Collaborators:
description
Transcript of Collaborators:
![Page 1: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/1.jpg)
Realistic Calculations of Neutrino-Realistic Calculations of Neutrino-Nucleus Reaction Cross sections Nucleus Reaction Cross sections
T.S. KosmasT.S. Kosmas
Division of Theoretical Physics, University of Ioannina,
Greece
Collaborators: Collaborators:
P. Divari, V. Chasioti, K. Balasi, V. Tsakstara, G. Karathanou, P. Divari, V. Chasioti, K. Balasi, V. Tsakstara, G. Karathanou, K. Kosta K. Kosta
MEDEX’07International workshop on DBD and Neutrino Physics Prague, Czech Republic, June 11 – 14, 2007
![Page 2: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/2.jpg)
OutlineOutline
• Introduction
• Cross Section Formalism 1. Multipole operators (Donnelly-Walecka method)
2. Compact expressions for all basic reduced matrix elements
• Applications – Results 1. Exclusive and inclusive neutrino-nucleus reactions
2. Differential, integrated, and total cross sections for the nuclei:
4040Ar, Ar, 5656Fe, Fe, 9898Mo, Mo, 1616OO 3. Dominance of specific multipole states – channels 4. Nuclear response to SN ν (flux averaged cross sections)
• Summary and ConclusionsSummary and Conclusions
![Page 3: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/3.jpg)
Charged-current reactions (l= electron, muon, tau)
Neutral-current reactions
Introduction
There are four types of neutrino-nucleus reactions to be studied :
![Page 4: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/4.jpg)
1-body semi-leptonic electroweak processes in nuclei
Donnely-Walecka method provides a unified description of semi-leptonic 1-body processes in nuclei
![Page 5: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/5.jpg)
The Effective Interaction Hamiltonian
(leptonic current ME)
Matrix Elements between initial and final Nuclear states are needed for obtaining a partial transition rate :
The effective interaction Hamiltonian reads
(momentum transfer)
![Page 6: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/6.jpg)
One-nucleon matrix elements (hadronic current)
Polar-Vector current:
2). Assuming CVC theory
Axial-Vector current:
1). Neglecting second class currents :
3). Use of dipole-type q-dependent form factors
4. Static parameters, q=0, for nucleon form factors
(i) Polar-Vector
(i) Axial-Vector
![Page 7: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/7.jpg)
Non-relativistic reduction of Hadronic Currents
The nuclear current is obtained from that of free nucleons, i.e.
The free nucleon currents, in non-relativistic reduction, are written
α = + , -, charged-current processes, 0, neutral-current processes
![Page 8: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/8.jpg)
Multipole Expansion – Tensor Operators
The ME of the Effective Hamiltonian reads
Apply multipole expansion of Donnely-Walecka in the quantities :
The result is (for J-projected nuclear states) :
![Page 9: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/9.jpg)
The basic multipole operators
are defined as
(V – A Theory)
The multipole operators, which contain Polar Vector + Axial Vector part,
The multipole operators are : Coulomb, Longitudinal, Tranverse-Electric, Transverse-Magnetic for Polar-Vector and Axial-Vector components
![Page 10: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/10.jpg)
Nucleon-level hadronic current for neutrino processes
For charged-current ν-nucleus processes
For neutral-current ν-nucleus processes
The form factors, for neutral-current processes, are given by
The effective nucleon level Hamiltonian takes the form
![Page 11: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/11.jpg)
Kinematical factors for neutrino currents
Summing over final and averaging over initial spin states gives
![Page 12: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/12.jpg)
Neutral-Current ν–Nucleus Cross sectionsIn Donnely-Walecka method [PRC 6 (1972)719, NPA 201(1973)81]
==============================================================================================================
where
The Coulomb-Longitudinal (1st sum), and Transverse (2nd sum) are:
![Page 13: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/13.jpg)
The seven basic single-particle operators
Normal Parity Operators
Abnormal Parity Operators
![Page 14: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/14.jpg)
Compact expressions for the basic reduced MEFor H.O. bases w-fs, all basic reduced ME take the compact forms
The Polynomials of even terms in q have constant coefficients as
Advantages of the above Formalism :(i) The coefficients P are calculated once (reduction of computer time)(ii) They can be used for phenomenological description of ME(iii) They are useful for other bases sets (expansion in H.O.
wavefunctions)
Chasioti, Kosmas, Czec.J. Phys.
![Page 15: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/15.jpg)
Polynomial Coefficients of all basic reduced ME
![Page 16: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/16.jpg)
Nuclear Matrix Elements - The Nuclear Model Nuclear Matrix Elements - The Nuclear Model
The initial and final states, |Ji>, |Jf>, in the ME <Jf ||T(qr)||Ji>2 are determined by using the QQuasi-particle RPA (QRPA)RPA (QRPA)
1). Interactions:1). Interactions:• Woods Saxon+Coulomb correction (Field)• Bonn-C Potential (two-body residual interaction)
2). Parameters:2). Parameters:• In the BCS level: the pairing parameters gn
pair , gppair
• In the QRPA level: the strength parameters gpp , gph
j1, j2 run over single-particle levels of the model space (coupled to J)D(j1, j2; J) one-body transition densities determined by our model
3). 3). Testing the reliability of the MethodTesting the reliability of the Method::• Low-lying nuclear excitations Low-lying nuclear excitations (up to about 5 MeV(up to about 5 MeV)• magnetic momentsmagnetic moments (separate spin, orbital contributions)
![Page 17: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/17.jpg)
Particle-hole, gph, and particle-particle gpp parameters for 16O ,40Ar, 56Fe, 98Mo
H.O. size-parameter, b, model space and pairing parameters, n, p pairs for 16O ,40Ar, 56Fe, 98Mo
![Page 18: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/18.jpg)
experimental theoretical
Low-lying Nuclear Spectra (up to about 5 MeV)
98Mo
![Page 19: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/19.jpg)
experimental theoretical
Low-lying Nuclear Spectra (up to about 5 MeV)
40Ar
![Page 20: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/20.jpg)
State-by-state calculations of multipole contributions to dσ/dΩ
56Fe
![Page 21: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/21.jpg)
Angular dependence of the differential cross-section
56Fe
![Page 22: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/22.jpg)
Total Cross section: Coherent & Incoherent contributions
g.s. g.s.
g.s. f_exc
56Fe
![Page 23: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/23.jpg)
Dominance of Axial-Vector contributions in σ
56Fe
![Page 24: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/24.jpg)
Dominance of Axial-Vector contributions in σ_tot
40Ar
![Page 25: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/25.jpg)
Dominance of Axial-Vector contributions in σ
16O
![Page 26: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/26.jpg)
Dominance of Axial-Vector contributions in σ
98Mo
![Page 27: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/27.jpg)
State-by-state calculations of dσ/dΩ
40Ar
![Page 28: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/28.jpg)
Total Cross section: Coherent + Incoherent contributions
40Ar
![Page 29: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/29.jpg)
State-by-state calculations of dσ/dΩ
16O
![Page 30: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/30.jpg)
16O
Coherent and Incoherent
![Page 31: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/31.jpg)
State-by-state calculations of dσ/dΩ
98Mo
![Page 32: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/32.jpg)
Angular dependence of the differential cross-section
98Mo
![Page 33: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/33.jpg)
98Mo
Angular dependence of the differential cross section for the excited states J=2+, J=3-
![Page 34: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/34.jpg)
Coherent and Incoherent
98Mo
![Page 35: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/35.jpg)
Nuclear response to the SN-ν for various targets
Assuming Fermi-Dirac distribution for the SN-ν spectra
Using our results, we calculated for various ν–nucleus reaction channels
normalized to unity as
α = 0, 3
2.5 < Τ < 8
Results of Toivanen-Kolbe-Langanke-Pinedo-Vogel, NPA 694(01)395
56Fe
===========================================================
![Page 36: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/36.jpg)
Flux averaged Cross Sections for SN-ν
α = 0, 3
2.5 < Τ < 8 (in MeV)
A= <σ>_A
V= <σ>_V
5656Fe Fe
![Page 37: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/37.jpg)
Flux averaged Cross Sections for SN-ν
A= <σ>
V= <σ>
α = 0, 3
1616O O
2.5 < Τ < 8 (in MeV)
![Page 38: Collaborators:](https://reader036.fdocuments.in/reader036/viewer/2022070417/5681548d550346895dc29c69/html5/thumbnails/38.jpg)
SUMMARY-CSUMMARY-CONCLUSIONSONCLUSIONS• Using H.O. wave-functions, we have improved the Donnelly-Walecka formalism : compact analytic expressions for all one-particle reduced ME as products (Polynomial) x (Exponential) both functions of q.
• Using QRPA, we performed state-by-state calculations for inelastic ν–nucleus neutral-current processes (J-projected states) for currently interesting nuclei.
• The QRPA method has been tested on the reproducibility of : a) the low-lying nuclear spectrum (up to about 5 MeV) b) the nuclear magnetic moments
• Total differential cross sections are evaluated by summing-over-partial-rates. For integrated-total cross-sections we used numerical integration.
• Our results are in good agreement with previous calculations (Kolbe-Langanke, case of 5656Fe,Fe, and Gent-group, 1616OO).
• We have studied the response of the nuclei in SN-ν spectra for Temperatures in the range : 2.5 < T < 8 and degeneracy-parameter α values : α = 0, 3
Acknowledgments: Acknowledgments: I wish to acknowledge financial support from the ΠΕΝΕΔ-03/807, Hellenic G.S.R.T. project to participate and speak in the present workshop.