A CLUSTER MODEL OF 6He AND 6Li By Jeremy Robert Armstrong AN ABSTRACT OF A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 2007 Professor Vladimir G. Zelevinsky ABSTRACT A CLUSTER MODEL OF 6He AND 6Li By Jeremy Robert Armstrong Light nuclei provide an ideal testing ground for few-body theories. Helium-6 is particularly interesting in that it shows an extended neutron system similar to a halo, is loosely-bound, and is a Borromean system. Lithium-6 is also loosely bound, and is a difficult challenge for many theories. An alpha plus two nucleon cluster model using the Brink formalism in secondary quantization was developed for calculating various properties of 6He and 6Li. The formalism includes a fully microscopic alpha particle, and allows for the exact treatment of Fermi statistics and the correct construction of eigenstates of angular momentum. Both nuclei were studied as the superposition of two configurations: an alpha plus two nucleon cluster and a nucleon-alpha-nucleon chain, or cigar configuration. The variational principle was used to obtain the binding energies of the nuclei and weights of both configurations. For 6He calculations were made to determine the excitation energy of the 2+ excited state, the B(E2) for the 0+ 2+ transition and the charge and matter radii. For 6Li, the excitation energy of → the 2+ and 3+ excited states, charge radius, electric quadrupole moment and magnetic dipole moment were calculated. Finally, the lifetime for the 6He Gamov-Teller beta decay was calculated. The results were obtained with the use of three nucleon-nucleon interactions: the Volkov V1 and V2 interactions, and the Minnesota potential. Results were compared with experimental data and the results of other theoretical models. In spite of the deliberate simplicity of the model, it describes the main physical properties of the nuclei on a level comparable with much more sophisticated theories. A CLUSTER MODEL OF 6He AND 6Li By Jeremy Robert Armstrong A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 2007 ABSTRACT A CLUSTER MODEL OF 6He AND 6Li By Jeremy Robert Armstrong Light nuclei provide an ideal testing ground for few-body theories. Helium-6 is particularly interesting in that it shows an extended neutron system similar to a halo, is loosely-bound, and is a Borromean system. Lithium-6 is also loosely bound, and is a difficult challenge for many theories. An alpha plus two nucleon cluster model using the Brink formalism in secondary quantization was developed for calculating various properties of 6He and 6Li. The formalism includes a fully microscopic alpha particle, and allows for the exact treatment of Fermi statistics and the correct construction of eigenstates of angular momentum. Both nuclei were studied as the superposition of two configurations: an alpha plus two nucleon cluster and a nucleon-alpha-nucleon chain, or cigar configuration. The variational principle was used to obtain the binding energies of the nuclei and weights of both configurations. For 6He calculations were made to determine the excitation energy of the 2+ excited state, the B(E2) for the 0+ 2+ transition and the charge and matter radii. For 6Li, the excitation energy of → the 2+ and 3+ excited states, charge radius, electric quadrupole moment and magnetic dipole moment were calculated. Finally, the lifetime for the 6He Gamov-Teller beta decay was calculated. The results were obtained with the use of three nucleon-nucleon interactions: the Volkov V1 and V2 interactions, and the Minnesota potential. Results were compared with experimental data and the results of other theoretical models. In spite of the deliberate simplicity of the model, it describes the main physical properties of the nuclei on a level comparable with much more sophisticated theories. ACKNOWLEDGMENTS I would like first to acknowledge and thank my advisor and friend, Vladimir Zelevinsky. He was patient enough to let me learn as much as I could on my own, without unleashing all his knowledge at once, which would have been too much I am sure. I loved all our discussions no matter what the subject, whether they were on physics, Slavic linguistics, or Soviet film. I hope we will still be able to continue these discussions after I leave. I would also like to thank the members of my guidance committee: Filomena Nunes, Krzysztof Starosta (Dzi¸ekuj¸e), C.P. Yuan, and Carlo Piermarocchi for their suggestions and comments, and even suggesting that I take more classes, because those classes turned out to be some of the most interesting courses I took at MSU. Next, I would like to thank the cyclotron and physics and astronomy department for providing a great environment to work in and a very good learning environment. I would especially like to mention Shari Conroy, who was able to answer any admin- istrative question and make everything run smoothly. My collaborator, Alexander Sakharuk, also deserves to be mentioned, as he definitely helped to clarify things when the results of the calculations were mystifying, and injected some energy into the project. I should also thank professors Paul Mantica and Dave Morrissey for be- ing very influential in me coming to Michigan State, and helping me through my transition between departments. There are a lot of thank yous for my fellow graduate students and post-docs. Starting with my chemistry coevals: Bryan Tomlin, Sean Liddick, and Debbie Davies. It was always fun with you guys, and distracting each other enough to make sure the early days of grad school were tolerable. The closest I came to a midnight shift were some of the late night discussions with Bryan, but it didn’t feel much like work! From my early days, I would also like to thank Declan Mulhall, who suggested that Vladimir “is a real gas to work for,” Heather Zwahlen for getting me running again, Nathan iii Frank for organizing the physics soccer team, Michelle Ouellette for always interesting discussions, Michal Mocko for trying to make sure I got at least some work done, Chandana Sumithrarachchi for a great trip to Notre Dame and our many cultural discussions, and Aws Abdo for being a very good mentor. I should of course, mention my office mate of many years now Ivan Brida, for always providing an interesting perspective on everything, our strict lunch schedule, entertainment, and LATEXhelp. For example, when he wanted to demonstrate how to label and then cite an equation, his example equation was labeled “Ivan is stupid.” I duly followed his suggestion, and referenced the equation “Ivan is stupid” with great enthusiasm. With entertainment and Ivan in mind, I should also acknowledge Wes Hitt, Terrance Strother, Jon Cook, Bryan Tomlin (again), and Andy Rogers for countless moments of levity involving my office-mate. Of the more recent arrivals, I would like to acknowledge Angelo Signoracci (keep the mantra of “she doesn’t care” alive!), Rhiannon Meharchand for making me social again, Phil Voss for the softball, Ania Kwiatkowski for her attempts at making me salsa, my other office-mate Biruk Gebremariam for many interesting talks about soccer and many other subjects, and Arnau Rios, for LATEXhelp and for being a very gracious host for Giuseppe’s coffee breaks. Finally, Francesco would like to say a big “Grazie!” to Giuseppe Lorusso, Stefano Di Chiara, Alfredo Estrade, and Christine Gockel for all the good times, races, and travels. Last, but certainly not least, I would like to thank my family: my father, James, my mother Claire, and my sister Heather. Their love and support made navigating the vicissitudes of graduate school much easier. Mom and Dad, thanks for everything! Heather, you’re a dork, but you sure make every day more fun with all your messages. Thank you all again for all that you did to help me along my way! iv Contents 1 Introduction 1 1.1 EarlyAttempts .............................. 2 1.2 MoreAdvancedClusterModels . 4 1.3 MotivationforthePresentStudy . 6 1.4 Constitutionofthepresentwork. 7 2 Formalism 9 2.1 InitialRemarks .............................. 9 2.2 Non-orthogonalOrbitals . 11 2.3 Examples ................................. 12 2.4 Angularmomentumprojection. 18 3 Skeletons of the Six-particle Systems 21 3.1 Helium-6.................................. 21 3.1.1 Alpha-dineutronconfiguration . 21 3.1.2 Cigarconfiguration . 26 3.1.3 Interferenceterm ......................... 30 3.2 Lithium-6 ................................. 35 3.2.1 Alpha-deuteronconfiguration . 36 3.2.2 Cigarconfiguration . 44 3.2.3 Interferenceterm ......................... 47 4 Gaussian Approximation 49 4.1 Helium-6.................................. 49 4.1.1 Alpha-dineutronconfiguration . 50 4.1.2 Kineticenergy........................... 54 4.1.3 Interaction............................. 59 4.1.4 Totalenergy............................ 70 4.1.5 Meansquareradius........................ 72 4.2 Cigarconfiguration ............................ 79 4.2.1 Kineticenergy........................... 82 4.2.2 Interaction............................. 84 4.2.3 Totalenergy............................ 90 4.2.4 Meansquareradius........................ 93 4.3 Electromagnetictransitions . 95 4.3.1 Background ............................ 96 v 4.3.2 Applied to 6He .......................... 97 4.4 Interferenceterm ............................. 112 4.4.1 Kineticenergy........................... 115 4.4.2 Interaction............................