department of physics university of jyvÄskylÄ research report no. 2/2012 supernova ... ·...

DEPARTMENT OF PHYSICSUNIVERSITY OF JYVÄSKYLÄ

RESEARCH REPORT No. 2/2012

SUPERNOVA-NEUTRINO INDUCED REACTIONS ON

MOLYBDENUM VIA NEUTRAL-CURRENTS

BY

EMANUEL YDREFORS

Academic Dissertationfor the Degree of

Doctor of Philosophy

To be presented, by permission of the

Faculty of Mathematics and Natural Sciences

of the University of Jyväskylä,

for public examination in Auditorium FYS1 of the

University of Jyväskylä on March 2, 2012

at 12 o'clock noon

Jyväskylä, FinlandMarch 2012

Preface

The work presented in this thesis was performed at the Department of Physics at theUniversity of Jyväskylä during the years 2008 to 2011. The research has been foundedby the University of Jyväskylä, the Graduate School in Particle and Nuclear Physics(GRASPANP) and the Ellen and Artturi Nyyssönen Foundation.

I would like to thank my supervisor Prof. Jouni Suhonen for introducing me to therealm of theoretical nuclear physics and for the excellent guidance of the researchproject. I also want to acknowledge Dr. Mika Mustonen for many fruitful discussionsconcerning nuclear theory and neutrino physics. I also want to thank my examinersProf. Cristina Volpe and Prof. Jonathan Engel for reviewing the manuscript.

Lastly, I want to express my gratitude to my family and several other friends for alltheir support and encouragement.

Jyväskylä, January 2012

Emanuel Ydrefors

i

ii Preface

Abstract

In this thesis which consists of an introductory part and six publications the neutral-current neutrino-nucleus scatterings o the stable molybdenum nuclei are studiedusing microscopic nuclear models. The main focus is on supernova neutrinos and thenuclear responses to them for the aforementioned nuclei are computed by folding thecomputed cross sections with realistic neutrino-energy spectra.

In this work both the coherent and incoherent contributions to the cross sections areconsidered. The contributions from the prominent multipole channels to the incoherentcross sections are also discussed in great detail.

iii

Author's address Emanuel YdreforsDepartment of PhysicsUniversity of JyväskyläFinland

Supervisor Professor Jouni SuhonenDepartment of PhysicsUniversity of JyväskyläFinland

Reviewers Professor Cristina VolpeInstitut de Physique NucléaireUniversity of OrsayFrance

Professor Jonathan EngelDepartment of Physics and AstronomyUniversity of North CarolinaUSA

Opponent Professor Karl-Heinz LangankeGSI Helmholtzzentrum für SchwerionenforschungGermany

iv

Publications

I. E. Ydrefors, M. T. Mustonen and J. Suhonen, MQPM description of the struc-

ture and beta decays of the odd A=95, 97 Mo and Tc isotopes, Nucl. Phys. A842 (2010) 33-47.

II. E. Ydrefors and J. Suhonen, Theoretical estimates of responses to supernova

neutrinos for the stable molybdenum isotopes, AIP conf. proc. 1304 (2010) 432-436.

III. E. Ydrefors, K. G. Balasi, J. Suhonen and T. S. Kosmas, Nuclear Responses toSupernova Neutrinos for the Stable Molybdenum Isotopes, in: J. P. Greene (Ed.),Neutrinos: Properties, Reactions, Sources and Detection, Physics Research andTechnology, Nova Science Publishers, 2011.

IV. E. Ydrefors, K. G. Balasi, T. S. Kosmas and J. Suhonen, The nuclear response

of 95,97Mo to supernova neutrinos, Nucl. Phys. A 866 (2011) 67-78. Erratum toappear in Nucl. Phys. A.

V. K. G. Balasi, E. Ydrefors and T. S. Kosmas, Theoretical study of neutrino

scattering o the stable even Mo isotopes at low and intermediate energies,Nucl. Phys. A 868-869 (2011) 82-98.

VI. E. Ydrefors and J. Suhonen, The nuclear response of molybdenum to supernova

neutrinos. Accepted for publication in AIP conf. proc.

The author of the present thesis has performed the numerical calculations for thelisted publications and written the rst draft of publications I, II, III, IV and VI andparts of the manuscript for V. The author has derived the expressions for the neutral-current transition densities in the microscopic quasiparticle-phonon model (MQPM).The author has implemented computer codes for calculation of transition densitiesand the cross sections for neutral-current neutrino scattering o nuclei.

v

vi Publications

Contents

1 Introduction 1

2 Neutral-current neutrino scattering o nuclei 3

2.1 The semi-leptonic weak Hamiltonian . . . . . . . . . . . . . . . . . . . 32.2 Cross-section formula for neutral-current ν-nucleus scattering . . . . . 5

3 Quasiparticle description of open-shell nuclei 7

3.1 Nuclear mean eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Quasiparticle random-phase approximation . . . . . . . . . . . . . . . . 83.3 Microscopic quasiparticle-phonon model . . . . . . . . . . . . . . . . . 9

4 Results and Discussion 11

4.1 Nuclear structure of the stable molybdenum isotopes . . . . . . . . . . 114.2 Neutral-current neutrino-nucleus cross sections . . . . . . . . . . . . . 14

5 Conclusions 25

A Neutral-current transition densities in the MQPM 31

vii

viii Contents

1 Introduction

The history of neutrino physics began already in 1930 when W. Pauli proposed theneutrino to explain the missing energy in nuclear beta decay. The neutrino and itsproperties have since then been a rapidly expanding research eld with e.g. the obser-vations of the non-zero neutrino mass and neutrino avour oscillations [1], see [2] fora review. There are however still many open questions such as e.g. the unknown valueof the neutrino mass, the Dirac-or-Majorana nature of the neutrino and the value ofthe third neutrino mixing angle.

From the astrophysical point of view neutrinos and neutrino-nucleus reactions playimportant roles in e.g. supernova explosions and for the nucleosynthesis of the heavyelements [3, 4]. Neutrinos interact only weakly with matter and neutrinos from astro-physical sources can therefore be detected by Earth-bound detectors by using neutral-current and charged-current neutrino-nucleus reactions. Supernova neutrinos thereforeconstitute valuable probes of both the unknown supernova mechanisms and neutrinoproperties [5]. One example of a possibility for future measurements is the MOON (MoObservatory of Neutrinos) project [6]. In this experiment the neutrinos will be mea-sured by using both charged-current and neutral-current neutrino-nucleus scattering.Observations of the neutral-current induced neutrino-nucleus reactions are importantin order to test the eects of neutrino oscillations on the measured energy spectra ofthe electrons produced by inversed beta decays [7].

Nuclear responses to neutrinos are important both for measurements and as inputs insupernova simulations [8]. Neutral-current neutrino-nucleus scattering is independentof the neutrino avour and can therefore be adopted also for detection of muon and tauneutrinos. Neutral-current detectors such as the OMNIS (Observatory for MultiavorNeutrInos from Supernova) [9] or the LAND (Lead Astronomical Neutrino Detector)[10] could therefore be used to test current supernova models and to study neutrinooscillations. The purpose of this work is therefore to study neutral-current neutrinoscattering o open-shell nuclei. The calculations presented in this thesis are based onthe Donnelly-Walecka method [11, 12] for the treatment of semi-leptonic processes innuclei. In its original form the theory was written in the isospin formalism. As part ofthis thesis the theory therefore has been formulated in the proton-neutron formalismwhich is more suitable for the treatment of heavy open-shell nuclei.

As the main application of our theory framework the cross sections for the neutrinoscatterings o the stable (A = 92, 94, 95, 96, 97, 98, 100) molybdenum isotopes are

1

2 Introduction

calculated. It should however be noted that the tools presented in this thesis could beapplied to other nuclear systems as well.

In addition, the 100Mo nucleus is a famous double-beta-decay emitter and has beenused by several current experiments e.g. the NEMO (Neutrino Ettore Majorana Ob-servatory) [13]. Thus nuclear-structure calculations in the Mo region are welcome frommany points of view. The obtained nuclear wave functions are essential in studies ofweak processes in 100Mo and in the other stable molybdenum nuclei. In the case of theneutrino-nucleus scattering the cross sections are derived from these wave functionsand then folded with the appropriate neutrino spectra in order to obtain realisticestimates for the nuclear responses to supernova neutrinos.

An accurate description of the nal and initial nuclear states is a key ingredient inrealistic neutrino-nucleus scattering calculations. In the following work the quasipar-ticle random-phase approximation (QRPA) [14] is adopted for the even-even isotopes.The possibility of using large valence spaces means that the QRPA and its exten-sions, e.g. the MAVA model [15], constitute therefore the most feasible models forthe construction of vibrational states in heavy open-shell nuclei [16]. For the oddnuclei (95Mo and 97Mo) the nuclear states are constructed by using the microscopicquasiparticle-phonon model (MQPM) [17].

2 Neutral-current neutrino

scattering o nuclei

The calculations in this thesis are based on the so-called Donnelly-Walecka formalismfor the treatment of semi-leptonic processes in nuclei. The general theory is outlinedin the comprehensive work by J. D. Walecka [18] and the application of the formalismto neutral-current reactions in nuclei was reviewed in [12]. An extensive review of thetheory in the form used in the present computations is given in [19]. In this chapter theformalism for the treatment of neutral-current neutrino-nucleus scattering thereforeis only briey summarized.

2.1 The semi-leptonic weak Hamiltonian

The Feynman diagram for the neutral-current neutrino scattering o a nucleus withproton number Z and mass number A is shown in Fig. 2.1. In this process whichproceeds via the exchange of a neutral Z0 boson the nuclear nal state is eitherthe same as the initial state (coherent scattering) or an excited state of the nucleus(incoherent scattering which is indicated by an asterisk). In the gure kµ and k′µ arethe four momenta of the incoming and outgoing neutrinos and pµ and p′µ representthe four momenta of the initial and nal nuclear states. Here the same conventions asin [19] are adopted. The four-momentum vector for the incoming neutrino thereforeis written as kµ = (Ek,−k) where Ek is the energy and k is the three momentum. Forthe outgoing neutrino we have similarly k′µ = (Ek′ ,−k′). The four momenta pµ andp′µ are dened correspondingly.

In the computations of this thesis the transferred four momentum qµ fulls −qµqµ M2

Z where MZ is the mass of the Z0 boson. The complicated coupling in Fig. 2.1can therefore to an excellent approximation be replaced by an eective four-pointinteraction. The nuclear matrix element of the eective weak Hamiltonian can thenbe cast into the form

〈f |He|i〉 =G√

2

∫lµd

3xe−iq·x〈f |J µ(x)|i〉, (2.1)

where for the neutral-current processes the coupling constant is G = GF = 1.1664 ×

3

4 Neutral-current neutrino scattering o nuclei

lepton current jleptµ

ν ′

k′µ

νkµ (A, Z)

pµ

(A, Z)∗p′µ

qµ

Z0

hadron current Jµ

Figure 2.1: Neutral-current neutrino scattering o a nucleus (A,Z). The transferred fourmomentum is here qµ = k′µ − kµ.

10−5 GeV. In (2.1) the hadron current J µ(x) is of the V− A structure

J µ(x) = JV,µ(x)− JA,µ(x), (2.2)

and lµ is dened by〈f |jleptµ (x)|i〉 = lµe

−iq·x, (2.3)

where jleptµ (x) represents the lepton current. At the origin (x = 0) the single-nucleonmatrix elements of the hadron currents JV,µ(x) and JA,µ(x) can be expressed in thegeneral forms

p(n)〈p′σ′|JV,µ(0)|pσ〉p(n)=u(p′, σ′)

V

[F

NC;p(n)1 (Q2)γµ − i

mNσµνqνF

NC;p(n)2 (Q2)

]u(p, σ), (2.4)

and

p(n)〈p′σ′|JA,µ(0)|pσ〉p(n) =u(p′, σ′)

VF

NC;p(n)A (Q2)γ5γ

µu(p, σ), (2.5)

where |pσ〉p(n) denotes the state vector of a free proton (neutron) with three-momentump and spin σ and mN denotes the nucleon mass. In the present theory the nucleonsare treated as point-like particles. The quark degree of freedom is then approximatelytaken into account by introducing the nucleon form factors FNC;p(n)

1 (Q2), FNC;p(n)2 (Q2)

and FNC;p(n)A (Q2) in (2.4) and (2.5). In this work the adopted expressions for the nu-

cleon form factors are those of [19, 20, 21].

2.2 Cross-section formula for neutral-current ν-nucleus scattering 5

2.2 Cross-section formula for neutral-current ν-nucleus

scattering

The double-dierential cross section for neutral-current neutrino-nucleus scatteringfrom an initial state i with angular momentum Ji to a nal state f characterized byangular momentum Jf can by a multipole expansion of 2.1 be written [18, 19](

d2σi→fdΩdEexc

)ν/ν

=G2

F|k′|Ek′

π(2Ji + 1)

(∑J>0

σJCL +∑J>1

σJT

), (2.6)

where

σJCL = (1 + cos θ)∣∣(Jf‖MJ(q)‖Ji)

∣∣2 +(1 + cos θ − 2b sin2 θ

)∣∣(Jf‖LJ(q)‖Ji)∣∣2

+ qEexc(1 + cos θ)2Re

(Jf‖MJ‖Ji)∗(Jf‖LJ(q)‖Ji), (2.7)

and

σJT =(1− cos θ + b sin2 θ

)(∣∣(Jf‖T magJ (q)‖Ji)

∣∣2 +∣∣(Jf‖T el

J (q)‖Ji)∣∣2)

∓ Ek + Ek′

q(1− cos θ)2Re

(Jf‖T mag

J (q)‖Ji)(Jf‖T elJ (q)‖Ji)∗

. (2.8)

Here Eexc = Ek −Ek′ is the excitation energy of the nuclear nal state and θ denotesthe angle between the incoming and outgoing neutrinos. The magnitude of the three-momentum transfer is furthermore given by

q = |q| =√E2

exc + 2EkEk′(1− cos θ) (2.9)

and b = EkE′k/q

2 has also been introduced. In (2.8) the minus sign is used for neutrinosand the plus sign for anti-neutrinos. The operatorsMJ(q), LJ(q), T mag

J (q) and T elJ (q)

are the ones dened in e.g. [19].

In (2.7) and (2.8) one-body forms are assumed for the operators and they can conse-quently be expanded as [22]

TJM = J−1∑ab

(a‖TJ |b)[c†acb]JM , (2.10)

and therefore(Jf‖TJ‖Ji) = J−1

∑ab

(a‖TJ‖b)(Jf‖[c†acb]J‖Ji), (2.11)

where the index a holds the single-particle quantum numbers na, la and ja. Herec†a is the particle creation operator and the time-reversed annihilation operator cais dened through the magnetic quantum number mα as cα = (−1)ja+mαc−α where

6 Neutral-current neutrino scattering o nuclei

−α = (a,−mα). In (2.11) both a and b are either proton or neutron indices and thesummations run over all single-particle orbitals in the chosen valence space. Expres-sions for the reduced transition densities (Jf‖[c†acb‖Ji) for the MQPM are given inappendix A and the transition densities in the QRPA are discussed in great detail in[22].

3 Quasiparticle description of

open-shell nuclei

In the included publications of this thesis the quasiparticle random-phase approxima-tion (QRPA) is adopted to construct the initial and nal nuclear states of the even-even isotopes. For the odd-mass nuclei (95Mo and 97Mo) the microscopic quasiparticle-phonon model (MQPM) is used. In this chapter the quasiparticle mean-eld is rstintroduced. The formalisms of the QRPA and the MQPM are then briey summarized.More comprehensive treatments are given in [22] (QRPA) and in [17] (MQPM).

3.1 Nuclear mean eld

In this thesis nuclear systems consisting of up to 100 nucleons are considered. Thestandard way to solve this rather involved many-body problem is to use the so-calledmean-eld approximation. The dominant part of the nuclear interaction is then takeninto account by constructing a nuclear mean-eld of non-interacting particles. Thenuclear Hamiltonian with a two-body interaction V is hence written as

H =∑α

εαc†αcα + 1

4

∑αβγδ

vαβγδc†αc†βcδcγ, (3.1)

where vαβγδ = 〈αβ|V |γδ〉 − 〈αβ|V |δγ〉. Here the notation introduced in Sec. 2.2 isadopted and thus e.g. α = (a,mα) with a = (na, la, ja). The single-particle energies εain (3.1) can either be generated by the self-consistent Hartree-Fock (HF) procedure(see e.g. [22]) or by the use of a phenomenological nuclear potential. In this work theCoulomb-corrected Woods-Saxon potential has been adopted.

An appropriate description of the pairing correlations is essential for the description ofthe nuclear states in open-shell nuclei. Evidence for nuclear pairing comes from e.g theodd-even mass dierence and the fact that most open-shell nuclei have low-lyingstates of spherical shape. The foundations for the treatment of pairing phenomena infermionic systems were laid by Bardeen, Cooper and Schrieer [23]. The importanceof pairing correlations in nuclei were rst recognized by A. Bohr, B. R. Mottelsonand D. Pines [24]. The BCS theory constitutes today one of the standard tools intheoretical nuclear physics. The starting point in this formalism is the ansatz for the

7

8 Quasiparticle description of open-shell nuclei

BCS ground state|BCS〉 =

∏α>0

(ua − vac†αc†α

)|HF〉, (3.2)

where the amplitudes ua and va are solved iteratively from the BCS equations. Thequasiparticles are subsequently dened via the Bogoliubov-Valatin transformation

a†α =uac†α + vacα,

aα =uacα − vac†α.(3.3)

With (3.3) the nuclear Hamiltonian (3.1) in quasiparticle representation takes theform

H = H11 +H02 +H20 +H13 +H31 +H40 +H04, (3.4)

where Hij is proportional to a normal-ordered product of i quasiparticle creation op-erators and j quasiparticle annihilation operators. In (3.4) a constant term has beenomitted which anyway has no contribution to the excitation energies of interest forthe present work. By the BCS procedure, i.e. by the minimization of the energy ofthe BCS ground state (3.2), the terms H02 and H20 vanish. In (3.4) the diagonalterm H11 contains most of the nucleon-nucleon short-range interactions and by theBCS approximation therefore a so-called quasiparticle mean eld is introduced, i.e. ageneralization of the Hartree-Fock mean-eld of particles and holes. In practice theparameters of the BCS calculation however have to be tted to reproduce the exper-imental pairing gaps. In this work the emperical pairing gaps have been computedfrom the three-point formulae [22]

∆p =14(−1)Z+1(Sp(A+ 1, Z + 1)− 2Sp(A,Z) + Sp(A− 1, Z − 1)),

∆n =14(−1)N+1(Sn(A+ 1, Z)− 2Sn(A,Z) + Sn(A− 1, Z)),

(3.5)

by using the proton (Sp) and neutron (Sn) separation energies of [25]. In (3.5) Zdenotes the proton number and N the neutron number of the even-even referencenucleus.

3.2 Quasiparticle random-phase approximation

In the QRPA the excited states of the even-even nucleus are constructed by couplingtwo-quasiparticle operators to good angular momentum Jω and parity πω. The statevector for the excitation ω = (Jω, πω, kω) is then given by

|ω〉 = Q†ω|QRPA〉, (3.6)

whereQ†ω =

∑a6a′

σ−1aa′(Xωaa′[a†aa

†a′]JωMω

+ Y ωaa′ [aaaa′ ]JωMω

), (3.7)

3.3 Microscopic quasiparticle-phonon model 9

and |QRPA〉 denotes the QRPA vacuum. Here σaa′ =√

1 + δaa′ and kω enumeratesphonons with the same angular momentum and parity. In (3.7) the sum runs over allproton-proton and neutron-neutron congurations so that none of them are countedtwice. The QRPA equations can be expressed in the matrix form [22](

A B−B∗ −A∗

)(Xω

Yω

)= Eω

(Xω

Yω

), (3.8)

where A is the well-known quasiparticle Tamm-Danco (QTDA) matrix and B con-tains the induced ground-state correlations. In (3.8) Eω denotes the energy of theQRPA phonon, i.e. the excitation energy of the state |ω〉 in a nucleus. The eigenen-ergies (Eω) and the eigenvectors (Xω and Yω) are then solved from (3.8) by usinge.g the diagonalization method introduced by Ullah and Rowe [26].

3.3 Microscopic quasiparticle-phonon model

In the MQPM the building blocks are quasiparticles and QRPA phonons determinedin the even-even reference nucleus adjacent to the odd-A nucleus under discussion. Thestates of the odd-A nucleus are then constructed by using the creation operator

Γ†k(jm) =∑n

Xkna†njm +

∑aω

Xkaω

[a†aQ

†ω

]jm, (3.9)

where j denotes the angular momentum and m the z-projection of the state. Thequantum number k enumerates here states with the same angular momentum andparity. The MQPM equations of motion [17] then lead to a generalized symmetriceigenvalue problem of the form

Hψkjm = Ek

jNψkjm, (3.10)

where H and N are the Hamiltonian matrix and the overlap matrix respectively andthe components of the eigenvector ψk

jm are the amplitudes Xkn and Xk

aω of (3.9). Forthe explicit forms of the matrices H and N the reader is referred to [17]. In (3.10)Ekj denotes the energy of the MQPM state. In this work the overcompleteness and

non-orthogonality of the quasiparticle-phonon basis have been handled by using thediagonalization method established by Rowe [27].

10 Quasiparticle description of open-shell nuclei

4 Results and Discussion

In this chapter the performed calculations of the neutral-current neutrino-nucleuscross sections for the stable molybdenum isotopes upon which this thesis is basedare reviewed. The computed energy spectra and wave functions are rst considered inSec. 4.1. The results for the neutrino-nucleus cross sections are subsequently discussedin Sec. 4.2.

4.1 Nuclear structure of the stable molybdenum iso-

topes

In the nuclear-structure calculations carried out for the even-even Mo isotopes in[28] and for the odd nuclei (95Mo and 97Mo) in [29] we adopted a single-particlevalence space composed of the full 3~ω and 4~ω harmonic-oscillator shells plus the0h11/2 and 0h9/2 orbitals both for protons and neutrons. The single-particle energieswere rst computed from the Coulomb-corrected Woods-Saxon potential by usingthe parametrization by Bohr and Mottelson [30]. In the next step the quasiparticleswere dened by performing BCS calculations for protons and neutrons separately. Inthese computations the monopole part of the interaction was scaled by a constantgpairp (gpairn ) for protons (neutrons) so that the energy of the lowest quasiparticle statewas roughly equal to the emperical pairing gap ∆p (∆n) computed from (3.5). Inthe present calculations the Bonn one-boson-exchange potential [31] was used as thetwo-body interaction. The agreement between the computed BCS spectra and theavailable experimental level data could be improved by slightly adjusting the single-particle energies for some of the states in the vicinity of the respective Fermi surfaces.The excited states of the even-even nuclei were subsequently constructed by using theQRPA. In the QRPA calculations the particle-particle and particle-hole parts of theQRPA matrix were scaled by the parameters gpp and gph respectively. For the odd-Anuclei the calculation proceeds further by the use of the MQPM. For the MQPMcalculations no further adjustments of parameters were made.

In Fig. 4.1 the calculated excitation energies of 94Mo and 100Mo are shown togetherwith the available data [32, 33]. It is seen from the gure that the results are insatisfactory agreement with the experimental energies. The obtained results are alsosupported by the higher-QRPA calculations performed in [34]. Here it should be noted

11


0.0

0.5

1.0

1.5

2.0

2.5

3.0

Ener

gy(M

eV)

0+ 0.00

2+ 0.89

4+ 1.76

2+ 2.04

2+ 2.383− 2.466+ 2.474+ 2.47

94Mo QRPA

0+ 0.00

2+ 0.87

4+ 1.570+ 1.742+ 1.862+ 2.07

2.124+ 2.29

(6+) 2.322+ 2.39

94Mo exp.

0+ 0.00

2+ 0.54

0+ 1.49

4+ 1.74

3− 1.98

0+ 2.216+ 2.292+ 2.315− 2.347− 2.43

100Mo QRPA

0+ 0.00

2+ 0.540+ 0.69

2+ 1.064+ 1.14

2+ 1.460+ 1.50

(3+) 1.61

(2+) 1.77

(4+) 1.776+ 1.853− 1.91

(1, 2+) 1.980+ 2.04

100Mo exp.

Figure 4.1: Comparison between the QRPA computed energy spectra and the experimentaldata of [32, 33] for 94Mo and 100Mo.

that it was concluded in [34] that the 2+2 and 4+

2 states in 94Mo and the 0+2 state in

100Mo are of two-phonon character.

In Fig. 4.2 the MQPM results for the energy spectra of the odd isotopes (95Mo and97Mo) are compared with the experimental energies of [35, 36]. It can be concludedthat the agreement with the data is excellent except for the energy of the 3/2+

1 state in95Mo. In [29] the quality of the MQPM wave functions were tested also by computinglog ft values of beta-decay transitions from 95Tc to 95Mo and between 97Tc and 97Mo.The obtained results were in fair agreement with experimental data and the nuclearwave functions can thus be considered as a suitable starting point for neutrino-nucleusscattering calculations. The distribution of the computed states in 95Mo are displayedin Fig. 4.3. It is seen in the gure that the states are quite irregularly distributed withrespect to the excitation energy Eexc. This is explained by the cut of the quasiparticle-phonon basis. The same pattern is repeated for 97Mo and those results are thereforenot shown here.

4.1 Nuclear structure of the stable molybdenum isotopes 13

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Ener

gy(M

eV)

5/2+ 0.00

1/2+ 0.81

3/2+ 0.86

3/2+ 0.97

7/2+ 0.99

5/2+ 1.02

1/2+ 1.31

9/2+ 1.32

7/2+ 1.38

5/2+ 1.61

3/2+ 1.87

11/2+ 1.93

9/2+ 1.93

5/2+ 1.93

95Mo MQPM

5/2+ 0.00

3/2+ 0.20

7/2+ 0.77

1/2+ 0.79

3/2+ 0.82

9/2+ 0.95

1/2+ 1.04

5/2+ 1.06

7/2+ 1.07

3/2+, 5/2+ 1.09

1/2+ 1.30

(3/2+, 5/2+) 1.32

(3/2) 1.37

+ 1.38

(5/2)+ 1.43

95Mo exp.

5/2+ 0.00

1/2+ 0.69

3/2+ 0.74

3/2+ 0.80

7/2+ 0.81

5/2+ 1.06

1/2+ 1.10

9/2+ 1.28

7/2+ 1.28

3/2+ 1.62

5/2+ 1.66

7/2+ 1.67

11/2+ 1.69

9/2+ 1.69

97Mo MQPM

5/2+ 0.00

3/2+ 0.48

7/2+ 0.66

1/2+ 0.68

5/2+ 0.72

3/2+ 0.72

(3/2+, 5/2+) 0.75

(1/2+) 0.80

1/2+ 0.89

(3/2+, 5/2+) 0.99

7/2+ 1.02

3/2+ 1.09

11/2− 1.12

9/2+ 1.12

3/2+, 5/2+ 1.12

97Mo exp.

Figure 4.2: Comparison between the MQPM calculated energy spectra and experimentaldata [35, 36] for 95Mo and 97Mo.

0

50

100

Num

ber

ofst

ates

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0Eexc [MeV]

Figure 4.3: Distribution of the calculated states in 95Mo. In the gure the height of the bink corrreponds to the number of states having excitation energies Eexc ∈ [k∆E, (k + 1)∆E]where k = 0, 1, . . . and ∆E = 0.5 MeV. The fraction of states which contributes signicantlyto the neutral-current neutrino-nucleus scattering cross sections is shown in red.


4.2 Neutral-current neutrino-nucleus cross sections

In the papers [28, 38] we subsequently computed the cross sections for the neutral-current neutrino-nucleus scattering o the even-even and odd molybdenum isotopesrespectively by using the formalism outlined in [19]. In the calculations the double-dierential cross section (2.6) was rst calculated for all nal states f , scatteringangles θ and neutrino energies Ek. The total cross section, σ(Ek), was then computedby numerical integration over the scattering angle and subsequently adding up all theindividual contributions of the included nuclear nal states.

In Figs. 4.4 and 4.5 the coherent and incoherent contributions to the reactions94Mo(ν, ν ′)94Mo∗ and 95Mo(ν, ν ′)95Mo∗ as functions of the energy of the incomingneutrino are shown. From the gures it can be concluded that for both reactionsthe total cross section is dominated by the coherent, g.s. −→ g.s., transition for theneutrino energies relevant to supernova neutrinos. The same characteristics were alsoobtained for the other molybdenum isotopes.

10−1

100

101

102

103

104

σ(E

k)[1

0−

42cm

2]

0 20 40 60 80 100 120Ek [MeV]

incoherenttotalcoherent

Figure 4.4: Coherent, incoherent and total cross sections as functions of the energy of theincoming neutrino for the neutral-current neutrino-nucleus scattering o 94Mo.

4.2 Neutral-current neutrino-nucleus cross sections 15

100

101

102

103

104

σ(E

k)[1

0−

42cm

2]

0 20 40 60 80 100 120Ek [MeV]

incoherenttotalcoherent

Figure 4.5: Coherent, incoherent and total cross sections as functions of the energy of theincoming neutrino for the reaction 95Mo(ν, ν ′)95Mo∗.

In [38, 39] we studied the contributions from the possible multipole channels of (2.6) tothe incoherent neutrino-nucleus cross sections. The results for 94Mo, 95Mo and 100Mofor the energies Ek = 13.2 MeV and Ek = 60.0 MeV of the incoming neutrino arepresented in Figs. 4.6 and 4.7 respectively. In the gures the contributions from theaxial-vector and vector components of the nuclear current (2.2) to each multipole arealso shown. In Fig. 4.6 and Fig. 4.7 the contribution ("interference") coming from themixing of the axial-vector and vector parts of the nuclear current is also displayed. Itis visible in Fig. 4.6 that in the low-energy region excitations by the 1+ multipole arethe most prominent ones for the neutral-current neutrino-nucleus scattering o 94Mo,95Mo and 100Mo. The same conclusions holds also for the other molybdenum isotopes.The conclusion which we stated in [39] concerning the large contribution from the0+ multipole to the cross sections for 98Mo and 100Mo is therefore wrong. This wascaused by a minor bug found in our computer codes.

The contributions from the 1+ multipole for 98Mo and 100Mo are however smallerthan the ones for 94Mo and 96Mo. For small neutrino energies the 1+ transitions pro-ceed mainly through the operator T el,A

1 . The essential nuclear-structure informationis thus contained in the strength functions S1+(Eexc) which here have been dened asS1+(Eexc) = |√6π(1+

f ‖T el,A1 (q = 0.0)‖0+

g.s.)|2 for all 1+ nal states 1+f with excitation

energies Ef . The calculated strength functions for 94,96Mo and 98,100Mo are shown inFigs. 4.8 and 4.9 respectively. It can be concluded from the gures that for 94,96Momost of the 1+ strength is coming from transitions to states with Eexc < 10.0 MeV.On the other hand in 98Mo and 100Mo the bulk of the total strengths correspondsto states with Eexc > 10.0 MeV. The smaller 1+ contributions for 98,100Mo comparedto the ones of 94,96Mo are consequently explained by the fact that for the prominent1+ excitations in the heaver isotopes (98Mo and 100Mo) the energy of the outgoing


neutrino Ek′ is smaller leading to a suppression of cross sections because of the phasespace factor |k′|Ek′ in (2.6). From Fig. 4.7 it can furthermore be concluded that forlarger neutrino energies the picture is more complicated with many contributing mul-tipoles. In this region of neutrino energies transitions of axial-vector nature dominatethe cross sections.

0.0

1.0

2.0

3.0

4.0

5.0

σin

coh(E

k)[1

0−

42cm

2]

0− 0+ 1− 1+ 2− 2+ 3− 3+

Jπ

vectoraxial-vectorinterference

Ek =13.2 MeV

94Mo(ν, ν ′)94Mo

(a) 94Mo

0.0

3.0

6.0

9.0

12.0

σin

coh(E

k)[1

0−

42cm

2]

0− 0+ 1− 1+ 2− 2+ 3− 3+

Jπ


Ek =13.2 MeV

95Mo(ν, ν ′)95Mo

(b) 95Mo

0.0

1.0

2.0

σin

coh(E

k)[1

0−

42cm

2]

0− 0+ 1− 1+ 2− 2+ 3− 3+

Jπ


Ek =13.2 MeV

100Mo(ν, ν ′)100Mo

(c) 100Mo

Figure 4.6: Contributions of the dominant multipole channels to the incoherent neutrino-nucleus scattering o 94Mo, 95Mo and 100Mo for the energy Ek = 13.2 MeV of the incomingneutrino.


0.0

1.0

2.0

3.0

4.0

σin

coh(E

k)[1

0−

40cm

2]

0− 0+ 1− 1+ 2− 2+ 3− 3+ 4− 4+

Jπ


Ek =60 MeV

94Mo(ν, ν ′)94Mo

(a) 94Mo

0.0

1.0

2.0

σin

coh(E

k)[1

0−

40cm

2]

0− 0+ 1− 1+ 2− 2+ 3− 3+ 4− 4+

Jπ


Ek =60 MeV

95Mo(ν, ν ′)95Mo

(b) 95Mo

0.0

1.0

2.0

3.0

σin

coh(E

k)[1

0−

40cm

2]

0− 0+ 1− 1+ 2− 2+ 3− 3+ 4− 4+

Jπ


Ek =60 MeV

100Mo(ν, ν ′)100Mo

(c) 100Mo

Figure 4.7: Contributions of the dominant multipole channels to the incoherent neutrino-nucleus scattering o 94Mo, 95Mo, and 100Mo for the energy Ek = 60.0 MeV of the incomingneutrino.


0

1

2

3

4

5

6

7

S1+(E

exc)

0 5 10 15 20Eexc[MeV]

94Mo −→94 Mo∗

Σ =10.093

0

1

2

3

4

5

6

7

S1+(E

exc)

0 5 10 15 20Eexc[MeV]

96Mo −→96 Mo∗

Σ =10.348

Figure 4.8: Computed strength functions for transitions from the QRPA ground state toexcited 1+ states in 94Mo and 96Mo respectively.


0

1

2

3

4

5

6

7

S1+(E

exc)

0 5 10 15 20Eexc[MeV]

98Mo −→98 Mo∗

Σ =9.878

0

1

2

3

4

5

6

7

S1+(E

exc)

0 5 10 15 20Eexc[MeV]

100Mo −→100 Mo∗

Σ =10.019

Figure 4.9: Computed strength functions for transitions from the QRPA ground state toexcited 1+ states in 98Mo and 100Mo respectively.


The most important quantity from the experimental point of view is the averaged crosssection, 〈σ〉, which is obtained by folding the total cross section σ(Ek) with a givenprole of neutrino energies. In this thesis the main focus is on the supernova neutrinosand the spectrum of neutrino energies can then be approximated by a two-parameterFermi-Dirac distribution, i.e.

〈σ〉 =1

F2(α)T 3

∫dEkσ(Ek)E

2k

exp(Ek/T − α) + 1, (4.1)

where T denotes the neutrino temperature and α is the degeneracy parameter. In(4.1) the neutrino ux is normalized to unity by the constant F2(α). The values of Tand α employed in this work are presented in Table 4.1. In the table the correspondingaverage neutrino energies 〈Ek〉 are also given.

avour α T [MeV] 〈Ek〉 [MeV]

νe 2.1 3.6 13.0νe 3.2 3.8 15.4νµ, ντ , νµ, ντ 0.8 4.8 15.7

Table 4.1: Adopted values from [40] of the parameter α and the neutrino temperature T ,and the corresponding average neutrino energy 〈Ek〉 for the dierent neutrino avours.

The calculated averaged cross sections for the coherent neutral-current neutrino-nucleus scattering o the stable molybdenum isotopes are shown in Table 4.2. Theresults for the incoherent neutrino-nucleus scattering are similarly presented in Table4.3. From Table 4.1, Table 4.2 and Table 4.3 it is seen that both the incoherent andcoherent cross sections increase signicantly with increasing mean neutrino energy〈Ek〉. As is seen in Table 4.2 the coherent cross sections for the odd nuclei (95Moand 97Mo) are almost the same as the ones for their even-even partners (94Mo and96Mo respectively). On the contrary the incoherent cross sections for 95Mo and 97Moare slightly larger than for the even-even isotopes. The reason for this discrepancy iscurrently unknown.

It can also be concluded from Table 4.3 that the incoherent cross sections for 98Moand 100Mo are smaller than the ones of the other even-even isotopes. To explain thisbehaviour the 0+ and 1+ contributions to the averaged cross sections for the νe-scattering o the even-even nuclei are displayed in Fig. 4.10. As is visible in the gurethe 0+ contribution is rather small for all the even-even isotopes It is also seen inFig. 4.10 that the 1+ contribution is roughly the same for 92Mo, 94Mo and 96Mo butis notably smaller for 98Mo and 100Mo. This suppression of the 1+ transitions in theheavier isotopes (98Mo and 100Mo) was discussed earlier.

It should be noted that the computed cross sections in Table 4.2 and Table 4.3 arequite sensitive to the adopted values of T and α. In [38] we showed that for tau and


muon neutrinos the calculated averaged cross sections can dier by a factor of ∼ 2−3depending on the adopted neutrino spectra.

avour 〈σ〉A=92coh 〈σ〉A=94

coh 〈σ〉A=95coh 〈σ〉A=96

coh 〈σ〉A=97coh 〈σ〉A=98

coh 〈σ〉A=100coh

νe 1.81 1.96 2.00 2.12 2.11 2.28 2.45νe 2.40 2.60 2.66 2.81 2.81 3.03 3.26νµ, ντ 2.56 2.78 2.84 3.00 3.00 3.23 3.47νµ, ντ 2.56 2.78 2.83 3.00 3.00 3.23 3.47

Table 4.2: Computed averaged cross sections for the coherent neutral-current neutrino-nucleus scattering o the stable molybdenum isotopes in units of 10−39 cm2. The employedvalues of T and α are those given in Table 4.1.

avour 〈σ〉A=92incoh 〈σ〉A=94

incoh 〈σ〉A=95incoh 〈σ〉A=96

incoh 〈σ〉A=97incoh 〈σ〉A=98

incoh 〈σ〉A=100incoh

νe 11.6 11.8 16.2 11.9 16.0 9.94 8.59νe 17.3 17.6 22.1 17.7 21.8 15.1 13.1νµ, ντ 25.5 25.3 28.3 25.3 27.8 22.1 19.9νµ, ντ 22.7 22.7 25.5 22.8 25.1 20.0 17.7

Table 4.3: Computed averaged cross sections for the incoherent neutral-current neutrino-nucleus scattering o the stable molybdenum isotopes in units of 10−42 cm2. The employedvalues of T and α are those given in Table 4.1.

0

2

4

6

8

10

12

14

16

18

〈σ〉[1

0−42cm

2]

92 94 96 98 100A

0+

1+

Figure 4.10: Contributions from the 0+ and 1+ multipoles to the nuclear responses toelectron neutrinos for the even-even Mo isotopes.


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10N

orm

alize

ddσ/dE

exc

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Eexc [MeV]

7/2+47

5/2+47

3/2+2

95Mo(ν, ν′)95Mo∗

Figure 4.11: Dierential cross sections for incoherent neutral-current neutrino-nucleus scat-tering to nal states with excitation energy Eexc in

95Mo.

In Fig. 4.11 and Fig. 4.12 I show the dierential cross sections for supernova-νe scat-tering to various nal states in 95Mo and 97Mo respectively. As we concluded in [37]the singe-particle transitions 1d5/2 −→ 1d3/2 are important. These transitions cor-respond to the 3/2+

2 and the 3/2+1 states in 95Mo and 97Mo respectively. The other

prominent nal states in the gures are 3/2+, 5/2+ and 7/2+ states which are ofmainly three-quasiparticle character. The most important transitions are of the form1d5/2 −→ 0g9/2 ⊗ ω or of the form 1d5/2 −→ 0g7/2 ⊗ ω where ω stands for a QRPAphonon. In 97Mo there are also prominent transitions of either the non-spin-ip form1d5/2 −→ 1d5/2 ⊗ ω or the spin-ip form 1d5/2 −→ 1d3/2 ⊗ ω. The prominent nalstates can thus be interpreted as an odd neutron in the 1d orbital or as a 0g (or 1d)neutron coupled to a vibrating core. This is visualized in Fig 4.13 for 95Mo and thesame reasoning holds also for 97Mo.


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10N

orm

alize

ddσ/dE

exc

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Eexc [MeV]

7/2+48

5/2+50

3/2+1

5/2+2 5/2+

5

5/2+6 5/2+

10 5/2+31

97Mo(ν, ν′)97Mo∗

Figure 4.12: Dierential cross sections for incoherent scattering to nal states with excitationenergy Eexc in

97Mo.

Core

(94Mo)

unpaired nucleon cou-pled to the collectivecore by MQPM

unpaired nucleon

collective core ofpaired nucleonsdescribed by QRPA

ν

spin-fl

ip

coreexcitation

+

orbital transition

Core

(94Mo)

ν

Core

(94Mo) ν

unpaired nucleon cou-pled to the vibratingcore by MQPM

Figure 4.13: Neutral-current neutrino-nucleus scattering o 95Mo. In the gure the spin ofthe odd particle is indicated by an arrow.

5 Conclusions

In this thesis a comprehensive study of the neutral-current neutrino-nucleus scat-terings o the stable (A = 92, 94, 95, 96, 97, 98, 100) molybdenum isotopes was per-formed. Studies of weak interaction phenomena in these nuclei are of importance fore.g. the MOON experiment planned in Japan. In this work the cross sections for theaforementioned nuclear targets were computed for neutrino energies relevant for su-pernova neutrinos. The nuclear responses to supernova neutrinos were subsequentlycalculated by folding the obtained cross sections with a two-parameter Fermi-Diracdistribution. In the computations the nal and initial nuclear states of the even-evenisotopes were constructed within the context of the QRPA. For the odd nuclei (95Moand 97Mo) the nuclear states were generated by using the microscopic quasiparticle-phonon model.

In our calculations we have found that the coherent contribution dominates the crosssections for the neutrino energies which are relevant for supernova neutrinos. For thetypical energies of supernova neutrinos excitations of the 1+ multipole are the mostimportant for all the studied nuclei. For neutrino energies which correspond to thehigh-energy tail of the neutrino spectrum similar patterns are seen for all the studiedisotopes. At these energies the most important multipole is 1− but also several othermultipoles contribute signicantly to the cross sections.

For the odd nuclei we have found that the prevalent nal states can simply be thoughtof as an unpaired nucleon in a 1d orbital or as a 0g nucleon coupled to a vibratingcore. The coherent cross sections for the odd nuclei are roughly the same as for theeven-even isotopes. On the contrary the incoherent cross sections for 95Mo and 97Moare slightly larger than for even-even nuclei. The reason for this enhancement of thecross sections for the odd isotopes is currently unknown. One aspect which needs tobe studied more is how the cut of the quasiparticle-phonon basis aects the crosssections.

The computed averaged neutrino-nucleus cross sections turn out to be rather sensi-tive to the supernova model used. Depending on the adopted values of the neutrinoparameters the nuclear responses to tau and muon neutrinos might dier by a fac-tor of ∼ 2 − 3. This uncertainty is probably larger than the errors coming from thenuclear-structure calculations. Future experiments can thus presumably be used totest the validity of the current supernova models.

25

26 Conclusions

References

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[2] C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics,Oxford University Press, New York, 2007.

[3] K. Langanke and G. Martínez-Pinedo, Rev. Mod. Phys. 75 (2003) 819.

[4] K. Langanke, Acta Phys. Pol. B 39 (2008) 265.

[5] J. Gava and C. Volpe, AIP Conf. Proc. 1038 (2008) 193.

[6] H. Ejiri et al., Eur. Phys. J. Special Topics 162 (2008) 239.

[7] H. Ejiri. Private communication.

[8] K. Langanke, Nucl. Phys. A 834 (2010) 608c.

[9] P. F. Smith, Astro. Part. Phys. 8 (1997) 27.

[10] C. K. Hargrove et al., Astro. Part. Phys. 5 (1996) 183.

[11] J. S. O'Connell, T. W. Donnelly and J. D. Walecka, Phys. Rev. C 6 (1972) 719.

[12] T. W. Donnelly and R. D. Peccei, Phys. Rep. 50 (1979) 1.

[13] R. Arnold et al., Phys Rev. Lett. 95 (2005) 182302.

[14] M. Baranger, Phys. Rev. 120 (1960) 957.

[15] D. S. Delion and J. Suhonen, Phys. Rev. C 67 (2003) 034301.

[16] P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer, New York,1980.

[17] J. Toivanen and J. Suhonen, Phys. Rev. C 57 (1998) 1237.

27

28 References

[18] J. D. Walecka, Theoretical Nuclear and Subnuclear Physics, Imperial CollegePress, London, 2004.

[19] E. Ydrefors et al., Nuclear Responses to Supernova Neutrinos for the StableMolybdenum Isotopes, in: J. P. Greene (Ed.), Neutrinos: Properties, Reactions,Sources and Detection, Nova Science Publishers, 2011. Included in this thesis.

[20] W. M. Alberico, S. M. Bilenky and C. Maieron, Phys. Rep. 358 (2002) 227.

[21] W. M. Alberico et al., Phys. Rev. C 79 (2009) 065204.

[22] J. Suhonen, From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory,Springer, Berlin, 2007.

[23] J. Bardeen, L. N. Cooper and J. R. Schrieer, Phys. Rev. 108 (1957) 1175.

[24] A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936.

[25] G. Audi, A. H. Wapstra and C. Thibault, Nucl. Phys. A 729 (2003) 337.

[26] P. Ullah and D. J. Rowe, Nucl. Phys. A 163 (1971) 257.

[27] D. J. Rowe, J. Math. 10 (1969) 1774.

[28] K. G. Balasi, E. Ydrefors and T. Kosmas, Nucl. Phys. A 868-869 (2011) 82.Included in this thesis.

[29] E. Ydrefors, M. T. Mustonen and J. Suhonen, Nucl. Phys A 842 (2010) 33.Included in this thesis.

[30] A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. I, Benjamin, New York,1969.

[31] K. Holinde, Phys. Rep. 68 (1981) 121.

[32] D. Abriola and A. A. Sonzogni, Nucl. Data Sheets 107 (2006) 2423.

[33] B. Singh, Nucl. Data Sheets 109 (2008) 297.

[34] J. Kotila, J. Suhonen and D. S. Delion, Nucl. Phys. A 765 (2006) 354.

[35] T. W. Burrows, Nucl. Data Sheets 68 (1993) 635.

[36] A. Artna-Cohen, Nucl. Data Sheets 70 (1993) 85.

References 29

[37] E. Ydrefors and J. Suhonen, AIP conf. proc. 1304 (2010) 432. Included in thisthesis.

[38] E. Ydrefors et al., Nucl. Phys. A 866 (2011) 67. Included in this thesis. Erratumsubmitted to Nucl. Phys. A.

[39] E. Ydrefors and J. Suhonen, AIP conf. proc. (2011). In press. Included in thisthesis.

[40] M. T. Keil and G. G. Raelt, Astrophys. J. 590 (2003) 971.

30 References

A Neutral-current transition

densities in the MQPM

In this appendix the reduced neutral-current transition densities in the MQPM frame-work are presented. In the present notation the label p refers to a proton orbital andn to a neutron orbital. When the indices a and b are adopted both of them are ei-ther proton or neutron indices. Here only the expressions for a neutron-odd nucleusare given. It is however straightforward to obtain the corresponding expressions fora proton-odd nucleus by interchanging all neutron labels into proton labels and viceversa.

In the MQPM the basis set consists of one-quasiparticle states |ν〉 = |n;m〉 =|nnlnjn;m〉 and quasiparticle-phonon states |nω; jm〉. A quasiparticle state with an-gular momentum j and magnetic quantum number m is then given by

|n;m〉 = a†ν |QRPA〉 (A.1)

and a quasiparticle-phonon state is dened as

|nω; jm〉 = [a†nQ†ω]jm|QRPA〉, (A.2)

where |QRPA〉 is the QRPA vacuum. The QRPA creation operator Q†ω is here givenby

Q†ω = 12

∑aa′

σ−1aa′(X

ωaa′ [a†aa

†a′ ]JωMω + Y ω

aa′ [aaaa′ ]JωMω), (A.3)

where the auxiliary amplitudes are given by

Xωaa′ =

Xωaa′ if a < a′,

2Xωaa′ if a = a′,

(−1)ja+ja′+Jω+1Xωa′a if a > a′

(A.4)

and the amplitudes Y ωaa′ are dened correspondingly. In this way the restriction a 6 a′,

present in (3.7), is relaxed to simplify the calculations.

The reduced transition density for a transition between two one-quasiparticle statesis given by

(nf‖[c†acb]λ‖ni) = λ(uaubδanf δbi + (−1)ja+jb−λvavbδaniδbnf ). (A.5)

31

32 Neutral-current transition densities in the MQPM

For the reduced transition densities concerning transitions between a one-quasiparticlestate and a quasiparticle-phonon state the matrix elements take the forms

(nf‖[c†pcp′ ]λ‖niωi; ji)= −δninf δλJωi (−1)jnf+ji+λjiσ

−1pp′(vpup′Xωi

pp′ + upvp′Y ωipp′), (A.6)

(nf‖[c†ncn′ ]λ‖niωi; ji)

= −(−1)jnf+ji+λλjiJωi

[δnfni

δλJωi

Jωi2 σ−1nn′(vnun′Xωi

nn′ + unvn′Y ωinn′)

+(δn′nivnun′

jnf λ jijn′ Jωi jn

σ−1nnf

Xωinnf

+ δnnfδjij′n

jn′2 unvn′

× (−1)jn+jn′+λσ−1n′niY

ωin′ni

)− (−1)jn+jn′+λ

(δnnivnun′

jnf λ jijn Jωi jn′

× σ−1

n′nf Xωin′nf + δn′nf

δjijn

jn2 vnun′(−1)jn+jn′+λσ−1

nniY ωinni

)], (A.7)

(nfωf ; jf‖[c†pcp′ ]λ‖ni) = δnfniδλJωf jfσ−1pp′(upvp′X

ωfpp′ + vpup′Y

ωfpp′), (A.8)

and

(nfωf ; jf‖[c†ncn′ ]λ‖ni)

= jf λJωf

[δnfni

δλJωf

Jωf2 σ−1nn′(unvn′X

ωfnn′ + vnun′Y

ωfnn′)

+(δn′nfunvn′

×jf λ jnijn Jωf jn′

σ−1nniXωfnni + δnni

δjf jn′

jn′2 vnun′(−1)jf+jni+λσ−1

n′nf Yωfnnf

)− (−1)jn+jn′+λ

(δnnfunvn′

jf λ jnij′n Jωf jn

σ−1n′niX

ωfn′ni

+ δn′niδjf jn

jn2 vnun′(−1)jni+jf+λσ−1

nnfYωfnnf

)]. (A.9)

The reduced transition density between two quasiparticle phonon states can be writtenon the form

(nfωf ; jf‖[c†acb]λ‖niωi; ji)= uaub(nfωf ; jf‖[a†aab]λ‖niωi; ji) + (−1)ja+jb+λvavb(nfωf ; jf‖[a†baa]λ‖niωi; ji),

(A.10)

33

where

(nfωf ; jf‖[a†pap′ ]λ‖niωi; ji)

= δnfniλJωf Jωi jf ji

jf λ jiJωi jnf Jωf

∑b′

(−1)jf+jnf+jp+jb′σ−1pb′σ

−1p′b′

×(Jωf λ Jωi

jp′ jb′ jp

Xωfpb′X

ωip′b′ + (−1)λ

Jωf λ Jωijp jb′ jp′

Yωfp′b′Y

ωipb′

)(A.11)

and

(nfωf ; jf‖[a†nan′ ]λ‖niωi; ji)

= Jωf Jωi jf jiλ

[δn′ni(−1)jnf+Jωf+ji+λ

ji jf λjn jn′ Jωi

∑b′σ−1nb′σ

−1nf b′

×(jf Jωi jn

jb′ Jωf jnf

Xωfnb′X

ωinf b′ −

δjb′jf

jb′2 Y

ωinb′Y

ωfnf b′

)− δnnf (−1)Jωi+jni+jn+jn′+ji+λ

ji jf λjn jn′ Jωf

∑b′σ−1nib′σ

−1n′b′

×( ji Jωf jn′

jb′ Jωi jni

Xωfnib′X

ωin′b′ −

δjb′ji

jb′2 Y

ωinb′Y

ωfn′b′

)+ δninf

λ jf jijnf Jωi Jωf

∑b′

(−1)jn+jb′+jnf+jfσ−1nb′σ

−1n′b′

×(Jωf Jωi λ

jn′ jn jb′

Xωfnb′X

ωin′b′ + (−1)λ

Jωi Jωf λjn′ jn jb′

Y ωinb′Y

ωfn′b′

)+ σ−1

nniσ−1n′nf

((−1)Jωf+Jωi+jn′+λ

jn jni Jωfjn′ Jωi jnfλ ji jf

XωfnniX

ωin′nf

+ (−1)jn+jf+λδjn′jf δjnji

jn′2jn

2Y ωinniYωfn′nf

)]

+ δnnf δn′niδωiωf jf jiλ(−1)jn+ji+Jωf+λ

ji jf λjn jn′ Jωf

. (A.12)

In the case then the initial (i) and nal (f) states are the same also the termδabδλ0jav

2aji has to be added to (A.5), (A.6), (A.7), (A.8) , (A.9) and (A.10). This

is the case in e.g. coherent neutrino-nucleus scattering which is considered in thiswork.

34 Neutral-current transition densities in the MQPM

department of physics university of jyvÄskylÄ research report no. 2/2012 supernova ... ·...

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