Physics Department Institut für Kernphysik Pietralla Group Probing Nuclear Structure Relevant for Neutrinoless Double-Beta Decay with Nuclear Resonance Fluorescence Kernstrukturuntersuchungen mit Bezug zum neutrinolosen doppelten Betazerfall mit der Kernresonanzfluoreszenzmethode Zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation im Fachbereich Physik von Udo Friman-Gayer aus Miltenberg, BY, DE Tag der Einreichung: 19. November, 2019, Tag der Prüfung: 16. Dezember, 2019 1. Gutachten: Prof. Dr. Dr. h.c. mult. Norbert Pietralla 2. Gutachten: Prof. Dr. Wilfried Nörtershäuser Darmstadt – D 17 Probing Nuclear Structure Relevant for Neutrinoless Double-Beta Decay with Nuclear Resonance Fluorescence Kernstrukturuntersuchungen mit Bezug zum neutrinolosen doppelten Betazerfall mit der Kernresonanzfluoreszenzmethode Doctoral thesis in Physics by Udo Friman-Gayer 1. Review: Prof. Dr. Dr. h.c. mult. Norbert Pietralla 2. Review: Prof. Dr. Wilfried Nörtershäuser Date of submission: 19. November, 2019 Date of thesis defense: 16. Dezember, 2019 Darmstadt – D 17 Bitte zitieren Sie dieses Dokument als: URN: urn:nbn:de:tuda-tuprints-113857 URL: https://tuprints.ulb.tu-darmstadt.de/id/eprint/11385 Dieses Dokument wird bereitgestellt von tuprints, E-Publishing-Service der TU Darmstadt http://tuprints.ulb.tu-darmstadt.de tuprints@ulb.tu-darmstadt.de This work is licensed under a Creative Commons “Attribution-NonCommercial- NoDerivatives 4.0 International” license. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en https://tuprints.ulb.tu-darmstadt.de/id/eprint/11385 http://tuprints.ulb.tu-darmstadt.de tuprints@ulb.tu-darmstadt.de https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en Erklärungen laut Promotionsordnung §8 Abs. 1 lit. c PromO Ich versichere hiermit, dass die elektronische Version meiner Dissertation mit der schriftlichen Version übereinstimmt. §8 Abs. 1 lit. d PromO Ich versichere hiermit, dass zu einem vorherigen Zeitpunkt noch keine Promotion versucht wurde. In diesem Fall sind nähere Angaben über Zeitpunkt, Hochschule, Dissertationsthema und Ergebnis dieses Versuchs mitzuteilen. §9 Abs. 1 PromO Ich versichere hiermit, dass die vorliegende Dissertation selbstständig und nur unter Verwendung der angegebenen Quellen verfasst wurde. §9 Abs. 2 PromO Die Arbeit hat bisher noch nicht zu Prüfungszwecken gedient. Darmstadt, den 19. November, 2019 U. Friman-Gayer 3 Table of Contents Table of Contents 5 Nomenclature 9 1. Introduction 33 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure . . . . . . . . 33 1.1.1. A Simple Model: Hypothetical 0νββ Decay of the Dineutron 37 1.2. Current State of Nuclear Structure Input . . . . . . . . . . . . . . . . . 51 1.3. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2. Background 59 2.1. Nuclear Resonance Fluorescence . . . . . . . . . . . . . . . . . . . . . . 59 2.1.1. General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.1.2. Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2. Nuclear Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.2.1. Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.2.2. Interacting Boson Model . . . . . . . . . . . . . . . . . . . . . . 71 2.3. Scissors Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5 3. Experiments 83 3.1. High-Intensity γ-ray Source (HIγS) . . . . . . . . . . . . . . . . . . . . 83 3.2. Experimental Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.3. Experiments on 82Kr and 82Se . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4. Experiments on 150Nd and 150Sm . . . . . . . . . . . . . . . . . . . . . . 92 4. Analysis 93 4.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1.1. Propagation of Uncertainty . . . . . . . . . . . . . . . . . . . . . 93 4.1.2. Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1.3. Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 97 4.1.4. Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1.5. Numerical Evaluation and Visualization . . . . . . . . . . . . . 98 4.2. Simulation of the Experimental Setup . . . . . . . . . . . . . . . . . . . 98 4.3. Spectrum Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3.1. Original Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3.2. Detector Response . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3.3. Pileup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4. Sensitivity Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.2. Spectroscopic Sensitivity Limit . . . . . . . . . . . . . . . . . . . 106 4.5. Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5.1. Energy Calibration and Binning . . . . . . . . . . . . . . . . . . 108 4.5.2. Efficiency Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.3. Width Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6 TABLE OF CONTENTS 4.5.4. Photon Flux Calibration . . . . . . . . . . . . . . . . . . . . . . . 121 4.6. Derived Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.6.1. Quantum Numbers and Multipole Mixing Ratio . . . . . . . . 137 4.6.2. Branching Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.6.3. Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5. Results 143 6. Discussion 153 6.1. 82Se and 82Kr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2. 150Nd and 150Sm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7. Summary and Outlook 159 A. Spectroscopic Sensitivity Limit 161 A.1. Evaluation for Theuerkauf Lineshape Model . . . . . . . . . . . . . . . 167 B. Schematic Detector Setups 171 C. List of Gamma Energies 175 D. Spectra 185 Bibliography 351 Acknowledgements 371 Academic CV 375 TABLE OF CONTENTS 7 Nomenclature This section contains a list of all symbols and indices which are used in this work. The nomenclature complies with the following general rules: • Operator- and operator-like symbols are typed in bold characters. • Operator-like quantities are displayed with a caret, i.e. t̂. • Lowercase operator symbols denote single-particle operators. An uppercase symbol denotes the corresponding manybody operator, which is a sum of the single-particle operators, e.g. T̂= ∑A a=1 t̂a. The nomenclature contains only the lowercase symbols. • The eigenvalues of operator-like quantities are denoted with the same symbol as the operator, but without the caret and as regular text. For example, the eigenvalue equation for the operator t̂z is: t̂z |tz〉= tz |tz〉. One exception of this rule is the definition lz ≡ ml of the magnetic quantum number for the eigenvalues of angular moment operators for historical reasons and better readability. • The absolute value |r| of a boldface symbol is displayed as regular text (|r| ≡ r). • The components of up-to three-dimensional operator- and vector-like symbols are denoted by the indices x , y and z as regular text. For example, a three- dimensional vector r has the components rx , ry , and rz. The components of vectors with arbitrary (nX) dimensions are denoted by numbers starting from 1, i.e. X= � X1, X2, ...XnX �T . 9 • Raising and lowering operators are denoted by an additional ’+’ and ’-’ sign, respectively, for example t̂+ and t̂−. • General indices for a set of values, for example i for the set of states of a nucleus, are taken from the Latin or Greek alphabet. Additional indices are added as they follow in the respective alphabet, for example i is followed by j, k, .... The set itself is denoted by an uppercase letter, for example: k ∈ K. • The average/expectation value of a quantity is denoted in angular brackets, i.e. 〈X 〉 is the average value of X . • Nuclear reactions are denoted as ’(in, out)’, where ’in’ is the incoming pro- jectile and ’out’ a set of ejectiles, in accordance with the Evaluated Nuclear Structure Data File (ENSDF) manual [1]. • Particles are denoted by the symbols recommended by the Particle Data Group (PDG) [2]. In particular, the word ’photon’ and the symbol γ will be used. The word ’gamma ray’ was avoided, if possible, due to its conflicting definitions in the literature. 2νββ Two-neutrino double beta α j Expansion parameter for the representation of a nucleon-nucleon pair with J = 0 and MJ = 0 in terms of shell-model orbitals. ex/y/z Unit vector in x-, y- or z direction. jγ(Eγ, r , t) Photon flux density, i.e. directional number of photons per area and time unit with an energy Eγ at a space point r . P Vector of parameters. ra Single-particle position operator. rab Distance vector of two particles a and b. X Vector of input quantities. Y Vector of output quantities. 10 Nomenclature Ȧ(t) Time-dependent activity of a radioactive source in decays per time interval. dIi→ j→k dΩ Energy-integrated differential cross section for the absorption of a photon by a nucleus in the state i to an excited state j, and the subsequent decay to a state k. †( j j′; J , MJ ) Operator which creates a pair of nucleons from orbitals j and j′ with a total angular momentum J and a z projection MJ . ↠i , âi Creation/annihilation operator for a fermion in the state i. D̂† Operator which creates a pair of nucleons with J = 2 and a z projec- tion MJ in a given valence space. d̂µ, d̂† µ Annihilation/creation operator for a d boson state with a z compo- nent µ. f̂ Single-particle F -spin operator. Ĥ Hamilton operator. k̂a Single-particle kinetic energy operator k̂a = p̂2 a/2ma. l̂ Orbital angular momentum operator. Ô(0ν) Operator for neutrinoless double-beta decay. Ô(0ν)F Fermi-part of the operator for neutrinoless double-beta decay. Ô(0ν)GT Gamow-Teller part of the operator for neutrinoless double-beta decay. P̂v Valence-space projection operator. Q̂χ Quadrupole operator in the IBM with the parameter χ. Q̂v Complement of the valence-space projection operator, i.e. Q̂v = 1− P̂v . Nomenclature 11 ŝ , ŝ† s-boson annihilation/creation operator. Ŝ† Operator which creates a pair of nucleons with J = 0 and MJ = 0 in a given valence space. ŝa Single-particle spin operator. t̂ Single-particle isospin operator. The tz = 1/2 and tz = −1/2 eigen- states of this operator are the proton and the neutron, respectively. Ûa Mean-field potential energy of particle a. V̂2N ab General two-body potential between particles a and b. ˆ̃dµ Modified annihilation operator for a d boson state with a z compo- nent µ ( ˆ̃dµ = (−1)µd̂−µ). ħh Reduced Planck constant (6.582119596× 10−16 eVs [3]). 〈mν〉 Effective light neutrino mass in 0νββ decay. 〈Wi→ j→kεd(Eγ)〉 Abbreviation for the energy-dependent solid-angle integral over the product of the angular distribution of an NRF cascade i→ j→ k and the full-energy peak efficiency of a detector d. This integral appears in the expression for the number of NRF events which are counted by a detector. 〈X 〉t Time average of a quantity X . O (x) ’Big O notation’ or ’Landau symbol’ which indicates the limiting behavior of a function when x is ’small’ according to some criterium. B(σL; i→ j) Lower limit for the reduced transition probability B(σL; i→ j). In principle, all branching transitions of an excited state need to be known in order to determine the ’true’ reduced transition probabili- ties. This notation indicates that some of them may not have been observed. 12 Nomenclature P Projection operator onto states with good seniority. Ci(x) Cosine integral Ci(x) = − ∫∞ x cos(t) t dt. sc±(X ) Upper (’+’) and lower (’-’) limit of the shortest coverage interval of the quantity X . Si(x) Sine integral Si(x) = ∫ x 0 sin(t) t dt. dAi→k,d/dt Count rate of events corresponding to a transition from a state i to a state j observed by a detector d. gV/gA Ratio of the vector- and axial-vector coupling constants of the weak interaction (the experimental value of the inverse ratio is gA/gV = 1.27641(45)stat.(33)syst. [4]). N(Em) Original energy spectrum before the application of the detector response. See also the definition of N(Em). A Mass number (A= N + Z). A(Eγ) Number of events contained in a lineshape with the centroid energy Eγ. a0→ j→k,dd ′ Asymmetry of the number of counted events associated with a tran- sition from a state j to a state k between two detectors d and d ′. Ai→k,d Number of events corresponding to a transition from a state i to a state j observed by a detector d. ann Neutron-neutron s-wave scattering length. app Proton-proton s-wave scattering length. B(AZ X ) Binding energy of the nucleus A Z X . B(E, pB) Normalized probability distribution of the continuous background in a spectrum, which may depend on a vector of parameters pB. Nomenclature 13 c Speed of light (c = 2.99792458× 108 ms−1 [3]). D(Em, En) Detector response matrix which connects the energy bin En of the original spectrum to the energy bin Em of the detected spectrum. DX Set of all detectors in a setup X . Eγ Energy of a photon. Ebeam Nominal beam energy of the HIγS beam, which can be seen as the centroid of the approximately Gaussian beam profile. eπ, eν Charges of proton- and neutron bosons in the E2 operator of the IBM-2. Ei Excitation energy of state i. Em Energy of the m-th bin. ee even-even eo, oe even-odd, odd-even f (X , P) Arbitrary function f of a vector of input quantities X and parameters P. Fmax Maximum projection of the F -spin for a given number of proton- and neutron bosons: Fmax = 1/2 � Nν,s + Nν,d + Nπ,s + Nπ,d � . gπ, gν Proton- and neutron g factors in the M1 operator of the IBM-2. gπl , gνl Orbital g factors in the M1 operator for protons and neutrons. Their bare values are given by gπl = 1 and gνl = 0 (see, e.g., Sec. V.B.III in [5]). gπs , gνs Spin-g factors in the M1 operator for protons and neutrons with their bare values gπs = 5.5856946893(16) and gνs = −3.382608545(90) [6]. 14 Nomenclature G0ν Integrated kinematical/phase space factor in 0νββ decay. gX (ξ) Probability distribution function of the vector X . The symbol ξ denotes a vector of variables for all possible values of X . gi→ j Ratio of the number of Jz substates of the initial and final state for a transition from state i to j (’spin factor’ J j+1/Ji+1). H(r, 〈E〉) Neutrino potential in the Fermi- and Gamow-Teller part of the 0νββ decay operator, which depends on the distance r between two nucleons and the average energy difference 〈E〉 between an excited state of the intermediate nucleus and the mean value of the initial and final state of the 0νββ decay. Irel(Eγ) Relative intensity of a photon with the energy Eγ which is emitted by a radioactive source. Irel = 1 means that each decay of the source creates exactly one photon of this energy. Ii→ j→k Total cross section for the absorption of a photon by a nucleus in the state i to an excited state j, and the subsequent decay to a state k. Ii→ j Total cross section for the absorption of a photon by a nucleus in the state i to an excited state j. LC Threshold for a 95% confidence for the absence of activity. Li→ j Multipole order of an EM transition between states i and j (L ∈ N). The index for the corresponding transition is dropped if the initial and final state are clear from the context. Lαk Associated/generalized Laguerre polynomial. M Total mass of a system of particles. M(AZ X ), M(Z , A) Mass of the nucleus A Z X . M(Em, En) Pileup matrix which connects the energy bin En of the original spectrum to the energy bin Em of the pileup spectrum. Nomenclature 15 M (0ν) Nuclear matrix element for neutrinoless double-beta decay. me Electron mass (9.1093837015(28)×10−31 kg [6]). ml z projection of the orbital angular momentum operator (’magnetic quantum number’). mn Neutron mass (1.00866491582(49) u [7]). mn Proton mass (1.00782503224(09) u [7]). N (Em) Content of the bin of the spectrum N with the centroid energy Em. May also be denoted as Nm for brevity or to emphasize that a channel-energy mapping does not exist. In particular, the bare letter N denotes an actually observed spectrum. When N is split up into different contributions, subscripts, superscripts, or diacritics are used in this work. N Neutron number or principal quantum number of the harmonic oscillator. N(µ,σ) Normal distribution with mean value µ and standard deviation σ. n2S+1lJ Spectroscopic notation, which contains the radial (n), total spin (S), angular momentum (l) and total angular momentum (J) quantum number of a manybody state. N (0)(Em) Energy spectrum that would be detected if every single event could be resolved. Corresponds to ’zero-order pileup’. N (p)(Em) p-th order pileup correction to the spectrum N (0)(Em) with p ≥ 1. Nγ(Eγ) Number of photons with an energy Eγ. Nγ Total number of photons which hit the target. Nfit(Em) Fit function which is assumed to describe the shape of the spectrum N(Em). 16 Nomenclature NB Number of background events. ND Threshold for a 95% confidence for the presence of activity OR number of nucleon-nucleon pairs with J = 2 and a z component MJ . Nd Number of d bosons. Nm See N(Em). NR Number of randomly sampled values. NS Number of events caused by artificial activity OR number of nucleon- nucleon pairs with J = 0 and MJ = 0. Ns Number of bins of spectrum s OR number of s bosons. NT Normalization factor of the Theuerkauf line shape model. Nt Number of target nuclei. nt(r ) Number of target nuclei per unit volume at a point r . NB,Σ Total number of background events in a spectrum. Nnl Normalization constant of the harmonic oscillator wave function. oo odd-odd P(E, pP) Normalized probability distribution (’line shape’) of a peak, which may depend on a vector of parameters pP . Pγ(Eγ) Polarization factor which depends on the beam energy (Pγ ∈ [−1, 1]). Pν Legendre polynomial of degree ν. Pµν Unnormalized Legendre polynomial of degree ν and order µ. PT (Em,µ,σ, t l , t r) Theuerkauf model for the line shape. Nomenclature 17 Qββ Q value of a neutrinoless or a two-neutrino double-beta decay. Q0νββ ≈Q2νββ for negligible neutrino masses. Rnl(r) Radial part of the harmonic oscillator wave function. tdead Dead time of a detector. tlive Live time of a detector. tstart Point in time when an experimental run starts. tstop Point in time when an experimental run stops. t l Left-tail parameter of the Theuerkauf line-shape model. t r Right-tail parameter of the Theuerkauf line-shape model. T1/2 Half life of a decay process (T1/2 = ln(2)τ). VT (θ ) Factorized part of the transversal inelastic electron scattering cross section which almost only depends on the scattering angle if the electron energy is much larger than the excitation energy of the nucleus. Wi→ j→k,polarized(θ ,ϕ) Termwhich is added to (or subtracted from, depending on the EM character) the unpolarized angular distributionWi→ j→k,unpolarized(θ ) to take into account the excitation from i to j by a completely po- larized photon beam. Wi→ j→k,unpolarized(θ ) Angular distribution of the emitted photon in the transition from state j to k during the two-step cascade between states i, j and k. The index ’unpolarized’ indicates that the excitation is assumed to be caused by an unpolarized photon beam. Normalized to 4π. 18 Nomenclature Wi→ j→k(θ ,ϕ,δi→ j ,δ j→k, Pγ) Angular distribution of the emitted photon in the tran- sition from state j to k during the two-step cascade between states i, j and k. It depends on the angular momentum- and parity quan- tum numbers of the involved states and the multipole mixing ratios δi→ j and δ j→k. The excitation is assumed to be caused by a photon beam which has an energy-dependent polarization Pγ. Normalized to 4π. Xrand Random value drawn from the probability distribution gX of the quantity X . y (i) Auxiliary, energy-dependent quantity in the integrated phase space factor. Ylml (θ ,ϕ) Spherical harmonics. Z Proton number. 0νββ Neutrinoless double-beta N Set of positive integer numbers including zero. β j j′ Expansion parameter for the representation of a nucleon-nucleon pair with J = 2 and a z component MJ in terms of shell-model orbitals. n̂d Single-d boson number operator. CCD Charge-coupled device COM Center of mass DIN Deutsches Institut für Normung ENSDF Evaluated Nuclear Structure Data File EW Electroweak exp experimental Nomenclature 19 F Fermi FEP Full-energy peak FFT Fast Fourier transform FWHM Full width at half maximum GEANT4 Geometry and Tracking 4. Software framework for the simulation of the passage of particles through matter [8–10]. GT Gamow-Teller HIγS High-Intensity γ-ray Source HO Harmonic oscillator HPGe High-purity Germanium IBM Interacting boson model IBM-2 Proton-neutron version of the interacting boson model INT Intrinsic LCB Laser Compton backscattering MC Monte Carlo n Radial quantum number of the harmonic oscillator i.e. number of nodes of the radial wave function OR index for an arbitrary nucleon, i.e. n ∈ {ν,π}. NME Nuclear matrix element NRF Nuclear resonance fluorescence ODR Orthogonal distance regression 20 Nomenclature PDF Probability distribution function PDG Particle Data Group PVC Polyvinyl chloride resi residual sc Scissors mode SM (Nuclear) Shell model stat. Statistical contribution to the total uncertainty. syst. Systematic contribution to the total uncertainty. theo. Theory contribution to the total uncertainty. u Atomic mass unit (931.49410242(28)MeV c−2. [6]) B(σL; i→ j) Reduced transition probability for the σL transition from state i to j. This quantity is given in units of eV fm(2L+1) in the SI [3] system of units. For conversion to the commonly used cgs [11] system of units, use the relations e2 cgs = 1.4399764MeVfm and µN ,cgs = 0.10515446efm. α Fine-structure constant (0.0072973525693(11)≈ 1/137 [6]). ββ Double beta χ2 Chi square statistic. Measure for the agreement between a model and measured data. χ Spin part of the total wave function. χ2 red Reduced chi square. χπ,χν Strength of the term in the proton- or neutron-quadrupole operator which controls the γ-softness. Nomenclature 21 dΩ Solid angle element. In spherical coordinates: dΩ= sin(θ )dθdϕ. ∆E General notation for an energy difference. ∆EN (Em) Width of a single bin of spectrum N . In the case of equidistant binning, the dependence on the centroid energy Em is omitted. ∆N(Em) Residual of the bin with centroid energy Em after subtracting a fit function Nfit(Em) from a spectrum N(Em). ∆T ’Shaping time’ of a trapezoidal filter. δ(x) Delta function. δp Pairing energy in the semiempirical mass formula. δL,i→ j Multipole mixing ratio in the convention of Krane, Steffen and Wheeler [12]: Ratio of the reduced transition widths of EM character and multipole order σL and σ′(L + 1) with σ 6= σ′ for a transition from state i to state j. ε(Eγ, r ) Energy-dependent full-energy peak efficiency for the detection of photons from an isotropic source at r . The parameter r may be omitted if the origin of the photons is clear. εint Intrinsic detection efficiency of a detector. εd Single-d boson energy in the IBM-2 Hamiltonian. γγ Double gamma Γ Gamma function. γ1 Auxiliary, Z-dependent quantity in the integrated phase space factor. Γi Total width of a state i. 22 Nomenclature Γi→ j Partial transition width from the state i to the state j. For the EM process of NRF, the detailed balance theorem holds (chapter X.2.E in [13]) and implies that Γi→ j = Γ j→i . σ̂a Single-particle Pauli matrix, which is related to the single-particle spin operator via ŝa = ħh/2σ̂a. κ Strength of the quadrupole-quadrupole interaction in the IBM-2 Hamiltonian. λ Overall strength of the Majorana operator in the IBM-2 Hamiltonian. λ Reduced wave length, i.e. de-Broglie wave length of a particle divided by 2π. If indexed with a transition label, i.e. λi→ j ≡ λ j→i , it denotes the reduced wave length which corresponds to the energy difference of the excited states. 〈Φ(Eγ)〉t Energy-dependent time-averaged photon flux. µ Mean value of a quantity OR the reduced mass of a system of parti- cles. µ(Eγ) Energy-dependent mass attenuation coefficient of a material. ν Symbol for the neutron OR a parameter of the quantum-mechanical harmonic oscillator wave function, which can be interpreted as the inverse squared length scale of the oscillator OR the seniority quantum number. ω Oscillator frequency multiplied by 2π. Φ Basis states which do not have to be eigenstates of the full Hamilto- nian of a system. Φγ(Eγ, t) Energy- and time-dependent photon flux on target, i.e. number of photons with an energy Eγ which hit the target per time interval at a time t. Nomenclature 23 π Symbol for the proton. πi Parity quantum number of state i. Ψ Eigenstates of the full Hamiltonian for a given system. ψnlml (r,θ ,ϕ) Harmonic oscillator wave function. 〈σN 〉 Standard deviation of the bin contents of spectrum N in an energy range where the expectation value for each bin is approximately equal. ρ Density of a material. σL Notation for an electromagnetic transition of character σ ∈ {E, M} and multipole order L ∈ N. σ Electromagnetic character. Either E for ’electric’ or M for ’magnetic’. σ(E) Energy-dependent width of a peak with a centroid energy E and a normal-distributed line shape. σ f (X , P) Uncertainty of the function value f (X , P) σN (Em) Uncertainty of the bin of the spectrum N with the centroid energy Em σX Uncertainty of the quantity X . Depending on the context, this symbol either denotes the 68.27% shortest coverage interval or the standard deviation. σi→ j(Eγ) Cross section for the absorption of a photon with an energy Eγ by a nucleus in a state i, leaving it in a state j. σX ,± Upper (’+’) and lower (’-’) limit of the uncertainty of the quantity X . τ Lifetime of excited nuclear states or, in general, the inverse of the decay parameter of an exponential function. 24 Nomenclature θ Polar angle in spherical coordinates. ϕ Azimuthal angle in spherical coordinates. ξ Isospin part of the total wave function. ξ1,2,3 Strength of the different terms in the Majorana operator in the IBM-2 Hamiltonian. dε dΩ′ (Eγ,θ ′,ϕ′, r ) Energy-dependent full-energy peak efficiency for the detection of photons, emitted by a source at r in a direction given by θ ′ and ϕ′. 0 Index for the ground state of a nucleus. � ↠b̂†... � i Coupling of a set of single-particle creation operators â†, b̂†, ... to a creation operator for a multi-particle state with a set of quantum numbers summarized by the index i. a, b, ... Indices for single particles in an A-body system, i.e. 1≤ a, b, ...≤ A. i, j, ... Indices for states of a nucleus, both excited states and the ground state. m, n, ... Indices for the bins of a spectrum. Nomenclature 25 Abstract Neutrinoless double-beta (0νββ) decay is a hypothetical second-order process of the weak interaction, which, if observed, would reveal neutrinos as the first example of so-called Majorana particles which are their own antiparticles. Furthermore, since the decay rate for 0νββ decay is directly related to the effective mass of the electron neutrino, it would allow for a direct determination of the neutrino mass. However, an obstacle for the planning of future 0νββ-decay searches and for a quantitative extraction of the neutrino mass are currently the poorly constrained nuclear matrix elements which mediate the decay process. These matrix elements have the be supplied by nuclear theory, which is challenged with the phenomena of nuclear shape evolution and shape coexistence that prevail in regions of the nuclear chart where most 0νββ decay candidates are located. A major problem is the lack of sensitive experimental data, which are required to fix the parameters of effective theories. Based on a previous successful study, the nuclear structure of the candidate pairs 82Se/82Kr and 150Nd/150Sm was investigated in this work using the method of nu- clear resonance fluorescence. The observables of interest were the decay channels of a low-lying collective nuclear excitation, the scissors mode, which are expected to be highly sensitive to the location of the candidate pairs in the phase diagram of nuclear shapes. The scissors mode can be studied selectively and with a high degree of model indepedence with the chosen method. The experiments were performed at the High-Intensity Gamma-Ray Source which currently provides the most intense, linearly polarized, quasi-monochromatic photon beam at the ener- gies of interest. Using the high sensitivity of the polarized beam, magnetic dipole excitations, which are the manifestations of the scissors mode in even-even nuclei, were identified and their decay behavior was characterized. A known drawback of 27 experiments with monoenergetic photon beams, namely the lack of a photon-flux calibration, was solved in the present work without any additional instrumentation by calibrating the flux on the nonresonant scattering of photons on the targets. For this purpose, a detailed Monte-Carlo particle simulation application was developed. For all nuclei of interest, decay branches on the order of few percent could either be observed, or constrained to such small values. Two effective nuclear models, the shell model and the interacting boson model, which are also frequently used to predict 0νββ decay matrix elements, were used for a preliminary interpretation of the data. For the nucleus 82Se, the shell model gave a good description of the energies of excited 1+ states and the total observed strength. The good agreement allowed for an interpretation of the structure of the wave functions of the scissors mode candidates, which advised against a simple relation between the measured quantities and the shape coexistence in that nucleus. For the higher-mass isotopes, a careful parameter adjustment in the framework of the interacting boson model was able to reproduce the entire low-energy structure of 150Nd and, with minor exceptions also of 150Sm. The new parameter sets in this model were used together with our collaborator to update previous predictions of nuclear matrix elements for 0νββ decay. Note that the analysis of the data on the A= 150 nuclei was done by Jörn Kleemann. This work presents only his main results. 28 Abstract Zusammenfassung Der neutrinolose doppelte Betazerfall (0νββ-Zerfall) ist ein hypothetischer Prozess zweiter Ordnung in der schwachen Wechselwirkung, der, im Falle einer experi- mentellen Beobachtung, Neutrinos als das erste Beispiel von sogenanntenMajorana- Teilchen identifizieren würde, welche ihre eigenen Antiteilchen sind. Außer- dem würde es eine direkte Bestimmung der Neutrinomasse ermöglichen, da die Zerfallsrate für den 0νββ-Zerfall im direkten Zusammenhang mit der effektiven Masse des Elektron-Neutrinos steht. Ein Hindernis für die Planung von zukünftigen Experimenten, die nach dem 0νββ-Zerfall suchen, und für eine quantitative Bes- timmung der Neutrinomasse sindmomentan die nicht ausreichend eingeschränkten Kern-Matrixelemente die dem Zerfallsprozess innewohnen. Diese Matrixelemente müssen von der Kernstrukturtheorie bereitgestellt werden, die vor dem Prob- lem der Beschreibung von Phänomenen wie der Entwicklung von Kerngestalten und der Koexistenz von Kerngestalten steht, welche in Regionen der Nuklidkarte vorherrschen, in denen auch die Kandidatenpaare für den 0νββ-Zerfall zu finden sind. Ein Hauptproblem ist der Mangel an aussagekräftigen experimentellen Daten, die benötigt werden um die freien Parameter von effektiven Theorien einzuschränken. Basierend auf einer vorherigen Studie wurde die Kernstruktur der Kandidaten- paare 82Se/82Kr und 150Nd/150Sm in dieser Arbeit mit der Methode der Kernres- onanzfluoreszenz untersucht. Zerfallskanäle einer niedrigliegenden, kollektiven Kernanregung, der Scherenmode, waren die wichtigen Observablen, denn sie sind erwartungsgemäß höchst sensitiv auf die Lage der Kandidatenpaare im Phasendi- agramm der Kerngestalten. Mit der gewählten Methode kann die Scherenmode selektiv und mit einem hohen Grad an Modellunabhängigkeit untersucht werden. Die Experimente wurde an der High-Intensity Gamma-Ray Source durchgeführt, 29 die momentan im interessanten Energiebereich die intensivsten linear polarisierten quasi-monochromatischen Photonenstrahlen zur Verfügung stellt. Durch die hohe Sensitivität des polarisierten Strahls konnten magnetische Dipolübergänge, als welche sich die Scherenmode in gerade-gerade - Kernen manifestiert, identifiziert und ihr Zerfallsverhalten charakterisiert werden. Ein bekannter Nachteil von Mes- sungen mit monoenergetischen Photonenstrahlen, der Mangel an Möglichkeiten zur Kalibrierung des Photonenflusses, wurde in der vorliegenden Arbeit dadurch umgangen, dass, ohne Zuhilfenahme weiterer Messaufbauten, der Photonenfluss anhand der nichtresonanten Streuung von Gammastrahlung an der Probe kalibriert wurde. Zu diesem Zweck wurde eine detaillierte Monte-Carlo Simulationsanwen- dung entwickelt. Für alle betrachteten Kerne konnten Zerfallskanäle in der Größenordnung von weni- gen Prozenten entweder beobachtet, oder auf solch kleine Werte eingeschränkt werden. Zwei effektive Kernmodelle, das Schalenmodell und das Modell wechsel- wirkender Valenzbosonen, die auch häufig benutzt werden um 0νββ - Zerfallsma- trixelemente vorherzusagen, wurden für eine vorläufige Interpretation der Daten benutzt. Das Schalenmodell lieferte eine gute Beschreibung der Anregungsen- ergien von 1+-Zuständen und der gesamten beobachteten Stärke für den Kern 82Se. Die gute Übereinstimmung erlaubte eine Interpretation der Struktur der Wellenfunktionen der mutmaßlichen Scherenmodenfragmente, die jedoch gegen einen einfachen Zusammenhang zwischen den Messgrößen und der Koexistenz von Kerngestalten in diesem Kern spricht. Bei den Isotopen mit höherer Masse konnte eine sorgfältige Anpassung der Parameter des Modells wechselwirkender Valenzbosonen die gesamte Niederenergie-Kernstruktur von 150Nd und, mit Ab- strichen, auch die von 150Sm reproduzieren. Die neuen Parametersätze für dieses Modell wurden in Zusammenarbeit mit unserer Kollaborateurin benutzt, um bish- erige Vorhersagen von Kernmatrixelementen für den 0νββ-Zerfall auf den neusten Stand zu bringen. Es wird angemerkt, dass die Analyse der Daten für die Kerne mit der Massenzahl 150 von Jörn Kleemann durchgeführt wurde. In dieser Arbeit werden lediglich seine Hauptergebnisse aufgeführt. 30 Zusammenfassung Epigraph Aber noch schlimmer wurde es, wenn er auf die Wissenschaft zu sprechen kam, - an die er nicht glaubte. Er glaube nicht an sie, sagte er, denn es stehe dem Menschen völlig frei, an sie zu glauben oder nicht. Sie sei ein Glaube, wie jeder andere, nur schlechter und dümmer als jeder andere, und das Wort ”Wissenschaft” selbst sei der Ausdruck des stupidesten Realismus, der sich nicht schäme, die mehr als fragwürdigen Spiegelungen der Objekte im menschlichen Intellekt für bare Münze zu nehmen oder auszugeben und die geist- und trostloseste Dogmatik daraus zu bereiten, die der Menschheit je zugemutet worden sei. Ob etwa nicht der Begriff einer an und für sich existierenden Sinnenwelt der lächerlichste aller Selbstwidersprüche sei? Aber die moderne Naturwissenschaft als Dogma lebe einzig und allein von der metaphysischen Voraussetzung, daß die Erkenntnisformen unserer Organisation, Raum, Zeit und Kausalität, in denen die Erscheinungswelt sich abspiele, reale Verhältnisse seien, die unabhängig von unserer Erkenntnis existierten. Diese monistische Behauptung sei die nackteste Unverschämtheit, die man dem Geiste je geboten. Polemik der Figur Leo Naphta in T. Mann, “Der Zauberberg“, 1. Auflage, Fischer E-Books (2009) 31 1. Introduction 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure Recently, the experimental observations of two-neutrino double-beta (2νββ) decay [14] and double-gamma (γγ) decay [15–17] have been complemented by two more nuclear decays which are mediated by the electroweak (EW) interaction at second order: the competitive version of γγ decay [18] and the two-neutrino double electron capture (2νECEC) decay [19]. From the point of view of nuclear structure physics, the transition rates of these decays provide a quasi1 model- independent access to nuclear matrix elements with the schematic structure [see, e.g. [21] (2νββ) [18] (γγ), [22] (2νECEC)], ∑ n wn(E f − En, En − Ei , ...)〈 f � �Ô � �n〉〈n � �Ô′ � � i〉, (1.1) which is unsurprisingly very similar to the expressions found in textbook-second- order perturbation theory (see, e.g., chapter XVI in [23]). In Eq. (1.1), the transition rate between the initial (i) and the final ( f ) state of the nuclear system is given by a sum over transition matrix elements of operators Ô and Ô′ to interme- diate states (n). The symbols wn denote weighting factors which depend on the energy differences between the nuclear states and potentially also other variables. Accessing matrix elements like 〈 f � �Ô � �n〉 by direct reactions is challenging, and 1There are non-negligible discrepancies between predictions of the phase-space factors in the EW theory (for ββ decay, see, e.g., [20]). But generally, the nuclear matrix elements are by far the most uncertain parameters. For 0νββ decay, this will be discussed below. 33 probably also model-dependent to a higher degree (see, e.g., the proposed exper- imental study of ββ-decay-analog matrix elements by double charge exchange reactions by the NUMEN project [24] and a related theoretical investigation [25]). While the aforementioned decay processes are in agreement with the Standard Model of particle physics [26], the motivation for this work is the lepton-number violating process of neutrinoless double-beta (0νββ) decay. Recent review articles about the topic, on which the following introduction of 0νββ decay will be based, have been published by Vergados, Ejiri, and Šimkovic [27] (theory), and Avignone, Elliott, and Engel [28] (theory and experiment). After a first theoretical study by Goeppert-Mayer on the possibility of a double-beta (ββ) decay process [29], Racah proposed a ’neutrino capture after beta decay’ [27] version as a test [30] of Majorana’s theory of the neutrino [31]. The latter, as an alternative to the Dirac-Fermi theory of the neutrino [32], assumed that it was its own antiparticle. This would facilitate a ’true’ (in addition to the sequential version of Racah [30]) 0νββ process as proposed later by Furry [33], based on [29]. Explicitly, the ββ-decay processes with and without emission of an electron antineutrino are denoted as [27]: A Z X →A Z+2 X ′ + 2e− + 2νe (1.2) A Z X →A Z+2 X ′ + 2e− (1.3) In the neutrinoless version [Eq. (1.3)], the two neutrinos form a virtual connection in the Feynman diagram of the process, but do not appear as real particles (Fig. 2 in [28]). In principle, ββ decay is possible for any situation where the binding energy B(AZ+2X ′′) of the nucleus A Z+2X ′′ is larger than the one of the nucleus A Z X ′. From the experimental point of view, a situation is preferred where the two-step transition between these nuclei via single β decays is energetically forbidden, i.e.: B(AZ+1X )< B(AZ X ′)< B(AZ+2X ′′) (1.4) For a chain of even-even (ee), odd-odd (oo), or even-odd/odd-even (eo, oe) isobars, B(AZ X ) is approximately proportional to −Z2 in proximity of the valley of stability according to the semiempirical mass formula (see, e.g., Sec. 3.3. in [34] or [35]). Therefore, it is much more likely that Eq. (1.4) is fulfilled for a sequence of 34 1. Introduction ee/oo isobars, where the odd-even staggering due to the nuclear pairing force is superimposed on the Z dependence. In this case, the half-life for 0νββ decay from a 0+ ground state of an ee nucleus to a 0+ state of the daughter nucleus is given by [27]2. � T (0ν)1/2 �−1 = G0ν(Qββ , Z) � � � � 〈mν〉 me � � � � 2 � �M (0ν) � � 2 (1.5) In Eq. (1.5), the symbol G0ν denotes the integrated phase space factor, which takes into account the residual interaction of the two emitted electrons with the Coulomb field of the daughter nucleus. It depends on the proton number Z of the daughter nucleus and the Q value of the decay, Qββ . For the assumed decay between 0+ states, the two electrons are in an s1/2 state, which makes the final result particularly simple3. In close analogy to the so-called Fermi theory [32] of single β decay, G0ν can be approximated as [Eqs. (3.5.17a - 3.5.21) for light neutrinos using the ’Fermi factor’ from (3.1.25, 3.1.26) [36]]: G0ν∝ ∫ d � p(1)e− · p (2) e− � dE(1)e− dE(2)e− ×δ    E(1)e− + E(2)e− − Qββ ︷ ︸︸ ︷ � M � A Z−2X � −M � A Z X ′′ �� c2    × p(1)e− p(2)e− � 2p(1)e− R �2(γ1−1) � 2p(2)e− R �2(γ1−1) (1.6) × � �Γ � γ1 + i y (1) �� � 2 � �Γ � γ1 + i y (2) �� � 2 Γ (2γ1 + 1)4 eπy(1) eπy(2) 2Equation (1.5) assumes the annihilation of two left-handed light Majorana neutrinos, which is regarded as the ’most popular’ mechanism by the authors of [27]. Other possibilities, like the exchange of right-handed or heavy neutrinos, some of which are not sensitive to the neutrino mass, are discussed in [27] as well. Nevertheless, all of them require nuclear structure input. 3The restriction to 0+→ 0+ transitions is a valid approximation for light neutrinos, which can be seen, e.g., by comparing the expressions for 0+→ 0+ and 0+→ 2+ transitions in [36]] 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 35 γ1 ≡ q 1− (αZ)2 (1.7) y (i) ≡ αZ E(i)e− p(i)e− c (1.8) In Eq. (1.6), p(i)e− and E(i)e− denote the linear momentum and the total energy of the i-th electron that is emitted in the decay. The symbol M(AZ X ) denotes the mass of the nucleus A Z X . For 0νββ decay, the difference of the rest energies of mother (A Z−2X ) and daughter (A Z X ′′) nucleus, obtained from the rest masses by multiplication with the squared speed of light c, is equal to Qββ [indicated by a brace in Eq. (1.6)]. The symbol Γ (z) denotes the gamma function. The integration contains a delta function δ, which ensures the conservation of energy. The symbol R denotes the radius of the daughter nucleus, which is assumed to be spherical with a well-defined boundary in this approximation [32], and the auxiliary symbols γ1 and y (i) are defined by Eqs. (1.7) and (1.8). In the latter two equations, α denotes the fine-structure constant. For medium-mass and rare-earth nuclei, the factor γ1 is on the order of unity, therefore the energy- and Z dependence of Eq. (1.6) is dominated by the exponential terms. Consequently, G0ν is expected to vary strongly with Z , and also at low Q values which are on the order of the rest energy of the electron mec 2. This behavior is confirmed by realistic calculations with different levels of approximative character[20, 37]4. The discrepancy of different predictions is mostly less than 30% [20, 38], depending on the nucleus of interest. This gives an estimate of the accuracy of G0ν. The second factor in Eq. (1.3) contains the ratio of the effective light neutrino mass in 0νββ decay [27], 〈mν〉= 3 ∑ k=1 � U (11) ek �2 mk, (1.9) and the electron mass. In Eq. (1.9), the quantities U (11)ek are matrix elements of the experimentally well-investigated [2] Pontecorvo-Maki-Nakagawa-Sakata matrix, which connect the light neutrino mass eigenstates mk to the flavor eigenstate ’e’ 4The authors of Ref. [20] claim a more precise calculation than [37]. Nevertheless, [37] is mentioned here as well, since the publication contains predictions for a larger set of nuclei and phase space factors for decays to excited states. 36 1. Introduction of the electron neutrino [39]. Of course, an observation of 0νββ decay alone would have a enormous impact on contemporary physics. In addition, the factor 〈mν〉/me adds the hope that a measurement of T (0ν)1/2 can be used to fix the currently unknown neutrino mass scale [27]. Due to the high interest in beyond-standard model physics, many groups around the world are searching for signals from 0νββ decay, using experimental setups with ever-increasing scale and finesse. For an overview of present and future efforts, see, e.g., a recent review article by Dolinski, Poon, and Rodejohann [40]. The currently highest lower limits at a 90% confidence level for several 0νββ-decay candidates are shown in Fig. 1.1. The last factor in Eq. (1.3) is the nuclear matrix element (NME) M (0ν), whose general structure was discussed at the beginning of this section (Eq. (1.1)). Since the most ’straightforward’ access to M (0ν) would be via the 0νββ decay itself (if 〈mββ 〉 was known) the NMEs have to be provided by nuclear theory at the moment, despite the ongoing experimental efforts on analog reactions (see, e.g., [24]). The NME is the quantity that connects the aforementioned exciting beyond-standard model physics to the main objective of this work, i.e. the nuclear structure of 0νββ decay candidates. In Sec. 1.1.1, a simplified model will be used to illustrate the influence of nuclear structure on M (0ν). In Sec. 1.2, a summary of state-of-the-art realistic calculations for M (0ν) will be presented. 1.1.1. A Simple Model: Hypothetical 0νββ Decay of the Dineutron Themain aspects of the interplay between nuclear structure - in particular quadrupole deformation - and 0νββ-decay will be illustrated by a simple model in this section5. It is mainly based on chapters 3, 4 and 13 of the textbook by Talmi [59]. The smallest system in which a 0νββ decay could occur is a two-nucleon system of protons (π) with an isospin quantum number of tz = +1/2 , and neutrons (ν) with tz = −1/2. The corresponding two-nucleon systems are the Tz = −1 (dineutron), 5This section was motivated by the search of the author for a simple explanation of commonly quoted properties of the NMEs like ’the matrix element is large if the overlap of the wave functions of the initial and final state is large’. After finding Eqs. (34-36) in [28] and realizing that the approximated operator for 0νββ decay is actually relatively simple, it was decided to attempt a simple calculation from which this section originates. The reader may feel free to skip it. 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 37 48Ca 76Ge 82Se 96Zr 100Mo 116Cd 128Te 130Te 136Xe 150Nd 0 Mother Nucleus 1022 1023 1024 1025 1026 1027 T(0 ) 1/ 2 lo we r l im it [y ] EL EG AN T IV NE M O- 3 CA ND LE S III GE RD A M AJ OR AN A CU PI D- 0 NE M O NE M O- 3 NE M O- 3 NE M O- 3 AM oR E Au ro ra NE M O- 3 LN GS CU OR E Ka m LA ND -Z en EX O- 20 0 NE M O- 3 Figure 1.1.: Experimental limits (90% confidence interval) for the 0νββ-decay half life of different mother nuclei. For each mother nucleus, up to three bars indicate the current half-life limits. The experimental collaborations which published the results are indicated by labels on top of the bars. The limits are from [41–43] (48Ca), [44, 45] (76Ge), [46–48] (82Se), [49] (96Zr), [50, 51] (100Mo), [52, 53] (116Cd), [54] (128Te), [55] (130Te), [56, 57] (136Xe), and [58] (150Nd). T = 0 (deuteron), and Tz = 1 (diproton) eigenstates of the total isopin operator T̂ . In the following, consider the hypothetical situation where both the diproton and the dineutron are more strongly bound than the deuteron. Furthermore, the diproton is assumed to be more strongly bound than the dineutron to facilitate a ββ decay 6. 6In reality, of course, the situation is quite adverse: The proton-neutron system (the deuteron) is 38 1. Introduction Defintion of Operators For the two-nucleon system, the intrinsic quadrupole moment of a state i, which will be used as a measure of the quadrupole deformation in this section7, is given by (see, e.g., Appendix ’Electromagnetic Transitions and Moments’ in [66]): Q = 〈i| √ √16π 5 � e1r 2 1 Y20 (θ1,ϕ1) + e2r 2 2 Y20 (θ2,ϕ2) � |i〉 (1.10) In Eq. (1.10), a = 1,2 are the single-particle indices. The quantities ea actually denote the electric charges of the particles. To be able to assign a quadrupole moment to the nuclear matter distribution instead of the charge distribution (otherwise, the dineutron would always have a deformation of zero in this model), eπ = eν = 1 will be used. The symbols ra denote the single-particle position operators, which depend, in particular, on the polar- θa and the azimuthal angle ϕa in spherical coordinates. Both are arguments of the spherical harmonic Y20. In the two-nucleon system, the Fermi (F) and Gamow-Teller (GT) parts of the 0νββ decay operator, which are expected to be the dominant contributions, are approximately given by (Eqs. (34) and (35) in [28]): Ô(0ν)F = H(r12, 〈E〉)τ̂+1 τ̂ + 2 (1.11) Ô(0ν)GT = H(r12, 〈E〉) (σ̂1 · σ̂2) τ̂ + 1 τ̂ + 2 (1.12) In Eqs. (1.11) and (1.12), the symbols τ̂+a denote the single-particle isospin raising operators, which convert neutrons into protons. The symbol (σ̂1 · σ̂2) denotes a scalar product of the two single-particle pauli matrices, which are proportional indeed weakly bound with an energy of about 2.22MeV [7]. On the other hand, the relatively large measured absolute values of the s-wave scattering lengths for the neutron-neutron (discrepant values of ann = −18.63± 0.10stat. ± 0.44syst. ± 0.30theo. fm [60] and ann = −16.06± 0.35 fm [61] coexist at the moment) and the proton-proton (app = −7.8063± 0.0026 fm [62]) system suggest that both systems are very weakly unbound. Due to the repulsive Coulomb interaction in the diproton, it can be expected to be less strongly bound than the dineutron. A recent theoretical investigation [63] even suggests that experimental results for ann can not exclude a very weakly bound system, since the experimental techniques are not sensitive to the sign of the scattering length. 7The deformation parameter β2 from the collective model [64], which is often used to quantify the quadrupole deformation of a nucleus, is approximately proportional to the quadrupole moment at typical deformations of nuclei (see, e.g., the discussion of Eq. (6.9) in [65]). 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 39 to the spin operators ŝa = ħh/2σ̂a. In the definition of ŝ , the symbol ħh denotes the reduced Planck constant. The function H(rab, 〈E〉), which appears in both operators, is the neutrino potential. It is approximately given by [Eq. (36) in [28]]: H(r12, 〈E〉)≈ 2R πr12 ∫ ∞ 0 d (qc) sin (qr12/ħh) qc + 〈∆E〉 = R πr12 {2Ci (〈∆E〉r12/ħhc) sin (〈∆E〉r12/ħhc) (1.13) + cos (〈∆E〉r12/ħhc) [π− Si (〈∆E〉r12/ħhc)]} 〈∆E〉 ≡ 〈E〉 − � Ei + E f � /2 (1.14) In Eq. (1.13), rab denotes the distance between two nucleons a and b. The quantity 〈∆E〉, defined in Eq. (1.14), is the difference between the average energy 〈E〉 of excited states in the intermediate nucleus A Z+1X ′ and the mean value of the energies of the initial (Ei) and final (E f ) state of the 0νββ decay of the nuclei A Z X and AZ+2X ′′, respectively. Actually, a sum over all intermediate state of the nucleus A Z+1X ′ would have to be performed (Eq. (24) in [28]), but their energies are often replaced by an effective value 〈E〉. This is the so-called closure approximation [see, e.g., Eq. (25) in [28] and [67, 68] for recent theoretical investigations]. The neutrino potential has been multiplied by the nuclear radius R to make it dimensionless. In the second equality of Eq. (1.13), the integration over the momentum variable qc has been executed, resulting in an expression that includes the sine- and cosine integrals Si(x) and Ci(x). While the neutrino potential in Eq. (1.13) depends on the distance of the nu- cleons r12, the quadrupole moment in Eq. (1.10) requires the knowledge of the single-nucleon wave functions. A model for which the transition between both representations of the same total wave function is very simple is the harmonic oscillator (HO). The Hamiltonian for the two-nucleon system is given by [see, e.g., 40 1. Introduction Eq. (13.1) in [59]]: ĤHO = 1 2m1 p̂2 1 ︸ ︷︷ ︸ k̂1 + 1 2 m1ω 2r 2 1 ︸ ︷︷ ︸ ĤHO,1 + 1 2m2 p̂2 2 ︸ ︷︷ ︸ k̂2 + 1 2 m2ω 2r 2 2 ︸ ︷︷ ︸ ĤHO,2 (1.15) In Eq. (1.15), p̂a denotes the momentum operator for particle a, and ma its mass. In the scope of this problem, the masses of the proton and the neutron will be assumed to be equal: mν = mπ ≡ m , in particular also m1 = m2 = m. Both quantities appear in the single-particle kinetic energy operator k̂a, whose definition is indicated by the innermost braces. The symbolω denotes the oscillator frequency, multiplied by 2π. Equation (1.15) is the form of ĤHO where the independence of the motion of the two particles is most evident, because it is a sum of two single-particle Hamiltonians, as indicated by the outermost braces. An equivalent formulation of the problem in terms of an independent motion of the center-of- mass (COM) and the intrinsic (INT) two-nucleon system is possible [see, e.g., Eqs. (13.2) and (13.3) in [59]]: ĤHO = 1 4M P̂2 + 1 2 Mω2R2 ︸ ︷︷ ︸ ĤHO,COM + 1 2µ p̂2 12 + 1 2 µω2r 2 12 ︸ ︷︷ ︸ ĤHO,INT (1.16) In Eq. (1.16), the symbols P̂ and R denote the momentum operator and the position of the center-of-mass, respectively. The symbols p̂12 and r12 denote the relative momentum operator and the distance vector between the two particles, respectively. The symbols M = 2m and µ= m/2 denote the total- and the reduced mass of the system. Two braces indicate the independent center-of-mass- and intrinsic parts of the Hamiltonian. 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 41 Wave Functions For any of the independent systems, the eigenstates of the Hamiltonian are given by [see, e.g., Eq. (3.7) and (4.3) in [59]]: ψnlml (r,θ ,ϕ) = 1 r Rnl (r)Ylml (θ ,ϕ) (1.17) = Nnl r l exp � −νr2 � L l+1/2 n−1 � 2νr2 � Ylml (θ ,ϕ) . In Eq. (1.17), the vector r and its components r, θ and ϕ may be replaced by r1, r2, r12, or R from Eqs. (1.15) and (1.16). The wave function ψnlml depends on the angular momentum quantum number l, its projection on the z axis ml , and the number of nodes n of the radial wave function Rnl . The radial wave function includes a normalization factor Nnl [Eq. (4.3) in [59]] and the associated Laguerre polynomials Lαk . To simplify the notation, the quantity ν= µω 2ħh (1.18) has been introduced, which can be interpreted as half the squared oscillator length. In Eq. (1.18), the symbol µ may be replaced by any of m, µ, and M which corresponds to the choice of the position vector. The angular part of the wave function is given by the 3D spherical harmonics Y , which depend on the quantum numbers l and ml . The eigenvalues of the harmonic oscillator are [Eq. (4.5) in [59]]: Enl = � 2 (n− 1) + l + 3 2 � ħhω= � N + 3 2 � ħhω (1.19) In Eq. (1.19), the principal quantum number N has been introduced. It is now assumed that the spatial dynamics of both the dineutron and the diproton are given by Eqs. (1.15) and (1.16). Furthermore, it is implicitly assumed that the energetic prerequisites for 0νββ decay [Eq. (1.4)] are fulfilled8. The aim is to study the impact of nuclear structure on 0νββ decay by calculating nuclear matrix elements between initial and final two-nucleon states. The lowest-lying states of ĤHO,INT in Eq. (1.16) for the Tz = ±1 systems are given by 11s0 and the triplet 23p0, 8This can be achieved, for example, by introducing artificial, isovector terms in the Hamiltonian that depend on powers of Tz . 42 1. Introduction 23p1, and 23p2 in the spectroscopic notation n2S+1lJ . Using the symbols ψnlml for the spatial-, χSSz for the spin-, and ξT Tz for the isospin wave function, their total wave functions Ψ are given by (only the J = 0 states are shown): |Ψ(11s0)〉= ψ100(R)ψ100(r12)×χ00 × ξ1(±1) (1.20) |Ψ(23p0)〉= 1 p 3 ψ100(R)ψ211(r12)×χ1(−1) × ξ1(±1) (1.21) − 1 p 3 ψ100(R)ψ210(r12)×χ10 × ξ1(±1) + 1 p 3 ψ100(R)ψ21(−1)(r12)×χ11 × ξ1(±1) In Eqs. (1.20) and (1.21), it was assumed that the center-of-mass motion remains in the 1s state. The numerical factors and phases in Eq. (1.21) are the usual Clebsch-Gordan coefficients (see, e.g., chapter 44 in [2] for tabulated values) for the coupling of angular momentum l = 1 and spin S = 1 to J = 0. The spatial wave functions for the 1s and 2p states can be expanded in single-particle wave functions [compare Eq. (13.10) and (13.11) in [59]]: ψ100(R)ψ100(r12) = ψ100(r1)ψ100(r2) (1.22) ψ100(R)ψ21ml (r12) = 1 p 2 � ψ100(r1)ψ21ml (r2)−ψ21ml (r1)ψ100(r2) � (1.23) The ’-’ sign on the right-hand side of Eq. (1.23) is required since the wave function on the left-hand side is antisymmetric. Quadrupole Deformation For the calculation of the quadrupole moments of the 1s and 2p states, it is evident from the definition in Eq. (1.10) that only the spatial part of the wave functions plays a role. Furthermore, since the quadrupole operator is separable in the same way as ψnlml , the matrix element is a product of integrals over the radial part (r2 a ) and the angular part Y20. For a single harmonic-oscillator wave function, the 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 43 integral over the radial part only depends on the principal quantum number [Eq. (4.10) in [59]]: 〈ψnlml |r2 a |ψnlml 〉= 1 2ν � N + 3 2 � . (1.24) Note that Eq. (1.24) defines the mean square radius. As already implied by Eq. (1.24) for the angular part, any Jz substate can be selected due to the Wigner- Eckart theorem for spherical tensor operators [see, e.g., chapter 8 of [59], in particular Eqs. (8.4) and (8.15)], and the matrix elements evaluate to: 〈ψnlml |Y20|ψnlml 〉= (2l + 1) √ √ 5 4π � l 2 l 0 0 0 �2 . (1.25) Eq. (1.25) contains a Wigner 3- j symbol, which vanishes for l = 0 and evaluates to p 2/15 for l = 1. Consequently, the quadrupole moments of the physical states in Eqs. (1.20)/(1.22) and (1.21)/(1.23) are (using the shorter, basis-independent notation ψnlml (r1)ψn′ l ′ml′ (r2)→ 〈ψnlm|〈ψn′ l ′ml′ |, |ψnlm〉|ψn′ l ′ml′ 〉): Q(11s0)∝ 〈ψ100|Y20|ψ100〉= 0 (1.26) Q(23p0)∝ � 〈ψ100|〈ψ211| − 〈ψ211|〈ψ100| − 〈ψ100|〈ψ210|+ 〈ψ210|〈ψ100| (1.27) + 〈ψ100|〈ψ21(−1)| − 〈ψ21(−1)|〈ψ100| � × � r 2 1 Y20(θ1,ϕ1) + r 2 2 Y20(θ2,ϕ2) � × � |ψ100〉|ψ211〉 − |ψ211〉|ψ100〉 − |ψ100〉|ψ210〉+ |ψ210〉|ψ100〉 + |ψ100〉|ψ21(−1)〉 − |ψ21(−1)〉|ψ100〉 � ∝ 2 ∫ dΩ � Y11Y20Y11 + Y10Y20Y10 + Y1(−1)Y20Y1(−1) � > 0 In the second step of Eq. (1.27), vanishing off-diagonal matrix elements have been neglected, and the equivalent integrations over Ω1 and Ω2 have been summed. Equations (1.26) and (1.27) show that the restriction to the first two states of the system was sufficient to study the influence of quadrupole deformation on the NME, since the s state is spherical and the p state has a finite quadrupole moment. 44 1. Introduction Nuclear Matrix Elements For a qualitative analyis of the NMEs the isospin-, spin-, and spatial part (in this order) can be considered separately due to the absence of couplings between the subspaces in the 0νββ-decay operators [Eqs. (1.20) and (1.21)]. Firstly, the isospin part can be neglected, since its action is always the same, turning the νν system into a ππ system: τ̂+1 τ̂ + 2 |1,−1〉 ∝ |1,+1〉 (1.28) Concerning the spin-part of the wave function, only the Gamow-Teller part of the 0νββ-decay operator [Eq. (1.12)] contains a spin-dependent factor σ̂1σ̂2. The wave functions χSSz are eigenfunctions of this operator, which can be seen by rewriting it in the following way: (σ̂1σ̂2) |χSSz 〉= 4ħh2 (ŝ1 ŝ2)|χSSz 〉= (1.29) 2ħh2 � Ŝ2 − ŝ2 1 − ŝ2 2 � |χSSz 〉= ¨ −3ħh2, S = 0 1ħh2, S = 1 In the last equality of Eq. (1.29), the eigenvalues for the two possibilities of parallel (S = 1) and antiparallel (S = 0) spins have been evaluated. This means that neither the Fermi- (a simple identity operator), nor the Gamow-Teller part of the 0νββ- decay operator connect states with different spin alignments, in particular not the 1s and 2p states. Note that this is a consequence of the extremely simple LS coupling in the two-body system that was considered here, where a given value of J can be traced back unambiguously to L and S. Nevertheless, it can be anticipated for systems with more than two particles that transitions between states with a large number of S = 0-nucleon pairs, i.e. in particular ground states of even-even nuclei (see also Sec. 2.2.2), are enhanced. At last, consider the spatial part of the NME. Obviously, the neutrino potential in Eq. (1.13) only depends on the distance r12 of two nucleons. Therefore, it can immediately be concluded from the orthogonality of the spherical harmonics [see, 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 45 e.g., chapter 3 in [59]]: ∫ 2π 0 dϕ ∫ π 0 dθY ∗lmYl ′m′ = δl l ′δmm′ , (1.30) that matrix elements between different l vanish. Since the type of deformation is encoded in the contributions of different spherical harmonics to a given state, this means that 0νββ decay favors transitions between similar shapes. This makes the process susceptible to phenomena like shape-phase transitions along a chain of nuclei [69] and shape coexistence [70, 71] within a single nucleus, both of which will be discussed in Sec. 1.2. While the spherical harmonics are related to the type of deformation, the radial wave functions can be a measure of the degree of deformation. For the harmonic oscillator wave functions, this can be seen from the mean square radius in Eq. (1.24), which increases monotonously with the principal quantum number. The distance dependence of the neutrino potential H(r12, 〈∆E〉) in Eq. (1.13) is shown in part (a) of Fig. 1.2 for realistic values of 〈∆E〉. Since it is nonnegative for all values of r12, H(r12) can be interpreted as an (unnormalized) weighting factor in integrals of the type Nnl Nn′ l ′ ∫ r 0 dr12Rnl H12Rn′ l ′ , (1.31) which favors small distances between nucleons. The lower three panels of Fig. 1.2 show the normalized radial probability density N2 nlR 2 nl r −2 for different low-lying eigenstates of the two-body system. In addition, the value of the integral in Eq. (1.31) is shown, which is asymptotic for r →∞ due to the exponential decay of the integrand [see Eq. (1.17)]. It can be seen that the radial integrals in the NMEs for 0νββ decay are largest for small nuclear deformations, i.e. configurations for which the average distance between nucleons is minimized. The factor 〈∆E〉 has a significant, but uniform impact on all the integrals. Based on the findings in the previous paragraphs, a simple study of the effects of shape evolution [69] can be performed: Assume that both the ground states of the dineutron Ψ(0)νν and the diproton Ψ(0)ππ are superpositions of the two lowest-lying 46 1. Introduction J = 0 states 11s0 and 23p0 9: |Ψ(0)nn 〉= Æ 1− βnn|11s0〉+ Æ βnn|23p0〉 (1.32) 0≤ βnn ≤ 1 (1.33) In Eq. (1.32), the index nn may stand for either νν or ππ. A parameter βnn, suggestively named like the nuclear deformation parameter, has been introduced to control the mixing of the 1s and 2p states. According to Eqs. (1.26) and (1.27), the admixture of a 2p component in the ground state introduces a finite quadrupole deformation: Q(Ψ(0)nn )∝ βnn (1.34) The NME for 0νββ-decay between the states in Eq. (1.32) consists of matrix elements of the type: M (0ν) = 〈Ψ(0)ππ(βππ)|Ô (0ν)|Ψ(0)νν (βνν)〉= (1.35) � Æ 1− βππ〈11s0|+ Æ βππ〈23p0| � Ô(0ν) � Æ 1− βνν|11s0〉+ Æ βνν|23p0〉 � = Æ 1− βππ Æ 1− βνν〈11s0|Ô(0ν)|11s0〉 + Æ 1− βππ Æ βνν〈11s0|Ô(0ν)|23p0〉 + Æ βππ Æ 1− βνν〈23p0|Ô(0ν)|11s0〉 + Æ βππβνν〈23p0|Ô(0ν)|23p0〉= Æ 1− βππ Æ 1− βνν〈11s0|Ô(0ν)|11s0〉+ Æ βππβνν〈23p0|Ô(0ν)|23p0〉. The last equality in Eq. (1.35) follows from the fact that the off-diagonal matrix elements vanish due to Eqs. (1.29) and (1.30). The symbol Ô(0ν) may stand for the Fermi- or the Gamow-Teller part of the complete 0νββ-decay operator [Eq. (33) [28]]: Ô(0ν)F+GT = Ô(0ν)GT − � gV gA �2 Ô(0ν)F (1.36) 9An operator that mixes states of equal J must be introduced in the Hamiltonian to create such a situation. For example, in Sec. 7.1 of Casten’s textbook [65] where the deformed (Nilsson) shell model is introduced, an anisotropic oscillator potential [Eq. (7.2) therein] can be used for this purpose. 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 47 In Eq. (1.36), the Fermi part is scaled by the squared ratio of the vector- and axial- vector coupling constants gV/gA of the weak interaction, which has a numerical value of about (gV/gA) 2 ≈ 0.61 [4]. The dependence of the matrix elements in Eq. (1.35) for Ô(0ν) = (gV/gA) 2 Ô(0ν)F and Ô(0ν) = Ô(0ν)GT on the mixing parameters βνν and βππ is shown in part (a) and (b) of Fig. 1.2, respectively. Due to the scaling by the eigenvalues of the operator (σ̂1σ̂2) [Eq. (1.29)] and the factor (gV/gA) 2 in front of the Fermi part, the Gamow-Teller matrix elements show a much larger variation over the parameter space. The opposite signs of the 1s- and 1p matrix elements in Eq. (1.35), which are a consequence of the different spin alignments, cause a cancellation of the matrix elements for strongly mixed configurations. The absolute values of the NMEs are largest for similar, pure configurations of the mother- and daughter nucleus. In part (c) of Fig. 1.3, the NME, i.e. the squared absolute value of the sum of part (a) and (b) is shown. Its general dependence on βνν and βππ is similar to the dominating Gamow-Teller component. The accelerated change of the NME towards pure configurations and βνν ≈ βππ suggests that the NMEs will be highly sensitive to shape evolution, which may occur suddenly or gradually via first- or second-order phase transitions [72, 73]. 48 1. Introduction 0.0 0.2 0.4 0.6 0.8 1.0 H (r, E ) ( fm ) a) E = 2 MeV E = 4 MeV E = 8 MeV 0.0 0.2 0.4 0.6 0.8 1.0 N 2 10 |R 10 /r| 2 (a rb . u ni ts ) b) 1s | 1s 0.0 0.2 0.4 0.6 0.8 1.0 N 2 21 |R 21 /r| 2 (a rb . u ni ts ) c) 2p | 2p 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r (fm) 0.0 0.2 0.4 0.6 0.8 1.0 N 2 20 |R 20 /r| 2 (a rb . u ni ts ) d) 2s | 2s 0 20 40 60 r 0N 2 10 R 2 10 H dr 0 20 40 60 r 0N 2 21 R 2 21 H dr 0 20 40 60 r 0N 2 20 R 2 20 H dr Figure 1.2.: (a): Distance dependence of the neutrino potential H in the 0νββ- decay operator for different values of 〈∆E〉 and R= 1 fm [Eqs. (1.11) - (1.14)]. (b−d): Probability density of the nucleon-nucleon distance r for a 1s-, 2p-, and 2s harmonic oscillator wave function with ν= 1 fm in red (left ordinate), and the corresponding radial integral over the neutrino potential [Eq. (1.31)] in the same line styles as part (a) [right ordinate with identical scale for (b− d)]. 1.1. Neutrinoless Double-Beta Decay and Nuclear Structure 49 0.0 0.2 0.4 0.6 0.8 1.0 (a rb . u ni ts ) a) 0.0 0.2 0.4 0.6 0.8 1.0 (a rb . u ni ts ) b) 0.0 0.2 0.4 0.6 0.8 1.0 (arb. units) 0.0 0.2 0.4 0.6 0.8 1.0 (a rb . u ni ts ) c) 2 1 0 1 2 3 (g V /g A )2 M 0 F (a rb . u ni ts ) 2 1 0 1 2 3 M 0 GT (a rb . u ni ts ) 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 |M 0 GT (g V /g A )2 M 0 F |2 (a rb . u ni ts ) Figure 1.3.: Dependence of matrix elements of the Fermi- (a) and Gamow-Teller part (b) of the NME on the mixing parameters βνν and βππ in the two- nucleon model of Sec. 1.1.1. The former is multiplied by a factor of (gV/gA)2 according to Eq. (1.36). To obtain parts (a) and (b), Eq. (1.35) was evaluated for 0≤ βνν,βππ ≤ 1. The distance of the contour lines is the same for (a) and (b). Part (c) shows the squared absolute value of the total NME. The parameters βνν and βππ are correlated with the 2p admixture, i.e. proportional to the quadrupole deformation. 50 1. Introduction 1.2. Current State of Nuclear Structure Input Section 1.1 ended with the problem of the determination of the NME M (0ν) from nuclear theory. Using a simple model, general properties of the NME were studied in Sec. 1.1.1. It turns out that these properties are reproduced by microscopic models for actual 0νββ-decay candidates. As an example, compare a calculation of the nuclear structure impact of deformation on the 0νββ decay between 150Nd and 150Sm within the framework of energy-density functional theory by Rodríguez and Martínez-Pinedo [74] (see also [75]) in Fig. 1.4 to the schematic calculation in Fig. 1.3. The preference for decay between similarly deformed, in particular weakly deformed, structures is apparent. With the deformation parameters of Fig. 1.4 fixed by their experimental values, the authors of Ref. [74] also give predictions for the 0νββ-decay NME. Since their calculation implicitly contained more parameters and model assumptions (see, e.g. [76, 77] about the density functional that was used in [74]), the question about the accuracy and precision of theoretical NMEs arises. Obviously, there are no measurements to validate the calculated NMEs. Therefore, an estimate of the predictive power must be obtained by comparing the predictions of different models. The top part of Fig. 1.5 shows a compilation of predicted NMEs by various effective theories of nuclear structure from a review article by Engel and Menéndez [21]. From this comparison, it can be seen that the uncertainty of the NMEs is much larger than the one of the phase space factors (see Sec. 1.1). To obtain the bottom part of Fig. 1.5, Eq. (1.5) was solved for the product T (0ν)1/2 |〈mν〉|2 to give an estimate of the impact of the NMEs on actually observable quantities. The predictions, which have an impact on the planning of future 0νββ searches and the extraction of meaningful information about the neutrino (Sec. 2.3 in [21]), vary by more than an order of magnitude at first glance. However, the predictions by different models are obviously correlated. It is evident that they could be brought into better agreement, at lowest order, by a simple multiplication with a constant factor. This is due to known deficiencies of the models, like the restriction to a valence space in the shell model and IBM, or the overestimation of pairing correlations in quasiparticle models (see Sec. 3.6 in [21]). Nevertheless, a systematic uncertainty inherent in all models is the general overprediction of weak decay rates. This problem, the so-called ’quenching’ of the bare axial-vector coupling constant gA (see Sec. 4 in [21]), was identified by comparison to experimental data on β- and 1.2. Current State of Nuclear Structure Input 51 < β 1 5 0 S m |M (0 ν) G T |β 1 5 0 N d > Figure 1.4.: Strength of the Gamow-Teller part of the 0νββ-decay operator as a function of the deformation (measured by the usual deformation parameter β) of the initial and final states of the nuclei 150Nd and 150Sm. Result of a realistic theoretical calculation within the frame- work of energy density functional theory. The right-hand ordinate indicates symbolically the matrix element whose absolute magnitude is indicated by the color code. A dark dot in the figure indicates the most probable deformations of both isotopes as returned by the model. Reprinted and modified figure with permission from [74] Copyright (2010) by the American Physical Society. 52 1. Introduction 2νββ decay (see Sec. 3.1 and 4 in [21]). The observed quenching of M1 strength for electromagnetic transitions of nuclei (see, e.g., Sec. V.C.2 in [5] or Sec. V.D.1 in [89]) has the same origin, summarized as ’quenching of spin matrix elements’ in a review article by Towner [90]. From a microscopic point of view, quenching is caused mainly by the renormalization of transition operators (see also Secs. 2.3 and 2.2.2) and meson-exchange/two-body currents between nucleons [90]. Modern ab-initio theory (see, e.g., a review article by Epelbaum, Hammer, and Meißner [91]) does not suffer from the quenching problem [92], and calculations have been performed for nuclei as heavy as 100Sn (ibid., and [93]). The many- body methods are, however, restricted to nuclei in proximity of shell closures at the moment [94, 95]. Except for 48Ca, for which an ab-initio description is possible [96, 97], the isotopes of interest for 0νββ decay studies are at least 6 nucleons/nucleon holes away from doubly magic nuclei, as can be seen in Fig. 1.6. In addition, many of them are located in regions of the chart of nuclides where collective behavior and phase transitions between spherical and deformed shapes are expected to occur (black ellipses in Fig. 1.6). For the description of these phenomena, effective models are invaluable [73] (in particular Sec. III therein), but they require precise and unambiguous experimental data to constrain their free parameters and benchmark their predictions (see Sec. VI.C in [28]). For example, there has recently been a lot of 0νββ-decay related experimental effort to study the low-energy structure of 76Ge/76Se [102–106]10, since germanium is one of the most promising materials for 0νββ-decay detection (see Fig. 1.1). The present work concentrates on the 0νββ-decay candidate pairs 82Se/82Kr and 150Nd/150Sm. The former is motivated by recent progress [68] in the framework of the nuclear shell model towards large-scale calculations based on realistic interac- tions [101], and beyond the closure approximation (see Sec. 1.1.1). Furthermore, due to pioneering work in the operation of cryogenic calorimeters [107], the CUPID-0 collaboration was able to improve the half-life limit for the 0νββ decay of 82Se by one order of magnitude [46] compared to previous results [47]. Note that CUPID-0 is a prototype for the planned large-scale CUPID experiment, which aims to be competitive with highest currently reported limits for other isotopes [108, 109]. The motivation for the investigation of the pair 150Nd/150Sm is its favorable 0νββ half-life, which is predicted to be about an order of magnitude lower than the ones for 76Ge and 136Xe (see Fig. 1.5), which hold the current 10The five publications are taken from the most recent of them, i.e. [106]. 1.2. Current State of Nuclear Structure Input 53 records for half-life upper limits (see Fig. 1.1). It should be noted that several of the calculations presented in Fig. 1.5 do not take into account an additional decay branch to the 0+2 state of 150Sm, which is predicted [110] to lead to a further significant reduction of the half-life. From the structural point of view, the stable even-even Nd and Sm isotopes are collective nuclei in the vicinity of a shape phase transition, which are known to be described well by the interacting boson model [111]. In particular, 150Nd is a textbook realization [112] of the X (5) critical-point symmetry [113] for the transition between spherical and rotational nuclei. Leading experts in the field, in a workshop on "Nuclear matrix elements for neutri- noless double beta decay" in 2005 [114] (see also Sec. VI.C in [28]), recommended to perform charge exchange- [24, 115], nucleon transfer- [116]11, muon capture- [117] and neutrino-nucleus scattering [118] experiments. These reactions, in combination with β-, 2νββ-, and 2νECEC decay studies, were found to be the most sensitive to 0νββ-decay related matrix elements. The authors of [114] also emphasize the importance of deformation for 0νββ decay, which was discussed above. In the present work, it was decided to employ the nuclear resonance fluores- cence method (see Sec. 2.1) to study the structure of 82Se/82Kr and 150Nd/150Sm. Compared to the experimental methods mentioned above, the involved matrix elements are less closely related to 0νββ decay. However, the experiments are more straightforward, and observables of interest can be extracted with a high degree of model independence. The focus was on decay channels of a low-lying collective dipole excitation, the so-called scissors mode, which are sensitive to the deformation and the coexistence of shapes in an atomic nucleus (see Sec. 2.3). The experimental study of this work is based on pioneering work by Beller et al. [110]. 11The given review article by Wimmer is more focused on experiments with radioactive beams. See also references therein. 54 1. Introduction Figure 1.5.: (Top) Predictions of NMEs M0ν for the 0νββ decay of different iso- topes. The displayed calculations were performed in nonrelativistic (’NR-EDF’ [78]) and relativistic (’R-EDF’ [79, 80]) energy density functional theory, quasiparticle random-phase approximation (’QRPA Jy’ [81], ’QRPA Tu’ [82, 83], ’QRPA CH’ [84]), the interacting boson model (’IBM-2’ [85]), and the shell model (’SM Mi’ [86], ’SM St-M, Tk’ [87, 88]). All predictions used a bare gA. Some of them have given uncertainty estimates from a variation of the interaction. (Bottom) Corresponding predictions of the product of the 0νββ decay half life T 0ν 1/2 and the unknown effective neutrino mass mββ . Figure from [21] (Fig. 5 therein). Reproduced with permission of IOP Publishing in the format Thesis/Dissertation via Copyright Clearance Center. 1.2. Current State of Nuclear Structure Input 55 48Ca→48Ti 76Ge→76Se 82Se→82Kr 110Pd→110Cd 96Zr→96Mo 124Sn→124Te 100Mo→100Ru 116Cd→116Sn 128Te→128Xe 130Te→130Xe 148Nd→148Sm 136Xe→136Ba 134Xe→134Ba 150Nd→150Sm 154Sm→154Gd 160Gd→160Dy 198Pt→198Hg 232Th→232U 238U→238Pu Figure 1.6.: Candidates for 0νββ decay on the chart of nuclides. The chart, which was taken from [98] and modified, shows all known isotopes with the neutron number on the abscissa and the proton number on the ordinate. Different colors correspond to different ranges of the isotopic lifetimes. Stable nuclei are shown in black. For better orientation, the classical magic numbers at 8, 20, 28, 50, 82, and 126 [99, 100] are indicated by white lines. Midshell regions, where nuclei are expected to be deformed in their ground states, are indicated by black circles in analogy to Fig. 2.1 in the nuclear structure textbook of Casten [65]. Compared to the latter figure, another circle has been added in the p f shell (Z ≥ 28 and 28 ≤ N ≤ 50) to indicate the region where the phenomena of shape coexistence and triaxial deformation appear [101]. The set of 0νββ decay candidates is from [85]. For pairs of isotopes that are shown in gray or black, dedicated experiments in search for their 0νββ decay have been or are still ongoing. The black-labeled isotopes are the main objective of this work. 56 1. Introduction 1.3. Outline The present work is structured as follows: This introduction is followed by an overview of the relevant formalism (Sec. 2). ’Formalism’ includes a discussion of the experimental method of nuclear resonance fluorescence (Sec. 2.1), the theoretical frameworks of the shell- and the interacting boson model (Secs. 2.2.1 and 2.2.2), and the nuclear structure phenomenon of interest (Sec. 2.3). The following section (Sec. 3) describes the High-Intensity γ-Ray Source, where all experiments of this work were performed (Sec. 3.1), the experimental setups (Sec. 3.2), and the dedicated experiments (Secs. 3.3 and 3.4). In the ’Analysis’ section (Sec. 4), the processing of the experimental data (spectra) is described. The section starts with general information about the treatment of uncertainty and parameter estimation (Sec. 4.1). After that, the Monte-Carlo simulation framework which was used for the analysis is introduced (Sec. 4.2). The remaining sections give give a detailed description and examples for all relevant analysis steps. Results for the observed transitions of 82Se/82Kr and 150Nd/150Sm are presented in Sec. 5. A discussion of selected results can be found in Sec. 6. In particular, the origin of the observed magnetic dipole strength in the A= 82 nuclei will be discussed. Furthermore, updated predictions of NMEs for the 0νββ decay of 150Nd, based on the improved data, will be given (Sec. 6.2). The main body of this work concludes with a summary and an outlook (Sec. 7). 1.3. Outline 57 2. Background This section introduces the experimental technique of nuclear resonance fluores- cence (NRF) that was used to study the low-energy nuclear structure of the 0νββ decay candidates. Furthermore, two theoretical models will be discussed which were used to interpret the experimental results. At last, the nuclear scissors mode will be introduced with a focus on its relation to nuclear shapes. 2.1. Nuclear Resonance Fluorescence The most recent review article about NRF and the application to nuclear structure studies, which was used as a guideline for this section, was published by Kneissl, Pitz, and Zilges [119]. A historical review article on the topic, which is more focused on the technique itself, was published by Metzger [120]. 2.1.1. General Properties Resonance fluorescence, in particular N-RF [(γ,γ′)], is the interaction of a system with discrete bound states (the nucleus) and the quantized EM radiation field (see, e.g. [121], in particular chapter V, §20 therein). In other terms, it is the resonant absorption of a real photon by a nucleus, which leaves the nucleus in an excited state, and the subsequent emission of potentially multiple photons in the de-excitation process [119]. The mechanism has the advantage that the (nuclear) transition matrix elements between bound states can be clearly separated from the well-known EM part of 59 the reaction [see Eq. (2.4), or chapter V, §20 in [121] for a detailed derivation], facilitating a model-independent determination of the former. In this sense, the extraction of certain matrix elements is more straightforward over a large range of excitation energies and target proton numbers than for EM-mediated processes which involve the Coulomb scattering of charged elementary ([122], in particular chapter 4.2 therein) or nuclear [123] particles, since they eventually need to take into account the distortion of the projectile’s wave function by the target (see, e.g. [124] for electron scattering or [125] for high-energy heavy-ion scattering). Another advantage for the present study is the high selectivity of NRF to electric (E1) and magnetic (M1) dipole excitations. This can be anticipated from single- particle (’Weisskopf’) estimates (chapter XII.6.A in [13]) for the EM transition rates between SM states, which decrease by orders of magnitude when the mul- tipole order L increases1. The suppression of magnetic character compared to electric character, which is also predicted by these estimates, is balanced by the collectiveness of the M1 scissors mode [89] which is of main interest in this work. To demonstrate the main disadvantage of the NRF method, it is instructive to compare the relative magnitude of the energy-’integrated’ cross sections for elastic photon scattering [Eq. (2.4) with i = k] and inelastic electron scattering [(e, e′)] for low momentum transfer [Eq. (4-15a) in [122]], given a particular reduced M1 transition width B(M1) [Eq. (2.6)] to an excited state at a typical energy for a fragment of the scissors mode [89] of E j = 3MeV. Neglecting all factors which are on the order of unity for a favorable choice of the target nucleus and experimental kinematics, i.e. branching ratios, J- and L-dependent factors, and recoil corrections, the energy-integrated cross sections are approximately: dI dΩ (γ,γ′) ≈ 8π λ3 i→ j B(M1)W (θ ,ϕ)≈ 10−6 fm−1 × B(M1)W (θ ,ϕ) (2.1) dσ dΩ (e,e′) = ∫ E j+∆E/2 E j−∆E/2 dσ dΩdE (E)dE ≈ dσ dΩdE (E0)∆E ≈ α2 ħhc B(M1)∆EVT (θ )≈ 10−13 fm−1 × B(M1)VT (θ ) (2.2) 1The lowest multipole order L = 0, however, is forbidden for transitions involving a single real photon, because due to their masslessness, the angular momentum of photons must always be a nonzero integer number (see, e.g. appendix 1 in [121]). 60 2. Background In Eq. (2.2), the symbol VT (θ ) denotes a scattering-angle dependent functionwhich is on the same order of magnitude as W (θ ,ϕ) (Eq. (2.11)) for certain scattering angles. It is assumed that the cross section is approximately constant over the integration range ∆E . Setting ∆E = 1 eV, which is a typical width of a Doppler- broadened nuclear resonance [120] and definitely fulfils the aforementioned assumption, a numerical value for comparison with the NRF value is obtained. Equations (2.1) and (2.2) suggest that, given particle beamswith similar intensities, the reaction rate in NRF should be orders of magnitude higher than in (e, e′). However, this naive estimate neglects that NRF is a resonant process, i.e. only photons with an energy in the range of the resonance can excite it. Typical photon beam performances (particles per time and energy interval) at 3MeV are on the order of 102 s−1eV−1 [126] for bremsstrahlung-generated photons and 104 s−1eV−1 [127] for laser Compton-backscattered (LCB) photons. On the other hand, electrons with any energy larger than the resonance energy can in principle excite it. Therefore, a typical electron beam current of 20µA and a spectral width of few keV [126], i.e. a particle current of 1014 s−1, more than compensates the difference between Eqs. (2.1) and (2.2). Similar arguments can be applied for the scattering of nuclear particles. These instrumental restrictions must be overcome by increased target masses (possible due to the strong penetration power of gamma- rays [128]) and measuring times, which restrict NRF to stable, and sufficiently abundant isotopes [119]. Depending on the observable of interest, NRF experiments are also complicated by the large photonic background that is caused by nonresonant atomic scattering processes on the target or the beamline (see also Sec. 4.2 for a more detailed discussion). At energies of few MeV of the present experiment, the Compton effect is the dominant source of nonresonant background (see, e.g., chapter 2.III in [129], in particular Fig. 2.18). The cross section for Compton scattering, evaluated at a scattering angle of 90◦ for simplicity and integrated over the energy in analogy to Eq. (2.2), is given by (ibid.): dσ dΩ Compton (90◦)≈ 1 2 Zα2 � ħhc mec2 �2 x2 + x + 1 (x + 1)3 ∆E ≈ 10−3 eVfm2 × Z (2.3) In Eq. (2.3), the symbol x is an abbreviation for the ratio of the initial photon energy, and the electron rest energy, i.e. x = E j/mec2. Considering that typical values of B(M1) in Eqs. (2.1) and (2.2) for low-lying collective M1 strength are 2.1. Nuclear Resonance Fluorescence 61 on the order of 1µ2 N ≡ 1.6× 104 eVfm3 [89], NRF stills seems to dominate. But again, any photon in the beam spectrum can be Compton-scattered, so indeed the number of nonresonantly scattered photons will dominate the event rate in an NRF experiment with currently available beams. This is different for the scattering of massive, charged, and comparably highly energetic particles, where the main sources of background are not alternative scattering processes, but radiative corrections, especially for the light electrons (chapter 1.2 in [122])2. At last, an important difference to the aforementioned techniques is that NRF does not measure the excitation directly. A photoabsorption cross section derived from an NRF experiment will only resemble the ’true’ cross section if all branching transitions to lower-lying excited states are known or can be excluded [see Eq. (2.4)]. Especially in recent discussions about low-lying electric dipole strength, this was identified as a serious issue [130]. The direct branching transitions are expected at lower energies where the intensity of the nonresonant background increases approximately exponentially (see spectra of this work in Sec. D), which prevents a firm constraint of single channels even if the ground-state transitions can be identified unambiguously [131]. A model-independent way to overcome is problem is the usage of the NRF-based self-absorption technique (for an introduc- tion to the technique, see [120], for experimental applications see, e.g. [132, 133]). Ultimately, the model-independence of NRF can be sacrifized to take into account unobserved branching transitions by various assumptions which are summarized in [134]. For the present study, the comparably straightforward access to branching ratios andmultipole mixing ratios [see Eq. (2.10)] and the selectivity to dipole excitations were seen as important advantages of NRF. Since several NRF cross sections of low-lying dipole excited states were already known from previous studies [135– 137], the the limited access to absolute cross sections in experiments with quasi- monochromatic photons could be partially compensated. Furthermore, a recently developed model-dependent method for the relative and absolute calibration of the beam photon flux was successfully applied (see Sec. 4.5.4). An access to absolute photoabsorption cross sections was found to be of minor importance, since the information about the relevant structure is contained in the branching ratios. Thus, 2For completeness, it should also be mentioned that the possibility of a coincident detection of different particle types, which of course complicates the experiment, is an invaluable technique to reduce experimental background. 62 2. Background the investment into large amounts of enriched materials and weeks of experiments for an NRF experiment were found to be worthwile. 2.1.2. Formalism In the following, consider the process in which a real photon is resonantly absorbed by a nucleus which is in an intrinsic state i (most probably the ground state, denoted as ’0’). The absorption leaves the nucleus in an excited state j, from which it decays to a lower-lying state k via emission of another photon. Although it is emphasized here that ’another’ photon is emitted, the process is often denoted as ’photon scattering’ [119] in analogy to actual scattering experiments. The energy-integrated differential cross section for this process is [119]3: dIi→ j→k dΩ dΩ= π2 � ħhc E j − Ei �2 ︸ ︷︷ ︸ λ2 i→ j 2J j + 1 2Ji + 1 ︸ ︷︷ ︸ gi→ j Γi→ j ︸ ︷︷ ︸ Ii→ j Γ j→k Γ j ︸︷︷︸ br. ︸ ︷︷ ︸ Ii→ j→k Wi→ j→k � θ ,ϕ,δi→ j ,δ j→k, Pγ � 4π dΩ. (2.4) In Eq. (2.4), the excitation energies and angular momentum quantum numbers of the states of the nucleus are denoted as Ei and Ji, respectively. The quantities Γi→ j and Γ j→k denote the partial transition widths for the excitation- (i→ j) and the decay transition ( j→ k), respectively, whose relation to the total width Γ j of the state j will be discussed below. The symbol Wi→ j→k(θ ,φ,δi→ j ,δ j→k, Pγ) denotes the angular distribution of the emitted photon in the transition from state j to k for the given sequence of states, which also depends on the multipole mixing ratios δi→ j and δ j→k of the excitation (i→ j) and the decay ( j→ k), and the polarization Pγ of the photon used for the excitation (see below)4. It is normalized to 4π and can be integrated out trivially in Eq. (2.4), resulting in the total cross section Ii→ j→k for the process, which is indicated by the lowermost brace. The next-to lowest 3Ref. [119] explicitly gives the cross section for the excitation from the ground state (i ≡ 0), but this assumption is not necesssary (see, e.g. chapter V, §17 and §20 in [121]). 4The dependence of the angular distribution on the multipole mixing ratios and the polarization of the beam will often not be shown explicitly in other sections for the sake of brevity. 2.1. Nuclear Resonance Fluorescence 63 brace indicates the definition of the total cross section Ii→ j for photoabsorption from state i to j which differs from Ii→ j→k by the branching ratio (br.) Γ j→k/Γ j for the subsequent decay to the state k. Equation (2.4) can be simplified further by introducing the definition of the reduced wavelength λi→ j of the excitation photon, and the ’spin factor’ [119] gi→ j which takes into account combinatorially the Jz substates of the initial and final states in the excitation transition. The total transition width Γ j is related to the lifetime τ j of the state j via [119]: Γ j = ħh τ j . (2.5) It is the sum of partial transition widths for all possible decay channels [119]: Γ j = ∑ k∈K Γ j→k = ∑ k∈K J j+Jk ∑ L=|J j−Jk| Γ j→k,σL (2.6) In Eq. (2.6), K is assumed to be the set of all lower-lying states which can be populated from state j. If all partial widths Γ j→k are be assumed to be due to EM transitions5, they can be further decomposed into partial widths Γ j→k,σL for the different multipole orders Li→ j 6 and corresponding EM characters σ. According to the EM selection rules [Eq. (2.12) and (2.15) in chapter XII.2.B [13]], the following combinations of σ and L are allowed for a given transition from j to k: � �J j − Jk � �≤ L j→k ≤ J j + Jk (2.7) σ j→k = ¨ E, π jπk = (−1)(L+1) M , π jπk = (−1)L . (2.8) 5Since the beam energies of the present experiment are much lower than the particle separation thresholds of the nuclei of interest [138, 139], decays of photoexcited states by particle emission can be neglected. On the other hand, even the lowest beam energy of 2.4MeV is high enough that the alternative process of the emission of a conversion electron from a dipole-excited state is negligible. This can be seen from estimates similar to the Weisskopf estimates from Sec. 2.1.1 (chapter XII.5.A in [13]). 6The index of L, which indicates the corresponding transition, will only be used if the initial and final state are not clear from the context. 64 2. Background The first selection rule, Eq. (2.7), which corresponds to the conservation of angular momentum, limits the possible values for the momentum transferred by/to a photon. The second selection rule, Eq. (2.8), determines the EM character based on the parity quantum numbers π j and πk of the initial and final states of the transition. The partial transitions widths are related to the nuclear matrix elements of EM transition operators [119]: Γ j→k,σL = 8π L + 1 L [(2L + 1)!!]2 λ −(2L+1) j→k gk→ j 1 2J j + 1 � �〈Ψk ÔσL Ψ j〉 � � 2 ︸ ︷︷ ︸ B(σL; j→k) ︸ ︷︷ ︸ B(σL;k→ j) (2.9) In Eq. (2.9), the reduced probability B(σL; j→ k) for the transition j→ k, which is indicated by the inner braces, has already been expanded to show its relationship to the reduced matrix element of the EM multipole operator ÔσL (see, e.g., the appendix ’Electromagnetic Transitions and Moments’ in [66] where also explicit expressions for the EM operators can be found). From the definition of the reduced probability for decay [B(σL; j → k)] it can be seen that it differs from the one for excitation [B(σL; k → j)] by the factor gk→ j. The definition of the latter is indicated by the outer braces in Eq. (2.9). As indicated by the triangle inequality for L in Eq. (2.7), a transition between two states may include multiple multipolarities. The relative magnitude of one multipole order (σL) and the next-higher one [σ′L′ = σ′(L + 1)] with different EM character (σ′ 6= σ) is quantified by the multipole mixing ratio (here in the convention of Krane, Steffen and Wheeler [12]): δ2 L,i→ j = Γi→ j,σ′(L+1) Γi→ j,σL = L (L + 2) (L + 1)2 (2L + 3)2 λ−2 i→ j B [σ′ (L + 1)] B (σL) (2.10) The angular distribution for the two-step process in Eq. (2.4) can be expanded in terms of Legendre polynomials Pν and unnormalized associated Legendre polyno- 2.1. Nuclear Resonance Fluorescence 65 mials Pµν [119]7: Wi→ j→k � θ ,ϕ,δi→ j ,δ j→k, Pγ � = Wi→ j→k,unpolarized � θ ,δi→ j ,δ j→k � (±)L′i→ j Pγ � Eγ � Wi→ j→k,polarized � θ ,ϕ,δi→ j ,δ j→k � = ∑ ν∈{0,2,4} Aν � JiJ j Li→ j L ′ i→ jδL,i→ j � × Aν � JkJ j Lk→ j L ′ k→ jδL,k→ j � × Pν [cos (θ )] (2.11) (±)L′i→ j Pγ � Eγ � ∑ ν∈{2,4} A′ν � JiJ j Li→ j L ′ i→ jδL,i→ j � × A′ν � JkJ j Lk→ j L ′ k→ jδL,k→ j � × P(2)ν [cos (θ )] cos (2ϕ) Eq. (2.11) consists of two terms. The first term, Wunpolarized(θ ) denotes the angular distribution for the excitation by an unpolarized photon beam. It depends on the polar angle θ with repect to the direction of propagation of the incoming photon beam. The second term, Wpolarized(θ ,ϕ) affects the angular distribution for a nonzero polarization of the beam, quantified by the photon-beam energy (Eγ)- dependent polarization factor Pγ with � �Pγ � �≤ 1. It introduces a dependence on the azimuthal angleϕ, which is the angle with respect to the electric field vector ~E of the photon beam, i.e. the polarization axis. The symbol (±)L′i→ j indicates that the sign of the polarization term is positive (negative) for an electric (magnetic) character of the alternative multipolarity of the excitation transition i→ j. The expansion coefficients Aν and A′ν, which depend on the sequence of angular momenta and 7Kneissl, Pitz and Zilges [119] assume that no larger momentum transfer than L = 2 occurs. This also imposes the restriction that no more than 2 multipoles may be mixed in a single transition. In accordance with the expected dominance of low multipolarities (see Sec. 2.1.1), and since only dipole- and quadrupole transitions are relevant in this work, the same equations are given here. The most general formalism can be found in [12]. 66 2. Background the corresponding EM multipoles, can be found in [12]. 2.2. Nuclear Structure Models This section introduces the nuclear shell model (SM) and the proton-neutron version of the the interacting boson model (IBM, or IBM-2 for the π-ν version), both of which were employed in this work to describe the structure of the 0νββ- decay candidate pairs 82Se/82Kr and 150Nd/150Sm. A review article on the SM was published by Caurier, Martínez-Pinedo, Nowacki, Poves, and Zuker [5], while for the IBM, the textbook by Iachello and Arima [140] is usually cited. The models will be motivated as subsequent approximations of the full A-body problem. 2.2.1. Shell Model The introduction of the SM approximation follows a recent review article by Coraggio, Covello, Gargano, Itaco, and Kuo [141] (in particular Sec. 3 therein), which is more specialized than [5]. It starts from a general Hamiltonian H of the A-body system: Ĥ = ∑ a p̂2 a 2m ︸︷︷︸ k̂a + ∑ a