Properties of One- and Two-Nucleon Halo Nuclei in Effective Field Theory A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Bijaya Acharya May 2015 © 2015 Bijaya Acharya. All Rights Reserved. 2 This dissertation titled Properties of One- and Two-Nucleon Halo Nuclei in Effective Field Theory by BIJAYA ACHARYA has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by Daniel R. Phillips Professor of Physics and Astronomy Robert Frank Dean, College of Arts and Sciences 3 Abstract ACHARYA, BIJAYA, Ph.D., May 2015, Physics Properties of One- and Two-Nucleon Halo Nuclei in Effective Field Theory (160 pp.) Director of Dissertation: Daniel R. Phillips Halo nuclei have much larger sizes than other nuclei with a similar number of protons and neutrons. This is the result of the fact that the probability distribution of one or more nucleons in halo nuclei extends to much larger distances than that of the rest of the nucleons. Halo Effective Field Theory exploits the separation of scales between the size of the halo to that of the “core” to derive the correlations between low-energy observables as systematic expansions in the ratio of the size of halo to that of the core, with error estimates. We use this approach to calculate the reduced transition probability for E1 excitation of the halo neutron of 19C to the continuum up to next-to-next-to leading order in this expansion. By comparing the prediction with the Coulomb dissociation data from RIKEN, accurate results for the one-neutron separation energy and asymptotic normalization coefficient are extracted. Good agreement with data is also found for the longitudinal momentum distribution of the Coulomb dissociation cross section. We then derive universal relations for the matter radii of two-neutron halos and apply them to 22C to analyze its matter radius, which we use as input to constrain 21C and 22C energies. Even after accounting for the combined theoretical and 1 − σ experimental errors, the matter radius value imposes an upper bound of 100 keV in the two-neutron separation energy of 22C. We also derive universal relations for the E1 excitation of two-neutron halo nuclei and discover a simple scaling of the B(E1) spectrum with the three-body binding energy when all two-body subsystems are in the unitary limit. We compare our results with the data for 11Li, and make predictions for the outcome of an ongoing experiment on 22C. 4 Acknowledgments This dissertation could not have been completed without the guidance, participation, suggestions and encouragement I received from many people over the last five years. I am grateful to my committee members, Profs. Daniel Phillips, Charlotte Elster, Prakash, Carl Brune and Alycia Stigall for being such wonderful mentors. I would like to express sincere appreciation to my advisor Prof. Daniel Phillips for introducing me to this exciting world of nuclear physics and for instilling in me the knowledge and skills needed to make contributions to it. I am grateful to Profs. Elster and Prakash, who have followed my research with keen interest and provided valuable suggestions. My interactions with Profs. Brune, Frantz, Hicks, Roche and Schiller at individual and group meetings, journal clubs and seminars helped me develop an understanding of the experiments that are relevant to my work in particular, and to nuclear physics in general. I will be forever indebted to the Physics and Astronomy faculty here at Ohio University for strengthening my academic foundation. I would like to thank my collaborators, Drs. Chen Ji, Philipp Hagen and Hans-Werner Hammer, for having the patience to work with me, and for sharing their valuable experience in the process. I am grateful to the nuclear theory group members, Arbin, Linda, Brian, Hadi, Xilin, Vasily, Mamun, Anton and Azamat; and my colleagues at the accelerator lab for invigorating discussions and constructive criticisms. Finally, I would like to thank all the staff at our department for their warmth, generosity and hospitality. 5 Table of Contents Page Abstract . 3 Acknowledgments . 4 List of Tables . 8 List of Figures . 9 1 Introduction . 14 1.1 Experimental studies . 16 1.1.1 Mass measurements . 17 1.1.2 Coulomb dissociation . 18 1.1.3 Measurement of charge radii in ion traps . 19 1.1.4 Matter radii from cross-section measurements . 19 1.1.5 Fragmentation reactions and momentum distributions . 21 1.2 Theories of halo nuclei . 21 1.2.1 The nuclear shell model . 21 1.2.2 Ab initio calculations with nucleons . 24 1.2.2.1 The no-core shell model (NCSM) . 27 1.2.2.2 The hyperspherical harmonics (HH) method . 28 1.2.2.3 Coupled cluster (CC) method . 28 1.2.3 Cluster models . 29 1.2.4 Halo effective field theory (Halo-EFT) . 30 1.2.4.1 Two-body systems in Halo-EFT . 30 1.2.4.2 Three-body systems in Halo-EFT . 33 1.3 This dissertation . 37 2 Effective Field Theory for Halo Nuclei: A Review of the Formalism . 39 2.1 Two-body systems . 41 2.2 Proton halo nuclei . 45 2.3 Electromagnetic observables in Halo-EFT . 48 2.4 Two-neutron halo with a spinless core . 52 2.5 Conclusion . 59 3 Coulomb Energy in a Square Well Potential Model . 61 3.1 Effective range for the square well . 63 3.2 Coulomb energy at first order in αem ..................... 64 3.2.1 Errors in EFT calculation . 66 3.3 Coulomb energy to all orders in αem ..................... 67 6 3.4 Conclusion . 71 4 Coulomb Dissociation of One-Neutron Halos: Applications to Carbon-19 . 73 4.1 Coulomb dissociation cross section of Carbon-19 in Halo-EFT . 74 4.1.1 Extracting parameters from Coulomb dissociation data . 77 4.1.1.1 Angular distribution . 77 4.1.1.2 Relative energy spectrum . 78 4.1.2 Longitudinal momentum distribution . 82 4.2 Electric charge radii of Carbon-18 and Carbon-19 . 82 4.3 Conclusion . 83 5 Two-Neutron Halos: Matter Radii and Efimov States . 85 5.1 The Jacobi coordinates and momenta . 85 5.2 The three-body free states: plane waves and spherical waves . 86 5.2.1 The two-body T-matrices in Jacobi representation . 88 5.3 The bound state . 88 5.4 Two-neutron halo: identical neutrons and the permutation operator . 90 5.4.1 Matter radii . 93 5.5 Carbon-22 . 95 5.5.1 Matter radius constraints on binding energy . 95 5.5.1.1 EFT errors . 97 5.5.2 On the possibility of excited Efimov states . 98 5.6 Conclusion . 100 6 Universal Relations for the Dipole Excitation of Borromean Halo Nuclei . 101 6.1 The wavefunction for the three-body unbound state . 101 6.2 The dipole matrix element . 102 6.2.1 Evaluation of the matrix element by distortion of the integration paths108 6.3 The B(E1) spectrum . 110 6.3.1 Unitary Limit . 111 6.3.2 Lithium-11 . 112 6.3.3 Coulomb dissociation of Carbon-22 . 116 6.3.4 Momentum distributions . 117 6.4 The photonuclear sum rules . 119 6.4.1 Direct derivation of the sum rules for neutron halo nuclei . 122 6.4.2 Numerical evaluations of the sum rules for two-neutron halo nuclei 123 6.5 Conclusion . 127 7 Summary and Outlook . 128 References . 131 Appendix A: Coulomb Dissociation Cross Section . 147 7 Appendix B: Numerical Techniques and Errors . 150 8 List of Tables Table Page 2 n;c 6.1 The value of bE and the maximum possible value of k0 for all the Zi (q; K; E) featuring in Eq. (6.23). 110 2 2 6.2 Power-law fits to mEB=(eZ) dB(E1)=dE in PWIA and with FSI included for various values of A.................................125 6.3 Integrated dipole strength in the unitary limit. The second and the third columns show, respectively, the integrated dipole strengths with dB(E1)=dE calculated in PWIA, and with FSI included. 125 6.4 Non-energy-weighted and energy-weighted integrals of the dipole response function for A = 20 at several values of EB and Enc. 126 p p B.1 A( mEB; mEB ; EB) evaluated at Z = 6, A = 20, Enc = 0:2 MeV by differentiation of the wavefunction as shown in Eq. (6.8) . 155 B.2 Dependence of In(Kn) on the integration path rotation angle for E = EB = 0:4 MeV, Z = 6, A = 20, with Enc fixed at 0.1 MeV. 159 B.3 Dependence of Ic(Kc) on the integration path rotation angle for E = EB = 0:4 MeV, Z = 6, A = 20, with Enc fixed at 0.1 MeV. 160 B.4 dB(E1)=dE calculated by numerically evaluating Eq. (6.32) with different number of Gauss-Legendre quadrature points. The calculation is performed at Z = 6, A = 20, EB = 0:4 MeV and Enc = 0:2 MeV. 160 9 List of Figures Figure Page 1.1 A low-mass region in the chart of nuclei (proton number versus neutron number). The black squares are stable nuclei. Several one-neutron, and two-neutron halo nuclei, along with the four-neutron halo nucleus 8He, are identified. Figure taken from Ref. [4]. 15 1.2 The interaction radii of light nuclei versus the atomic number. Figure taken from Ref. [1]. 20 1.3 The longitudinal momentum distributions of the fragments for 19C single- neutron removal on Beryllium target, measured at 57 MeV per nucleon at MSU [50], at 64 MeV per nucleon at RIKEN [47]; and on Carbon target, measured at 910 MeV per nucleon at GSI [45].