Mo nucleus. Dipole excitations of stable cadmium isotopes have been systematically studied by Kohstall et al. [135] using nuclear resonance fluorescence. Candidates for 1+ ms two-phonon MSSs have been assigned from explicit measurements of parity quantum numbers from Compton polarimetry. The nucleus 112 Cd features enhanced dipole excitations at 2931 keV and at 3231 2 + keV with spin and parity assignments 1(+) and 1+ and B(M1; 0+ 1 → 1 ) values of 0.137(5) µ N and 0.068(3) µ2N , respectively. The total M1 excitation strength distribution appears also to be fragmented with a total strength that again is smaller than that found near mass A ≈ 90.

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4.3.3. 114 Cd 114 Cd were measured by Bandyopadhyay Absolute M1 strengths for 2+ → 2+ 1 transitions in et al. [104] again from angular distributions and Doppler shifts in inelastic neutron scattering at + Lexington. In this nucleus too, they found almost pure M1 radiation for the 2+ 5,6 → 21 transitions + + from the 25 and 26 states at 2048 and 2219 keV. The reduced transition strengths amount to + + + 2 2 B(M1; 2+ 5 → 21 ) = 0.017(5) µ N and B(M1; 26 → 21 ) = 0.089(9) µ N , respectively. + The total strength of the M1 transitions to the 21 state and its energetic center of gravity are close to the values observed previously for the neighboring even–even isotope 112 Cd discussed above. Again, the one-phonon 2+ 1,ms one-phonon state appears to be fragmented but here in such + a fashion that the 26 state at 2219 keV keeps the dominant part of the total M1 strength that once more is reduced with respect to the values observed for the lighter nuclei at N = 52. Information on dipole excitations of 114 Cd is available from the NRF work by Kohstall et al. [135]. Tentative 1(+) assignments have been made for a group of three states at 3110, 3214, and 3220 keV excitation energy from Compton polarimetry. If they do indeed have a spin and 2 + parity of 1+ , they carry M1 excitation strengths of B(M1; 0+ 1 → 1 ) = 0.148(12) µ N , 0.019(2) 2 2 µ N , and 0.098(6) µ N , respectively. These states have been considered as candidates for being 112 Cd, the total M1 excitation fragments of the 1+ ms two-phonon MSS. Just like in the case of 114 strength in Cd appears to be fragmented with a similar total strength to the next lighter even–even isotope. The total low-energy (<4 MeV) M1 excitation strength in stable even–even cadmium isotopes is found [135] to range between 0.05 µ2N and 0.2 µ2N . It is considerably smaller than in the N = 52 isotones and close to the value of zero expected for ideal vibrators in the U (5) dynamical symmetry limit of the IBM-2. 4.4. Nuclei in the [50 < Z < 82, 50 < N < 82]-major shell 4.4.1. 128 Xe The nucleus 128 Xe has been studied in (α, nγ ) reactions by Wiedenh¨over et al. [38]. Gamma–gamma coincidences and their angular correlations yielded evidences for excited 2+ states and for the E2/M1 multipole mixing ratios of their decay γ -ray transitions to the 2+ 1 state. + + The 24 state at 2128 keV decays by a pure M1 transition to the first 21 state [38]. Moreover, this decay transition is Doppler shifted when observed under varying observation angles with respect to the beam axis, indicating the short lifetime of the level which must be shorter than the stopping time of the recoiling nuclei in the 125 Te target. This stopping time is of the order of a picosecond. Doppler shift analysis [38] resulted in an effective lifetime of τ (2+ ) = 170(70) fs for this level in this reaction which represents an upper limit for its true lifetime. The M1 decay 2 transition strength must, therefore, have a lower limit of B(M1, 2+ → 2+ 1 ) = 0.07(2) µ N . + + 128 This observation suggests this 24 state of Xe as its 21,ms state. The measurement of a similar 94 effective lifetime of about 200 fs for the well-identified 2+ 1,ms state of Mo with a true lifetime + 91 τ (23 ) = 51 fs (see above) in the Zr(α, nγ ) reaction also suggests a considerably smaller value 128 Xe than 170 fs. Consequently, the strength of the M1 for the true lifetime of the 2+ 4 state of + decay transition to the 21 state might be expected to be up to about three to four times larger than the present lower limit of 0.07 µ2N . We consider this state to be an excellent candidate for 128 Xe. being the 2+ 1,ms state of Two-phonon MSSs in the stable even–even isotopes of the xenon chain were recently reported from NRF experiments [136]. A strong dipole excitation at 2838 keV excitation

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267

energy was observed that dominates the dipole strength distribution up to 4 MeV. It features a + pronounced decay to the 2+ 2 state in accordance with the expectations for a two-phonon 1ms state. Unfortunately, a direct parity measurement is lacking. Assuming a positive parity of this J = 1 2 + state, the corresponding B(M1, 0+ 1 → 1 ) value of 0.819(14) µ N , deduced from the photon scattering cross section, would agree with predictions from the IBM-2 in the O(6) dynamical symmetry. 4.4.2. 126 Xe A situation similar to the previous one arises for the lighter even–even isotope 126 Xe. Gade et al. [113] have studied γ -rays from the 123 Te(α, n)126 Xe fusion evaporation reaction at 15.5 + MeV beam energy. Again, the 2+ 4 state at 2064 keV was found to decay to the 21 state by a + pure M1 transition. Doppler shift analysis indicates an effective lifetime of τeff (24 ) = 0.35(7) ps > τ (2+ 4 ) which represents an upper limit for the true lifetime. This, together with the branching ratio for decay to the ground state, converts into a reduced transition strength of + 2 B(M1; 2+ 4 → 21 ) > 0.025(6) µ N which might indicate this level as a candidate for being + 126 the one-phonon 21,ms state of Xe. The dipole excitation strength distribution of 126 Xe has been studied by von Garrel et al. [136] in NRF experiments up to 4 MeV. A strong dipole excitation has been observed at 2919 keV excitation energy. Unambiguous information on its parity quantum number is lacking. This state decays strongly to the 2+ 2 state with S O(5) quantum number τ = 2 which suggests positive parity. In that case, this 1+ state alone would carry an excitation strength of B(M1; 0+ 1 → 2 in good agreement [136] with the expectation for the M1 strength 1+ ) = 0.82(4) µ N 2919keV of the excitation to the two-phonon 1+ ms state for the O(6) dynamical symmetry limit of the IBM-2. 4.4.3. 134 Ba On the basis of the observation of small E2/M1 multipole mixing ratios for the 2+ → 2+ 1 134 Ba at 2029 and 2088 keV were initially proposed as fragments transitions the 2+ states of 3,4 of the 2+ ar et al. [21]. Subsequent lifetime measurements 1,ms state of this nucleus by Moln´ done by Fazekas et al. using DSAM-INS yielded lifetime values of τ (2+ 3 ) = 230(23) fs and + states. These lifetimes result, in combination τ (2+ ) = 85(7) fs [35] for these two short-lived 2 4 + + with the γ -ray decay branching ratios Iγ (2+ → 2+ 1 )/Iγ (2 → 01 ) = 4.785(3) and 2.51(8), as well as the multipole mixing ratios δ = −0.31(5) and 0.02(5), in M1 transition strength values + + + 2 2 of B(M1; 2+ 3 → 21 ) = 0.064(7) µ N and B(M1; 24 → 21 ) = 0.134(13) µ N , respectively. + The E2 strengths for decay to the ground state are B(E2; 2+ 3 → 21 ) = 0.43(5) W.u. and + + + + B(E2; 24 → 21 ) = 1.55(14) W.u. The total 2 → 21 M1 decay strengths of 0.20(2) µ2N and the E2 ground state decay strength of 2.0(2) W.u. match the expectation for the one-phonon 2+ 1,ms state. Two-phonon dipole states of 134 Ba have been investigated in photon scattering experiments [137,138]. A pronounced concentration of M1 excitation strength has been observed close to 3 MeV. The 1+ states that carry most of the M1 excitation strength can be considered as + the main fragments of the 1+ ms state. The most strongly excited 1 states were indeed observed + to decay to the symmetrical two-phonon 22 state in agreement with the expectation for a twophonon MSS.

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4.4.4. 136 Ba Photon scattering experiments [85] were used to identify both the one-phonon 2+ 1,ms state and strong dipole excitations. This experiment first demonstrated the viability of identifying the 2+ 1,ms state from lifetime information from nuclear resonance fluorescence and triggered a series of experiments for identification of the one-phonon 2+ 1,ms state of heavy vibrational nuclei. Fig. 4 + 136 0 shows the part of the Ba(γ ,γ ) spectra that contains the 2+ 4 → 01 γ -ray at the transition energy of 2129 keV. 136 Ba. It has a short lifetime of τ = 67(7) fs. The 2+ state at 2129 keV is the 2+ 1,ms state of + 4 2 Its ground state decay strength of B(E2; 24 → 0+ 1 ) = 90(10) e fm is weakly collective. It corresponds to 2.1 single-particle units and amounts to 11% of the E2 decay strength of the 2+ 1 symmetrical one-phonon state. This is three times larger than the excitation strength of the 2+ 2 state, which in the vibrator nucleus 136 Ba is interpreted as the 2+ member of the symmetrical two-quadrupole phonon triplet. From angular correlations of γ -decays after neutron capture it is known [139] that the 2+ 4 → + 21 transition has a pure M1 character with a very small E2/M1 mixing ratio δ = +0.005(9). From the γ -decay branching ratio and the lifetime observed in the (γ , γ 0 ) experiment an M1 + 2 decay strength of B(M1; 2+ 4 → 21 ) = 0.26(3) µ N has been deduced which corresponds to the + + M1 matrix element |h21 k M1 k 24 i| = 1.14(7) µ N . The spin and parity quantum numbers J π = 2+ , together with the small multipole mixing ratio δ2+ →2+ and with the absolute E2 and 1

M1 strengths of decay to the ground state and to the 2+ 1 state deduced from the photon scattering cross section and the γ -ray decay branching ratio, represent the complete set of signatures for 136 Ba. Given the uncertainties of about 10%, the identifying M1 the one-phonon 2+ 1,ms state of decay strength coincides with the total M1 decay strength of the fragmented 2+ 1,ms state in the + 134 136 neighboring isotope Ba discussed above. In Ba the 21,ms configuration is concentrated in + 136 Ba which could comparably strongly decay the 2+ 4 state. The existence of another 2 state of by E2 and M1 transitions to the ground state and to the 2+ 1 state, respectively, can be excluded in the excitation energy range from 2 to 4 MeV, due to the complete character of the photon scattering reaction with continuous-energy bremsstrahlung. The NRF experiment also yielded information on 1+ states with comparatively large M1 ground state excitation strengths [85]. One of them, the 1+ state at 2694 keV, is known to decay strongly to the 2+ 2 state, however with unknown multipolarity. Making the most reasonable + + assumption of dominant M1 character for that 1+ → 2+ 2 transition, the B(M1; 12694 → 22 ) 2 value amounts to 0.6(1) µ N . All data are in satisfactory agreement with a schematic IBM-2 calculation except for the fragmentation of the total M1 strength of excitation to two rather than one 1+ state [85]. 4.4.5. 138 Ce 138 Ce has very recently been studied by Rainovski et al. [94] The one-phonon 2+ 1,ms state of using the method of projectile Coulomb excitation at the Argonne National Laboratory, Argonne, IL. A beam of 138 Ce ions was delivered by the ATLAS accelerator with an intensity of about 1 particle nA. It has been extracted from the ATLAS-ECR ion source loaded with cerium material moderately enriched in the isotope 138 Ce which has only a low natural abundance of 0.25%. The ion beams with energies of 480 MeV and 400 MeV bombarded a 1 mg/cm2 thick carbon target for 15 h and 5 h, respectively. The γ -rays issued by the predominantly one-step Coulomb excited

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138 Ce from relevant strength distributions obtained from projectile Coulomb Fig. 22. Identification of the 2+ 1,ms state of + excitation. The top panel shows the 2 -to-ground state E2 distribution in Weisskopf units as a function of the initial + 2 138 Ce state’s energy. The bottom panel displays the 2+ → 2+ 1 M1 distribution in units of µ N suggesting the 24 state of + at 2.237 MeV excitation energy as the dominant fragment of the 21,ms state. From Rainovski et al. [94].

projectiles were detected with the Gammasphere array which consisted of 98 HPGe detectors arranged in 17 rings around the beam axis. Gammasphere was used in singles mode at an average counting rate of about 4000 events per second. A total of 2.4×108 events of γ -ray fold 1 or higher were collected at a beam energy of 480 MeV in 15 h beam time. The singles data make a simple measurement of Coulomb excitation cross sections relative to the one for the 2+ 1 state possible. Spin and multipolarity assignments have been obtained from single-γ -ray angular distribution analyses. An example is shown in Fig. 6. The data yield lifetime and decay information for the + + + + lowest six 2+ 1,2,3,4,5,6 states. The resulting 2 → 01 E2 and 2 → 21 M1 strength distributions are shown in Fig. 22. + + The 2+ 4 state at 2237 keV with B(M1; 24 → 21 ) = 0.122(10) dominates the M1 strength distribution. It is considered as the main fragment of the 2+ 1,ms one-phonon state [94]. However, the close-lying 2+ state acquires a considerable amount of M1 strength, too. This situation has 3 been interpreted [94] as a fragmentation of the MS one-phonon state in terms of a two-state scenario of mixing of this mode with a nearby symmetrical 2+ state. A mixing matrix element of 44(3) keV has been determined. Analogous considerations for the lighter N = 80 isotone 136 Ba yield a mixing matrix element of <10 keV. This difference by a factor of 4 between the mixing matrix elements of these neighboring even–even isotones has been interpreted in terms of the underlying shell structure [94]. The proton πg(7/2) sub-shell closure at Z = 58 (cerium) leads to one simple 2+ proton configuration in 136 Ba which couples to the neutron 2+ configuration in an antisymmetrical way in forming the main component of the 2+ 1,ms state. This seniority-2 + configuration cannot mix easily with more highly excited 2 proton configurations that must involve excitations of protons to higher-lying orbitals. This phenomenon, present in 136 Ba but absent in 138 Ce at the sub-shell closure, has been considered as shell stabilization of mixedsymmetry states [94].

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4.5. Nuclei in the [50 < Z < 82, 82 < N < 126]-major shell MS states of vibrational nuclei were first suggested by Hamilton et al. [19] based on the small E2/M1 multipole mixing ratios in N = 84 isotones. MS assignments have subsequently been made frequently on the unsafe basis of small multipole mixing ratios alone. We will instead restrict ourselves to those cases where the large M1 transition strengths have been measured 140 Ba, absolutely. From the initial identification of the 2+ 1,ms states in the N = 84 isotones 142 Ce, and 144 Nd the assignments in the last two have been supported by subsequent lifetime measurements. 4.5.1. 142 Ce 142 Ce at an excitation energy of Hamilton et al. [19] initially suggested the 2+ 3 state of + 2004.9(1) keV [36] as the 21,ms state of this nucleus based on the small E2/M1 multipole + + mixing ratio δ = 0.41(7) observed for the 2+ 3 → 21 transition. The large B(M1; 23 → + 2 21 ) = 0.26(5) µ N value was first measured by Vermeer et al. in Coulomb excitation [32]. The + corresponding M1 matrix element amounts to |h2+ 1 k M1 k 23 i| = 1.14(11) µ N and the decay of this state to the ground state was found to be weakly collective with a strength of B(E2; 2+ 3 → 2 fm4 = 3.2(5) W.u. These findings were supported and made more accurate 0+ ) = 140(22) e 1 by lifetime measurements by Vanhoy et al. [36] done with the DSAM-INS technique. From the + + latter experiments the lifetime τ (2+ 3 ) = 65(7) fs and the mixing ratio δ(23 → 21 ) = −0.26(15) + + were determined, which results in the updated values B(M1; 23 → 21 ) = 0.23(2) µ2N and + B(E2; 2+ 3 → 01 ) = 2.5(2) W.u. [36]. However, those experiments also showed the existence of another short-lived 2+ state of 142 Ce at an excitation energy of 2365 keV. This state, which 142 Ce, behaves similarly to the 2+ state and also exhibits the signature for is the 2+ 4 state of 3 + the 21,ms state. From the measured lifetime τ (2+ 4 ) = 23(3) fs the absolute transition strengths + + + 2 B(M1; 2+ 4 → 21 ) = 0.36(5) µ N and B(E2; 24 → 01 ) = 2.6(4) W.u. were determined [36]. This finding has been interpreted [36,140] as a fragmentation of the 2+ 1,ms state over these two + close-lying experimental 23,4 states. Two-phonon MS states of 142 Ce with positive parity have not been firmly identified up to now. A recent photon scattering experiment [140] identified 14 dipole excitations in the energy range from 2.1 to 3.9 MeV. Due to the lack of solid parity information for all of them but the lowest two below 2.4 MeV a more conclusive assignment of two-phonon 1+ ms states could not be made. From the observed γ -ray decay branching ratios of these dipole excitations to the ground state and to the 2+ 1 , one J = 1 state could be identified [140] which can be considered as a + promising candidate for being a first 1− ms state resulting from the coupling of the 21,ms state to the 3− 1 octupole vibration [62]. However, more data are needed for a more solid assignment. 4.5.2. 144 Nd 144 Nd at an excitation energy of 2072 keV as the Hamilton et al. [19] proposed the 2+ 3 state of + 21,ms state of this nucleus due to the small E2/M1 multipole mixing ratio δ = 0.31(11) observed + + + 2 for the 2+ 3 → 21 transition. A B(M1; 23 → 21 ) = 0.14(3) µ N value has been measured +19 by Hicks et al. in DSAM-INS where a lifetime value of τ (2+ 3 ) = 80−14 fs was found [103]. + + The corresponding M1 matrix element amounts to |h21 k M1 k 23 i| = 0.84(9) µ N and the decay of this state to the ground state was found to be weakly collective with a strength of + B(E2; 2+ 3 → 01 ) = 1.7(4) W.u.

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Again, like in the previous case of the isotone 142 Ce, these experiments showed the existence of another short-lived 2+ state at an excitation energy of 2369 keV. This state, which is the 144 Nd, exhibits the signatures for the 2+ 2+ 4 state of 1,ms state, too. From the measured lifetime + + 2 τ (24 ) = 56+22 fs the absolute transition strengths B(M1; 2+ −15 4 → 21 ) = 0.18(7) µ N and + B(E2; 2+ 4 → 01 ) = 0.7(2) W.u. were determined [103]. This finding has been interpreted as + indicating a fragmentation of the 2+ 1,ms state into these two close-lying experimental 23,4 states. Two-phonon MS states of 144 Nd could not be firmly identified from the neutron scattering experiments [103]. Investigations using photon scattering [141] identified 16 dipole excitations in the energy range from 2.1 to 3.9 MeV. Using the technique of γ -ray Compton polarimetry parity information for the strongest dipole excitations was determined. Positive parity could be + assigned to the 1+ state at 3213 keV. It carries an M1 excitation strength of B(M1; 0+ 1 →1 )= 2 0.171(12) µ N . This state is considered the best candidate for being the main fragment of the 1+ ms two-phonon MSS of 144 Nd. 4.5.3. 148 Sm The stable even–even nuclides of the samarium isotopic chain undergo a shape change from spherical vibrators near the N = 82 neutron shell closure to quadrupole deformed rotors beyond

148 Sm from relevant strength distributions obtained from NRF. The top panel Fig. 23. Identification of the 2+ 1,ms state of 2 shows the 2+ → 2+ 1 M1 distribution in units of µ N as a function of the initial state’s energy. The bottom panel displays the distribution for the strength of the E2 ground state excitation to these 2+ states in units of e2 fm4 . The arrow on the + M1 data bar for the 2+ 5 state indicates that it represents an experimental lower limit. Its large value suggests the 25 state + 148 at 2.146 MeV excitation energy is the dominant fragment of the 21,ms state of Sm. From Li et al. [114].

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148 Sm. From Li et al. [114]. Fig. 24. Decay properties of the 2+ 1,ms state of

neutron number N = 90. They serve, together with the stable nuclides of the neodymium isotopic chain, as the classic testing ground for the evolution of low-energy nuclear structure as a function of ground state quadrupole deformation. The deformation dependence of the strength of the M1 excitation to the 1+ scissors mode, formulated by Ziegler et al. [142] as the “δ 2 -law”, has been discovered in systematic photon scattering experiments at the S-Dalinac accelerator in Darmstadt. In the less deformed Sm isotopes with N < 90 the 1+ states should be considered as the fragments of the two-phonon 1+ ms state. Unfortunately, not much is known to date about the 2+ 1,ms one-phonon MSS of the even–even Sm isotopes. 148 Sm has recently been searched for in a photon scattering The one-phonon 2+ 1,ms state of experiment [114] on 148 Sm performed at the bremsstrahlung facility in Stuttgart [15,84]. A photon beam end point of E 0 = 3.2 MeV has been chosen for increased sensitivity to excitations between 2 and 3 MeV as compared to previous experiments [143]. A total of 16 γ -ray lines from 11 excited states of 148 Sm between 1.4 and 3.1 MeV were observed, two of them for the first time. 148 Sm at an excitation energy of 2146 keV a cross section of I For the 2+ s,1 = 4.2 eV b 5 state of + + has been measured [114] for the 0+ → 2 → 2 photon scattering cascade. This, together 1 5 1 with information on decay branching and multipole mixing ratios from the literature [144,145], + 4 2 allows us to determine its E2 excitation strength of B(E2; 0+ 1 → 25 ) = 130 − 195 e fm + + 2 and a lower limit of 0.12 µ N for its B(M1; 25 → 21 ) value. This information is compared to corresponding data for other 2+ states of 148 Sm in Fig. 23. The strong M1 transition with + + a matrix element of |h2+ 1 k M1 k 25 i| ≥ 0.76 µ N is the main signature for the 21,ms state. Fig. 24 summarizes the observations for this state. The existence of an enhanced E1 transition + to the 3− 1 octupole vibrational state has recently been identified as another signature for the 21,ms state in near-vibrational nuclei [146] and will be discussed below. It further supports [114] the + 148 Sm as the 2+ identification of the 2+ 5 state of 1,ms state. This is the first assignment of the 21,ms state in a nuclide of the Sm isotopic chain, as well as in a nuclide of the N = 86 isotonic chain, that is firmly based on information on absolute electromagnetic transition strengths. It further makes 148 Sm the heaviest nuclide to date for which the 2+ 1,ms state has been identified from large measured M1 transition rates. 4.6. A = 60 mass region Shortly after the tentative identification [19] of 2+ 1,ms one-phonon MSSs of vibrational nuclei near the N = 82 neutron shell closure the mass region near the doubly closed shell 56 Ni

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was examined for vibrational MS structures [82,147]. Lifetime measurements and microscopic calculations generally corroborate the proposals of a one-phonon 2+ state with significant MS character. These states are observed at about 3 MeV excitation energy, i.e., about 1 MeV higher than in the heavy vibrators. Already Collins et al. [147] have therefore given the caveat that “It is possible that other types of excitation may be involved. For instance in the neighboring semi-magic nuclei, 52 Cr and 54 Fe, only proton bosons are present and so low-lying mixedsymmetry states are ruled out, although both nuclei have 2+ states at about 2 MeV which decay to the first 2+ state with B(M1) ' 0.1 µ2N .” For that reason additional signatures of MS + structures besides a strong M1 transition from the one-phonon 2+ 1,ms state to the 21 state would be desirable. Candidates in addition to those discussed in this subsection were considered in Ref. [147]. 4.6.1. 54 Cr Lifetimes of low-lying 2+ states of 54 Cr have been measured by a G¨ottingen–Grenoble collaboration of Lieb et al. [31] using the technique of gamma-ray-induced Doppler broadening [106,107] following neutron capture. The 2+ 3 state at 3074 keV excitation energy was found to have a short lifetime of 13(2) fs. It decays predominantly to the 2+ 1 state by a pure M1 transition with an E2/M1 multipole mixing ratio of 0.02(5) [147] and with a strength + + 2 + of B(M1; 2+ 3 → 21 ) = 0.39(6) µ N [31]. It dominates the 2 → 21 M1 transition strength distribution and has, therefore, been interpreted [31,147] as the main fragment of the 2+ 1,ms one54 phonon MSS of Cr. Shell model results mapped onto a boson model space [148] support mixedsymmetry character residing in low-lying 2+ states. 4.6.2. 56 Fe Eid et al. [82] have measured precise E2/M1 multipole mixing ratios for 2+ → 2+ 1 transitions 56 in Fe. Small values for the quadrupole/dipole mixing ratios of δ = −0.18(1) and −0.19(1) + were found for the 2+ 2 and 23 states at 2658 keV and 2960 keV, respectively. The lifetimes adopted of τ = 30(2) and 40(5) fs [149] result in M1 transition strengths of 0.30(2) and 0.14(2) µ2N for these transitions. The total M1 strength was interpreted as originating from the mixing 56 + of the 2+ 1,ms one-phonon MSS of Fe with a nearby symmetrical 2 state [82]. This two-state mixing is supported by electron scattering data from the S-Dalinac in Darmstadt reported by Hartung et al. [33]. They have measured form factors for the lowest three 2+ states. The form + factors for the 2+ 2 and 23 states are almost identical owing to the common origin from the onephonon MSS mixed between them. Large-scale shell model calculations performed by the Tokyo 56 Fe. In group (Nakada et al. [150]) agree with a dominant MS structure for the 2+ 2 state of contrast to the mixing scenario given above, these calculations yield non-collective character for + the 2+ 3 state while MS structure is found instead for the 24 state [150]. The nucleus 56 Fe features a low-lying 1+ state at 3448 keV excitation energy. It dominates the M1 excitation strength distribution up to about 7 MeV [151,152]. Shell model calculations point to an orbital character of the M1 matrix element [152]. Gamma-ray transitions between that 1+ state and the two fragments of the proposed one-phonon 2+ 1,ms have not been observed so far. 4.6.3. 66 Zn + + A structure in 66 Zn comprising a one-phonon 2+ 1,ms state and two 1ms and 22,ms two-phonon + states has been discussed by Gade et al. [108]. The 24 state at 2938 keV dominates the 2+ → 2+ 1

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+ + 2 M1 strength distribution with a value of B(M1; 2+ 4 → 21 ) = 0.21(2) µ N . The 12 state at 4295 + keV has a short lifetime of τ = 6(2) fs and decays to the 22 state by an M1 transition of the + same size. It further decays to the proposed 2+ 1,ms state by a strong E2 transition. The 26 state at 3331 keV dominates the 2+ → 2+ 2 M1 strength distribution with an M1 strength of 0.12(3) µ2N . These observations correspond to the signatures for MS multiphonon structures listed above. Consequently, IBM-2 calculations are in fair agreement with these MS assignments [108].

4.7. F-vector E1 transitions Up to now we have discussed E2 and M1 data used for identification and characterization of nuclear structures with mixed symmetry. Recently, experimental information on E1 transitions from or to MSSs became available. Octupole vibration and quadrupole–octupole coupling are well known as major sources of E1 strength at low energies [153,154], in many cases with + sizeable E1 decay rates. In heavy vibrational nuclei the 3− 1 → 21 transition is known to set the + scale for E1 quadrupole–octupole collectivity [155]. The E1 strength relative to the 3− 1 → 21 transition is therefore of particular interest for the classification of E1 transitions between low+ energy states. F-vector E1 transitions between the 3− 1 octupole vibration and the 21,ms onephonon state indicated in Fig. 25 were discovered and observed to be stronger than representative F-scalar E1 transitions in the same nucleus. The sd f -IBM-2 [62], a recently formulated version of the interacting boson model, has been applied to these data [146] and thereby first numerical values of relevant model parameters were established. Fig. 26 shows the relevant parts of the γ -ray spectrum from the 91 Zr(α, nγ )94 Mo reaction + that made a measurement of the E1 branching ratio for the decay of the 3− 1 state to the 21,ms 94 and 2+ 1 states for Mo possible [146]. At γ -ray energies of 466.4 and 1662.6 keV, one clearly + observes the peaks corresponding to the 3− 1 → 2 transitions of interest. Their pure dipole character has been established from the measurement of γ γ angular correlations and γ -ray angular distributions [77]. Of particular interest are the relative E1 strengths of transition between the 3− 1 octupole + + vibration and the symmetrical and MS one-quadrupole phonon states, 21 and 21,ms , i.e. the

Fig. The

25. Sketch of a typical arrangement of the basic quadrupole and octupole excitations in a vibrational nucleus. − 2+ 1,ms state and the 31 octupole phonon state usually decay by strong magnetic and electric dipole transitions, + respectively, to the 21 state, which in most nuclei represents the isoscalar quadrupole phonon excitation of the nuclear + valence shell. The 3− 1 ↔ 21,ms transition is expected to show predominantly isovector character. From Ref. [146].

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Fig. 26. Spectrum of γ -rays from 94 Mo taken with the Cologne OSIRIS cube spectrometer in the 91 Zr(α, nγ )94 Mo + reaction at 15 MeV in coincidence with the 2+ 1 → 01 transition. The coincidence relations establish the 466.4 and + 1662.6 keV γ -rays as dipole transitions between the 3− 1 state and the symmetrical and MS one-phonon 2 states of 94 Mo. From Ref. [146].

relative 1F = 0 and 1F = 1E1 strengths. Hence, the E1 strength ratio R E1 =

+ B(E1; 3− 1 → 21,ms )

(56)

+ B(E1; 3− 1 → 21 )

has been considered [146]. Table 8 summarizes the data on the fundamental quadrupole and octupole one-phonon vibrations in those N = 52 and 84 isotones for which all three levels are known. The four measured R E1 values are included in Table 8. In all cases, R E1 is larger than 1. Except for 92 Zr, where the 2+ 1 state has predominantly neutron character [109] due to the + Z = 40 sub-shell closure, the enhancement of the F-vector 3− 1 → 21,ms E1 transition over the − + F-scalar 31 → 21 E1 transition is by about an order of magnitude. Occurrence of an enhanced F-vector E1 transition to the 3− octupole vibration has become an additional signature for the one-phonon 2+ 1,ms state [114]. Data on R E1 have constrained the parameters of the effective quadrupole–octupole coupled E1 transition operator of the sd f -IBM-2 that make the predictions for E1 decay properties of other quadrupole–octupole coupled structures possible. Following those predictions a tentative assignment of quadrupole–octupole coupled MSS with negative parity has become possible for the first time [140]. Table 8 + Data on 3− 1 → 21,ms transitions from Ref. [146] 92 Zr

94 Mo

E(2+ 1 ) (keV)

934.5

871.1

832.6

641.2

E(2+ 1,ms ) (keV) E(3− 1 ) (keV) R E1 B(E1) (mW.u.)

1847.3 2339.6 2.7(2) 0.80(10)

2067.4 2533.7 26.0(7) 1.31(21)

2284.2 2650.0 6.7(10) –

2004.9 1652.9 >13.7 1.86(22)

(1mW.u.(E1) = 0.065 A2/3 10−3 e2 fm2 .)

96 Ru

142 Ce

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5. Summary and outlook It has been the goal of this review to gather the various sources of experimental information on nuclear structures with mixed proton–neutron symmetry in vibrational nuclei. The physics of mixed-symmetry states originates from the interplay and the competition of three generic aspects of atomic nuclei viewed as examples of a two-fluid, strongly interacting many-body quantum system. The quantum nature induces shell structure and thereby defines a valence shell. Its properties govern the low-energy structure of the system. Many strongly interacting particles provide the possibility of collective excitations. The two equivalent quantum fluids, formed by protons and neutrons, induce the isospin degree of freedom. It results in the F-spin degree of freedom when applied to valence bosons. Mixed-symmetry states are a fascinating phenomenon because they represent collective, F-vector excitations of the valence shell and hence relate directly to the aforementioned three fundamental aspects of atomic nuclei. The data summarized in this article together with the abundant literature on the scissors mode in deformed nuclei (see e.g. Refs. [14–16,18,156–161] and references therein) represent solid proof for the existence of nuclear mixed-symmetry states. Most of their properties, found experimentally, correspond closely to the initial predictions made by the pioneers of the interacting boson model in the late 1970s and early 1980s [3–5,7,9,10]. Since then many theorists have contributed to the field with different nuclear structure models. An additional review of its theoretical aspects would have been beyond both the scope of this article and the competence of its authors. We therefore restricted the theoretical presentation to the basic definitions and concepts in terms of the interacting boson model. Experimental progress has gained from the application of new techniques and from the new combination of classical methods of nuclear spectroscopy. Due to the importance of these developments we have chosen to summarize the methods used for obtaining the most abundant data sets. We hope this serves the understanding of the capabilities and that of the limitations of present experiments. Both are needed for an appreciation of existing data and for future progress. Research of recent years has led to new questions and experimental challenges to which we refer in this brief outlook. The separation of mixed-symmetry states in spherical and weakly deformed transitional systems from those in deformed systems – which we made in this review – is merely dictated by convenience. It has been done in this article in order to restrict the topic to a manageable size without neglecting important details that are partly due to the somewhat different arsenals of experimental methods and tools. But a unified view of both weakly deformed and strongly deformed nuclei should be kept for a comprehensive picture of the phenomenon of mixed-symmetry states. One of the outstanding questions is that for the evolution of mixed-symmetry states along a shape transition path from spherical to deformed nuclei. For the 1+ state, data exist for the sequences of samarium, neodymium, barium and xenon nuclei obtained at the Darmstadt and Stuttgart photon scattering facilities [142,143,162,138,136]. The evolution of the 2+ 1,ms one-phonon state along either a spherical-to-axially deformed or along a spherical-to-γ unstable shape transition is unknown. Fig. 27 shows a possible scenario. It is still unclear whether and under what conditions the wave function of the 2+ 1,ms state is robust enough to survive in a bath of high 2+ level density such that it typically remains detectable in spectroscopy experiments. It is very encouraging that evidence for the existence of 2+ ms states in the strongly deformed nucleus 156 Gd has already been reported from pioneering electron scattering experiments [39]. In a next step their dominant M1 decays to the 2+ 1 state of the ground band needs to be established and their transition strength to be measured for corroborating those proposals.

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Fig. 27. Schematic of the expected evolution of the one-phonon 2+ 1,ms state in a shape phase transition. From Li et al. [114].

Further evidence for proving the specific mixed-symmetric coupling of proton and neutron configurations, complementary to that of strong M1 transitions, is also desirable. Measurements of magnetic moments are known to show a high sensitivity to proton and neutron contributions to the nuclear wave functions. The typical short lifetimes of mixed-symmetry states limit the available time for interaction with an external magnetic field. Detection of the resulting small effects represents a big experimental challenge. The direct measurement of transition densities, separately for protons and for neutrons, e.g., from high-resolution inelastic electron scattering, could give final insight into mixed-symmetry structure. However, such a program would require data with very high statistics that are difficult and time-consuming to obtain. So far, mixed-symmetry states of vibrational nuclei have only been identified in even–even nuclei. The coupling of an unpaired particle to mixed-symmetry states in an odd mass nucleus would be interesting to observe. Promising preliminary data on the N = 52 isotone 93 Nb have recently been reported [163]. The presence of the unpaired particle and the spin M1 transitions related to its angular momentum weaken, however, the sensitivity of strong M1 transitions as a clear signature for mixed-symmetry states in odd nuclei. Due to the sensitivity of mixed-symmetry states to the properties of the local valence shell, information on mixed-symmetry states would be very useful for nuclei where the shell structure deviates from that at stability for instance because of considerable neutron excess. Up to now mixed-symmetry states have been assigned on the solid basis of strong M1 transitions for stable nuclides, only. This is because electron scattering, photon scattering, or neutron scattering are still inapplicable for the investigation of radioactive nuclei. Besides that of Coulomb excitation, these techniques are the main methods for lifetime measurements of short-lived states in the femtosecond range. Development of a technique for investigating mixed-symmetry states in unstable nuclei will represent a major breakthrough in the field. Finally one may wonder about mixed-symmetry structures for degrees of freedom other than quadrupole shape. Strong M1 transitions between low-energy 4+ states in near-closed-shell nuclei have already tentatively been interpreted in terms of symmetric and mixed-symmetry hexadecapole excitations [110,147]. The sd f -IBM-2 predicts [62] symmetrical and mixedsymmetry octupole vibrations and additionally an entire class of states that show a mixedsymmetric coupling of quadrupole and octupole parts that are each symmetrical individually. Formally the concept of mixed symmetry might be extended beyond the consideration of shape vibrations to other degrees of freedom of excitation or to other quantum systems with similar generic features to atomic nuclei. This has been done already for the scissors mode of deformed

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quantum systems [164–175]. Similar extensions to spherical, two-fluid many-body quantum systems might be possible. Note added at deadline: After completion of our data compilation new work on mixedsymmetry states of vibrational nuclei was published in the literature [176–182] that we were unable to discuss in more detail in this review. Acknowledgments We thank O. Burda, R.F. Casten, C. Fransen, A. Gelberg, J.N. Ginocchio, J.D. Holt, F. Iachello, J. Jolie, R.V. Jolos, U. Kneissl, A. Leviatan, A. Linnemann, N. Lo Iudice, D. M¨ucher, P. von Neumann-Cosel, T. Otsuka, V.Yu. Ponomarev, G. Rainovski, A. Richter, C. Scholl, N.A. Smirnova, Ch. Stoyanov, V. Werner, S.W. Yates, and A. Zilges for useful discussions. This work was partially supported by the U.S. NSF under grant No. PHY-0555396 and by the DFG under contracts Br 799/12-1, Pi 393/2-1, and SFB 634. References [1] D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. LoIudice, F. Palumbo, O. Scholten, Phys. Lett. B137 (1984) 27. [2] A. Faessler, Nuclear Phys. 85 (1966) 653. [3] F. Iachello, Lecture Notes on Theoretical Physics, Rijksuniversiteit Groningen, 1976. [4] A. Arima, T. Otsuka, F. Iachello, I. Talmi, Phys. Lett. B 66 (1977) 205. [5] T. Otsuka, Boson model of medium-heavy nuclei, Ph.D. Thesis, University of Tokyo, 1978. [6] T. Suzuki, D.J. Rowe, Nuclear Phys. A299 (1977) 461. [7] T. Otsuka, A. Arima, F. Iachello, Nuclear Phys. A 309 (1978) 1. [8] N. Lo Iudice, F. Palumbo, Phys. Rev. Lett. 41 (1978) 1532. [9] F. Iachello, Nuclear Phys. A 358 (1981) 89. [10] F. Iachello, Phys. Rev. Lett. 53 (1984) 1427. [11] R.R. Hilton, Z. Phys. A316 (1984) 141; And conference contribution to Int. Conf. on Nuclear Structure (Dubna, June 1976) (unpublished). [12] A. Faessler, R. Nojarov, Phys. Lett. 166B (1986) 367. [13] K. Heyde, J. Sau, Phys. Rev. C 33 (1986) 1050. [14] A. Richter, Prog. Part. Nucl. Phys. 34 (1995) 261. [15] U. Kneissl, H.H. Pitz, A. Zilges, Prog. Part. Nucl. Phys. 37 (1996) 349. [16] D. Zawischa, J. Phys. G: Nucl. Part. Phys. 24 (1998) 683. [17] P. Van Isacker, K. Heyde, J. Jolie, A. Sevrin, Ann. Phys. (NY) 171 (1986) 253. [18] K. Heyde, P. von Neumann-Cosel, A. Richter (in preparation). [19] W.D. Hamilton, A. Irb¨ack, J.P. Elliott, Phys. Rev. Lett. 53 (1984) 2469. [20] P. Park, A.R.H. Subber, W.D. Hamilton, J.P. Elliott, K. Kumar, J. Phys. G: Nucl. Phys. 11 (1985) L251. [21] G. Moln´ar, R.A. Gatenby, S.W. Yates, Phys. Rev. C 37 (1988) 898. [22] S.T. Ahmad, W.D. Hamilton, P. Van Isacker, S.A. Hamada, S.J. Robinson, J. Phys. G 15 (1989) 93. [23] A. Giannatiempo, A. Nannini, A. Perego, P. Sona, G. Maino, Phys. Rev. C 44 (1991) 1508. [24] A. Giannatiempo, A. Nannini, A. Perego, P. Sona, D. Cutoiu, Phys. Rev. C 53 (1996) 2770. [25] M. Pignanelli, S. Micheletti, N. Blasi, R. De Leo, W.T.A. Borghols, J.M. Schippers, S.Y. Van der Werf, M.N. Harakeh, Phys. Lett. B202 (1988) 470. [26] R. De Leo, L. Lagamba, N. Blasi, S. Micheletti, M. Pignanelli, M. Fujiwara, K. Hosono, I. Katayama, N. Matsuoka, S. Morinobu, T. Noro, S. Matsuki, H. Okamura, J.M. Schippers, S.Y. Van der Werf, M.N. Harakeh, Phys. Lett. B226 (1989) 5. [27] R. De Leo, H. Akimune, N. Blasi, I. Daito, Y. Fujita, M. Fujiwara, S.I. Hayakawa, S. Hatori, K. Hosono, H. Ikegami, T. Inomata, I. Katayama, K. Katori, L. Lagamba, S. Micheletti, S. Morinobu, T. Nakagawa, S. Nakayama, A. Narita, T. Noro, R. Perrino, M. Pignanelli, H. Sakaguchi, J. Takamatsu, A. Tamii, K. Tamura, M. Tanaka, A. Terakawa, T. Tohei, M. Tosaki, T. Yamagata, A. Yamagoshi, M. Yosimura, M. Yosoi, Phys. Rev. C 53 (1996) 2718. [28] P.O. Lipas, P. von Brentano, A. Gelberg, Rep. Progr. Phys. 53 (1990) 1355.

N. Pietralla et al. / Progress in Particle and Nuclear Physics 60 (2008) 225–282 [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74]

279

P. Van Isacker, D.D. Warner, J. Phys. G: Nucl. Part. Phys. 20 (1994) 853. P. Van Isacker, Rep. Progr. Phys. 62 (1999) 1661–1717. K.P. Lieb, H.G. B¨orner, M.S. Dewey, J. Jolie, S.J. Robinson, S. Ulbig, Ch. Winter, Phys. Lett. B215 (1988) 50. W.J. Vermeer, C.S. Lim, R.H. Spear, Phys. Rev. C 38 (1988) 2982. G. Hartung, A. Richter, E. Spamer, H. W¨ortche, C. Rangacharyulu, C.W. de Jager, H. de Vries, Phys. Lett. 221B (1989) 109. J. Takamatsu, T. Nakagawa, A. Terakawa, A. Narita, T. Tohei, M. Fujiwara, S. Morinobu, I. Katayama, H. Ikegami, K. Katori, S.I. Hayakawa, Y. Fujita, M. Tosaki, S. Hatori, Phys. (Paris) Colloque 51 (1990) C6-423. B. Fazekas, T. Belgya, G. Moln´ar, A. Veres, R.A. Gatenby, S.W. Yates, T. Otsuka, Nuclear Phys. A 548 (1992) 249. J.R. Vanhoy, J.M. Anthony, B.M. Haas, B.H. Benedict, B.T. Meehan, S.F. Hicks, C.M. Davoren, C.L. Lundstedt, Phys. Rev. C 52 (1995) 2387. P.E. Garrett, H. Lehmann, C.A. McGrath, Minfang Yeh, S.W. Yates, Phys. Rev. C 54 (1996) 2259. I. Wiedenh¨over, A. Gelberg, T. Otsuka, N. Pietralla, J. Gableske, A. Dewald, P. von Brentano, Phys. Rev. C 56 (1997) R2354. D. Bohle, A. Richter, K. Heyde, P. Van Isacker, J. Moreau, A. Sevrin, Phys. Rev. Lett. 55 (1985) 1661. A. Arima, F. Iachello, Phys. Rev. Lett. 35 (1975) 1069. F. Iachello, A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge, 1987. A. Arima, F. Iachello, Ann. Phys. (NY) 99 (1976) 253. A. Arima, F. Iachello, Ann. Phys. (NY) 111 (1978) 201. A. Arima, F. Iachello, Ann. Phys. (NY) 123 (1979) 468. A.E.L. Dieperink, G. Wenes, Ann. Rev. Nucl. Part. Sci. 35 (1985) 77–105. R.F. Casten, D.D. Warner, Rev. Modern Phys. 60 (1988) 389. J. Jolie, Prog. Part. Nucl. Phys. 59 (2007) 337. D. Bonatsos, Interacting Boson Models of Nuclear Structure, Oxford University Press, Oxford, 1988. F. Iachello, P. Van Isacker, The Interacting Boson–Fermion Model, Cambridge University Press, Cambridge, 1991. R.F. Casten, P.O. Lipas, D.D. Warner, T. Otsuka, K. Heyde, J.P. Draayer, Algebraic Approaches to Nuclear Structure, Harwood, 1993. I. Talmi, Simple Models of Complex Nuclei, Harwood, 1993. A. Frank, P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics, Wiley, New York, 1994. R.F. Casten, Nuclear Structure from a Simple Perspective, Oxford University Press, New York, 2000. W. Greiner, J.A. Maruhn, Nuclear Models, Springer, 2006. W. Heisenberg, Z. Phys. 77 (1932) 1. H. Harter, P. von Brentano, A. Gelberg, R.F. Casten, Phys. Rev. C 32 (1985) 631. P. von Brentano, A. Gelberg, H. Harter, P. Sala, J. Phys. G: Nucl. Phys. 11 (1985) L85–L89. J. Jolie, P. Van Isacker, K. Heyde, A. Frank, Phys. Rev. Lett. 55 (1985) 1457. A. Leviatan, J.N. Ginocchio, Phys. Rev. C 61 (2000) 024305. O. Scholten, K. Heyde, P. Van Isacker, J. Jolie, J. Moreau, M. Waroquier, J. Sau, Nuclear Phys. A438 (1985) 41. A. Giannatiempo, G. Maino, A. Nannini, P. Sona, Phys. Rev. C 48 (1993) 2657. Nadya A. Smirnova, Norbert Pietralla, Takahiro Mizusaki, Piet Van Isacker, Nuclear Phys. A 678 (2000) 235. G. Siems, U. Neuneyer, I. Wiedenh¨over, S. Albers, M. Eschenauer, R. Wirowski, A. Gelberg, P. von Brentano, T. Otsuka, Phys. Lett. B320 (1994) 1. T. Otsuka, K.H. Kim, Phys. Rev. C 50 (1994) R1768. N. Pietralla, P. von Brentano, R.F. Casten, T. Otsuka, N.V. Zamfir, Phys. Rev. Lett. 73 (1994) 2962. N. Pietralla, P. von Brentano, T. Otsuka, R.F. Casten, Phys. Lett. B349 (1995) 1. N. Pietralla, T. Mizusaki, P. von Brentano, R.V. Jolos, T. Otsuka, V. Werner, Phys. Rev. C 57 (1998) 150. Y.V. Palchikov, P. von Brentano, R.V. Jolos, Phys. Rev. C 57 (1998) 3026. R.V. Jolos, Phys. Atomic Nuclei 64 (2001) 465. K.H. Kim, T. Otsuka, P. von Brentano, A. Gelberg, P. Van Isacker, R.F. Casten, in: G. Moln´ar (Ed.), Capture Gamma-Ray Spectroscopy and Related Topics, vol. I, Springer, Budapest, 1996, p. 195. N. Yoshida, A. Gelberg, T. Otsuka, I. Wiedenh¨over, H. Sagawa, P. von Brentano, Nuclear Phys. A 619 (1997) 65. T. Otsuka, Hyperfine Interact. 75 (1992) 23. N. Pietralla, P. von Brentano, A. Gelberg, T. Otsuka, A. Richter, N. Smirnova, I. Wiedenh¨over, Phys. Rev. C 58 (1998) 191. N. Pietralla, C. Fransen, D. Belic, P. von Brentano, C. Friessner, U. Kneissl, A. Linnemann, A. Nord, H.H. Pitz, T. Otsuka, I. Schneider, V. Werner, I. Wiedenh¨over, Phys. Rev. Lett. 83 (1999) 1303.

280

N. Pietralla et al. / Progress in Particle and Nuclear Physics 60 (2008) 225–282

[75] N. Pietralla, C. Fransen, P. von Brentano, A. Dewald, A. Fitzler, C. Friessner, J. Gableske, Phys. Rev. Lett. 84 (2000) 3775. [76] C. Fransen, N. Pietralla, P. von Brentano, A. Dewald, J. Gableske, A. Gade, A. Lisetskiy, V. Werner, Phys. Lett. B 508 (2001) 219. [77] C. Fransen, N. Pietralla, Z. Ammar, D. Bandyopadhyay, N. Boukharouba, P. von Brentano, A. Dewald, J. Gableske, A. Gade, J. Jolie, U. Kneissl, S.R. Lesher, A.F. Lisetskiy, M.T. McEllistrem, M. Merrick, H.H. Pitz, N. Warr, V. Werner, S.W. Yates, Phys. Rev. C 67 (2003) 024307. ¨ [78] H. Uberall, Electron Scattering from Complex Nuclei, Academic Press, New York, 1971. [79] H. Theissen, Springer Tracts in Mod. Phys. 65 (1972). [80] T. de Forest, J.D. Walecka, Adv. Phys. 15 (1973) 1. [81] J. Heisenberg, H.P. Blok, Ann. Rev. Nucl. Part. Sci. 33 (1983) 569. [82] S.A.A. Eid, W.D. Hamilton, J.P. Elliott, Phys. Lett. 166B (1986) 267. [83] F.R. Metzger, Prog. Nucl. Phys. 7 (1959) 54. [84] Ulrich Kneissl, Norbert Pietralla, Andreas Zilges, J. Phys. G: Nucl. Part. Phys. 32 (2006) R217. [85] N. Pietralla, D. Belic, P. von Brentano, C. Fransen, R.-D. Herzberg, U. Kneissl, H. Maser, P. Matschinsky, A. Nord, T. Otsuka, H.H. Pitz, V. Werner, I. Wiedenh¨over, Phys. Rev. C 58 (1998) 796. [86] C.I. McClelland, C. Goodman, Phys. Rev. 91 (1953) 760. [87] T. Huus, C. Zupancic, Mat. Fys. Medd. Dan. Vid. Selsk. 28 (1) (1953). [88] K. Alder, A. Bohr, T. Huus, B.R. Mottelson, A. Winther, Rev. Modern Phys. 28 (1956) 432. [89] K. Alder, A. Winther, Coulomb Excitation, Academic Press, New York, 1966. [90] K. Alder, A. Winther, Electromagnetic Excitation, Theory of Coulomb Excitation with Heavy Ions, North Holland, Amsterdam, 1975. [91] J. de Boer, Treatise on Heavy Ion Science I, Plenum, New York, 1984, p. 293. [92] D. Cline, Ann. Rev. Nucl. Part. Sci. 36 (1986) 683. [93] N. Pietralla, C.J. Barton III, R. Kr¨ucken, C.W. Beausang, M.A. Caprio, R.F. Casten, J.R. Cooper, A.A. Hecht, H. Newman, J.R. Novak, N.V. Zamfir, Phys. Rev. C 64 (2001) 031301. [94] G. Rainovski, N. Pietralla, T. Ahn, C.J. Lister, R.V.F. Janssens, M.P. Carpenter, S. Zhu, C.J. Barton III, Phys. Rev. Lett. 96 (2006) 122501. [95] K. Siegbahn, Alpha-, Beta-, and Gamma-Ray Spectroscopy, North-Holland, Amsterdam, 1965. [96] K.S. Krane, R.M. Steffen, R.M. Wheeler, Nucl. Data Tables 11 (1973) 351. [97] L.W. Fagg, S.S. Hanna, Rev. Modern Phys. 31 (1959) 711. [98] A. Wolf, N.V. Zamfir, M.A. Caprio, Z. Berant, D.S. Brenner, N. Pietralla, R.L. Gill, R.F. Casten, C.W. Beausang, R. Kr¨ucken, K.E. Zyromski, C.J. Barton, J.R. Cooper, A.A. Hecht, H. Newmann, J.R. Novak, J. Cederkall, Phys. Rev. C 66 (2003) 024323. [99] A. Bohr, B. Mottelson, Nuclear Structure, Benjamin, Reading, 1969/1975. [100] P.E. Garrett, N. Warr, S.W. Yates, J. Res. Natl. Inst. Stand. Technol. 105 (2000) 141. [101] C.A. McGrath, P.E. Garrett, M.F. Villani, S.W. Yates, Nucl. Instrum. Methods Phys. Res. A 421 (1999) 458. [102] T. Belgya, G. Moln´ar, S.W. Yates, Nuclear Phys. A 607 (1996) 43. [103] S.F. Hicks, C.M. Davoren, W.M. Faulkner, J.R. Vanhoy, Phys. Rev. C 57 (1998) 2264. [104] D. Bandyopadhyay, C.C. Reynolds, C. Fransen, N. Boukharouba, M.T. McEllistrem, S.W. Yates, Phys. Rev. C 67 (2003) 034319. [105] H. Morinaga, T. Yamazaki, In-Beam Gamma-Ray Spectroscopy, North-Holland, Amsterdam, 1976. [106] M. Jentschel, H.G. B¨orner, H. Lehmann, C. Doll, J. Res. Natl. Inst. Stand. Technol. 105 (2000) 25. [107] H.G. B¨orner, M. Jentschel, Yad. Fiz. 64 (6) (2001) 1087; Phys. Atomic Nuclei 64 (2001) 1011. [108] A. Gade, H. Klein, N. Pietralla, P. von Brentano, Phys. Rev. C 65 (2002) 054311. [109] V. Werner, D. Belic, 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, S.W. Yates, Phys. Lett. B 550 (2002) 140. [110] C. Fransen, V. Werner, D. Bandyopadhyay, N. Boukharouba, S.R. Lesher, M.T. McEllistrem, J. Jolie, N. Pietralla, P. von Brentano, S.W. Yates, Phys. Rev. C 71 (2005) 054304. [111] H. Klein, A.F. Lisetskiy, N. Pietralla, C. Fransen, A. Gade, P. von Brentano, Phys. Rev. C 65 (2002) 044315. [112] S.W. Yates, J. Rad. Nucl. Chem. 265 (2005) 291. [113] A. Gade, I. Wiedenh¨over, J. Gableske, A. Gelberg, H. Meise, N. Pietralla, P. von Brentano, Nuclear Phys. A 665 (2000) 268. [114] T.C. Li, N. Pietralla, C. Fransen, H. von Garrel, U. Kneissl, C. Kohstall, A. Linnemann, H.H. Pitz, G. Rainovski, A. Richter, M. Scheck, F. Stedile, P. von Brentano, P. von Neumann-Cosel, V. Werner, Phys. Rev. C 71 (2005) 044318.

N. Pietralla et al. / Progress in Particle and Nuclear Physics 60 (2008) 225–282 [115] [116] [117] [118] [119] [120] [121] [122] [123]

[124]

[125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136]

[137] [138]

[139] [140] [141] [142] [143] [144] [145] [146]

281

P. Paradis, G. Lamoureux, R. Lecomte, S. Monaro, Phys. Rev. C 14 (1976) 835. N. Pietralla, Czechoslovak J. Phys. 52 (2002) C607. N. Pietralla, C. Fransen, P. von Brentano, U. Kneissl, H.H. Pitz, AIP Conf. Proc. 529 (2000) 74. A.F. Lisetskiy, N. Pietralla, C. Fransen, R.V. Jolos, P. von Brentano, Nuclear Phys. A 677 (2000) 100. N. Lo Iudice, Ch. Stoyanov, Phys. Rev. C 62 (2000) 047302. R. Zhang, Y. Luo, P. Ning, Europhys. Lett. 73 (2006) 520. A. Linnemann, C. Fransen, M. Gorska, J. Jolie, U. Kneissl, P. Knoch, D. M¨ucher, H.H. Pitz, M. Scheck, C. Scholl, P. von Brentano, Phys. Rev. C 72 (2005) 064323. N. Pietralla, C. Fransen, A.F. Lisetskiy, P. von Brentano, V. Werner, Proceedings of the 11th Int. Conference on Capture Gamma-Rays and Related Topics, Prague, 2002. C. Fransen, N. Pietralla, A.P. Tonchev, M.W. Ahmed, J. Chen, G. Feldman, U. Kneissl, J. Li, V.N. Litvinenko, B. Perdue, I.V. Pinayev, H.-H. Pitz, R. Prior, K. Sabourov, M. Spraker, W. Tornow, H.R. Weller, V. Werner, Y.K. Wu, S.W. Yates, Phys. Rev. C 70 (2004) 044317. V.N. Litvinenko, B. Burnham, M. Emamian, N. Hower, J.M.J. Madey, P. Morcombe, P.G. O’Shea, S.H. Park, R. Sachtschale, K.D. Straub, G. Swift, P. Wang, Y. Wu, R.S. Canon, C.R. Howell, N.R. Roberson, E.C. Schreiber, M. Spraker, W. Tornow, H.R. Weller, I.V. Pinayev, N.G. Gavrilov, M.G. Fedotov, G.N. Kulipanov, G.Y. Kurkin, S.F. Mikhailov, V.M. Popik, A.N. Skrinsky, N.A. Vinokurov, B.E. Norum, A. Lumpkin, B. Yang, Phys. Rev. Lett. 78 (1997) 4569. S.H. Park, V.N. Litvinenko, W. Tornow, C. Montgomery, Nucl. Instrum. Methods Phys. Res. 475 (2001) 425. N. Pietralla, Z. Berant, V.N. Litvinenko, S. Hartman, F.F. Mikhailov, I.V. Pinayev, G. Swift, M.W. Ahmed, J.H. Kelley, S.O. Nelson, R. Prior, K. Sabourov, A.P. Tonchev, H.R. Weller, Phys. Rev. Lett. 88 (2002) 012502. N. Pietralla, H.R. Weller, V.N. Litvinenko, M.W. Ahmed, A.P. Tonchev, Nucl. Instrum. Methods Phys. Res. 483 (2002) 556. N. Pietralla, M.W. Ahmed, C. Fransen, V.N. Litvinenko, A.P. Tonchev, H.R. Weller, AIP Conf. Proc. 656 (2003) 365. N. Lo Iudice, Ch. Stoyanov, Phys. Rev. C 69 (2004) 044312. N. Lo Iudice, Ch. Stoyanov, Phys. Rev. C 73 (2006) 037305. P. von Brentano, Annual Report, IfS Stuttgart (1999) (unpublished). A. Giannatiempo, A. Nannini, A. Perego, P. Sona, G. Maino, Phys. Rev. C 44 (1991) 1508. M. D´el`eze, S. Drissi, J. Kern, P.A. Tercier, J.P. Vorlet, J. Rikovska, T. Otsuka, S. Judge, A. Williams, Nuclear Phys. A551 (1993) 269. M. D´el`eze, S. Drissi, J. Jolie, J. Kern, J.P. Vorlet, Nuclear Phys. A554 (1993) 1. C. Kohstall, D. Belic, P. von Brentano, C. Fransen, A. Gade, R.-D. Herzberg, J. Jolie, U. Kneissl, A. Linnemann, A. Nord, N. Pietralla, H.H. Pitz, M. Scheck, F. Stedile, V. Werner, S.W. Yates, Phys. Rev. C 72 (2005) 034302. H. von Garrel, P. von Brentano, C. Fransen, G. Friessner, N. Hollmann, J. Jolie, F. K¨appeler, L. K¨aubler, U. Kneissl, C. Kohstall, L. Kostov, A. Linnemann, D. M¨ucher, N. Pietralla, H.H. Pitz, G. Rusev, M. Scheck, K.D. Schilling, C. Scholl, R. Schwengner, F. Stedile, S. Walter, V. Werner, K. Wisshak, Phys. Rev. C 73 (2006) 054315. H. Maser, N. Pietralla, P. von Brentano, R.-D. Herzberg, U. Kneissl, J. Margraf, H.H. Pitz, A. Zilges, Phys. Rev. C 54 (1996) R2129. M. Scheck, H. von Garrel, N. Tsoneva, D. Belic, P. von Brentano, C. Fransen, A. Gade, J. Jolie, U. Kneissl, C. Kohstall, A. Linnemann, A. Nord, N. Pietralla, H.H. Pitz, F. Stedile, C. Stoyanov, V. Werner, Phys. Rev. C 70 (2004) 044319. M. Al-Hamidi, et al., Russian Nucl. Phys. 57 (1994) 579. A. Gade, N. Pietralla, P. von Brentano, D. Belic, C. Fransen, U. Kneissl, C. Kohstall, A. Linnemann, H.H. Pitz, M. Scheck, N.A. Smirnova, F. Stedile, V. Werner, Phys. Rev. C 69 (2004) 054321. Th. Eckert, O. Beck, J. Besserer, P. von Brentano, R. Fischer, R.-D. Herzberg, U. Kneissl, J. Margraf, H. Maser, A. Nord, N. Pietralla, H.H. Pitz, S.W. Yates, A. Zilges, Phys. Rev. C 56 (1997) 1256; 57 (1998) 1007. W. Ziegler, C. Rangacharyulu, A. Richter, C. Spieler, Phys. Rev. Lett. 65 (1990) 2515. W. Ziegler, N. Huxel, P. von Neumann-Cosel, C. Rangacharyulu, A. Richter, C. Spieler, Nuclear Phys. A 564 (1993) 366. M.R. Bhat, Nucl. Data Sheets 89 (2000) 797. L.I. Govor, A.M. Demidov, Yu.K. Cherepantsev, Yad. Fiz. 60 (10) (1997) 1733; Phys. Atomic Nuclei 60 (1997) 1579. N. Pietralla, C. Fransen, A. Gade, N.A. Smirnova, P. von Brentano, V. Werner, S.W. Yates, Phys. Rev. C 68 (2003) 031305(R).

282

N. Pietralla et al. / Progress in Particle and Nuclear Physics 60 (2008) 225–282

[147] S.P. Collins, S.A. Eid, S.A. Hamada, W.D. Hamilton, F. Hoyler, J. Phys. G 15 (1989) 321. [148] Hitoshi Nakada, Takaharu Otsuka, Phys. Rev. C 55 (1997) 748; Hitoshi Nakada, Takaharu Otsuka, Phys. Rev. C 55 (1997) 2418. [149] National Nuclear Data Center, A = 56, adopted value from various sources. http://www.nndc.bnl.gov/useroutput/ AR 16921 1.html, 2006. [150] Hitoshi Nakada, Takaharu Otsuka, Takashi Sebe, Phys. Rev. Lett. 67 (1991) 1086. [151] F. Bauwens, J. Bryssinck, D. De Frenne, K. Govaert, L. Govor, M. Hagemann, J. Heyse, E. Jacobs, W. Mondelaers, V.Yu. Ponomarev, Phys. Rev. C 62 (2000) 024302. [152] F.W. Fearick, G. Hartung, K. Langanke, G. Martinez-Pinedo, P. von Neumann-Cosel, A. Richter, Nuclear Phys. A727 (2003) 41. [153] W. Donner, W. Greiner, Z. Phys. 197 (1966) 440. [154] P.A. Butler, W. Nazarewicz, Rev. Modern Phys. 68 (1996) 349. [155] N. Pietralla, Phys. Rev. C 59 (1999) 2941. [156] E. Lipparini, S. Stringari, Phys. Rep. 175 (1989) 103. [157] N. Pietralla, P. von Brentano, R.-D. Herzberg, U. Kneissl, J. Margraf, H. Maser, H.H. Pitz, A. Zilges, Phys. Rev. C 52 (1995) R2317. [158] P. von Neumann-Cosel, J.N. Ginocchio, H. Bauer, A. Richter, Phys. Rev. Lett. 75 (1995) 4178. [159] N. Pietralla, P. von Brentano, R.-D. Herzberg, U. Kneissl, N. Lo Iudice, H. Maser, H.H. Pitz, A. Zilges, Phys. Rev. C 58 (1998) 184. [160] J. Enders, H. Kaiser, P. von Neumann-Cosel, C. Rangacharyulu, A. Richter, Phys. Rev. C 59 (1999) R1851. [161] J. Enders, P. von Neumann-Cosel, C. Rangacharyulu, A. Richter, Phys. Rev. C 71 (2005) 014306. [162] J. Margraf, C. Wesselborg, R.D. Heil, U. Kneissl, H.H. Pitz, C. Rangacharyulu, A. Richter, H.J. W¨ortche, W. Ziegler, Phys. Rev. C 45 (1992) R521. [163] C.J. McKay, J.N. Orce, S.R. Lesher, D. Bandyopadhyay, M.T. McEllistrem, C. Fransen, J. Jolie, A. Linnemann, N. Pietralla, V. Werner, S.W. Yates, Eur. Phys. J. A 25 (Suppl. 1) (2005) 377. [164] E. Lipparini, S. Stringari, Phys. Rev. Lett. 63 (1989) 570. [165] D. Gu´ery-Odelin, S. Stringari, Phys. Rev. Lett. 83 (1999) 4452. [166] V.O. Nesterenko, W. Kleinig, F.F. de Souza Cruz, N. Lo Iudice, Phys. Rev. Lett. 83 (1999) 57. [167] L. Serra, A. Puente, E. Lipparini, Phys. Rev. B 60 (1999) R13966. [168] O.M. Marag`o, S.A. Hopkins, J. Arlt, E. Hodby, G. Hechenblaikner, C.J. Foot, Phys. Rev. Lett. 84 (2000) 2056. [169] Onofrio Marag`o, Gerald Hechenblaikner, Eleanor Hodby, Christopher Foot, Phys. Rev. Lett. 86 (2001) 3938. [170] P.-G. Reinhard, V.O. Nesterenko, E. Suraud, S. El Gammal, W. Kleinig, Phys. Rev. A 66 (2002) 013206. [171] M. Modugno, G. Modugno, G. Roati, C. Fort, M. Inguscio, Phys. Rev. A 67 (2003) 023608. [172] Kenichi Kasamatsu, Makoto Tsubota, Masahito Ueda, Phys. Rev. A 69 (2004) 043621. [173] M. Rodr´ıguez, P. Pedri, P. T¨orm¨a, L. Santos, Phys. Rev. A 69 (2004) 023617. [174] Keisuke Hatada, Kuniko Hayakawa, Fabrizio Palumbo, Phys. Rev. B 71 (2005) 092402. [175] Carlos Lobo, Yvan Castin, Phys. Rev. A 72 (2005) 043606. [176] J.N. Orce, J.D. Holt, A. Linnemann, C.J. McKay, S.R. Lesher, C. Fransen, J.W. Holt, A. Kumar, N. Warr, V. Werner, J. Jolie, T.T.S. Kuo, M.T. McEllistrem, N. Pietralla, S.W. Yates, Phys. Rev. Lett. 97 (2006) 062504. [177] E. Elhami, J.N. Orce, S. Mukhopadyay, S.N. Choudry, M. Scheck, M.T. McEllistrem, S.W. Yates, Phys. Rev. C 75 (2007) 011301(R). [178] J.N. Orce, C. Fransen, A. Linnemann, C.J. McKay, S.R. Lesher, N. Pietralla, V. Werner, G. Friessner, C. Kohstall, D. M¨ucher, H.H. Pitz, M. Scheck, C. Scholl, F. Stedile, N. Warr, S. Walter, P. von Brentano, U. Kneissl, M.T. McEllistrem, S.W. Yates, Phys. Rev. C 75 (2007) 014303. [179] T. Ahn, N. Pietralla, G. Rainovski, A. Costin, K. Dusling, T.C. Li, A. Linnemann, S. Pontillo, Phys. Rev. C 75 (2007) 014313. [180] S.R. Lesher, C.J. McKay, M. Mynk, D. Bandyopadhyay, N. Boukharouba, C. Fransen, J.N. Orce, M.T. McEllistrem, S.W. Yates, Phys. Rev. C 75 (2007) 034318. [181] N. Pietralla, G. Rainovski, T. Ahn, A. Costin, Nuclear Phys. A 788 (2007) 85c. [182] O. Burda, N. Botha, J. Carter, R.W. Fearick, S.V. F¨ortsch, C. Fransen, H. Fujita, J.D. Holt, M. Kuhar, A. Lenhardt, P. von Neumann-Cosel, R. Neveling, N. Pietralla, V.Yu. Ponomarev, A. Richter, O. Scholten, E. Sideras-Haddad, F.D. Smit, J. Wambach, Phys. Rev. Lett. 99 (2007) 092503.