{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23945"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23945","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Third-order charged particle beam optics","abstract":"The motion of a charged particle through a magnetic field configuration can be described in terms of deviation from a certain ideal trajectory. One uses power series expansion of the phase-space coordinates to obtain the transfer matrices for a particular optical system. In this thesis we present a complete third-order theory of computing transfer matrices and apply it to magnetic elements in an accelerator beam-line. A particular attention is devoted to studying particles' orbits in an extended fringing field of a dipole magnet. Analytical solutions are obtained up to the third order in the formalism of the matrix theory. They contain form factors describing the fall-off pattern of the field. These form factors are dimensionless line integrals of the field strength and its derivative. There is one such integral in the first-order solution, two in the second, and nine in the third. An alternate way of describing charged particle optics is also presented. It is based on a Hamiltonian treatment and uses certain symplectic operators, which are defined in terms of Poisson brackets, to parametrize the transfer map of a system. We apply this approach to the fringing field problem and obtain a third-order solution. We furthermore show how to convert this solution into conventional transfer matrices by examining the connection between the non-canonical matrix theory and the Hamiltonian description.","abstract_html":"The motion of a charged particle through a magnetic field configuration can be described in terms of deviation from a certain ideal trajectory. One uses power series expansion of the phase-space coordinates to obtain the transfer matrices for a particular optical system. In this thesis we present a complete third-order theory of computing transfer matrices and apply it to magnetic elements in an accelerator beam-line. A particular attention is devoted to studying particles' orbits in an extended fringing field of a dipole magnet. Analytical solutions are obtained up to the third order in the formalism of the matrix theory. They contain form factors describing the fall-off pattern of the field. These form factors are dimensionless line integrals of the field strength and its derivative. There is one such integral in the first-order solution, two in the second, and nine in the third. An alternate way of describing charged particle optics is also presented. It is based on a Hamiltonian treatment and uses certain symplectic operators, which are defined in terms of Poisson brackets, to parametrize the transfer map of a system. We apply this approach to the fringing field problem and obtain a third-order solution. We furthermore show how to convert this solution into conventional transfer matrices by examining the connection between the non-canonical matrix theory and the Hamiltonian description.","abstract_has_math":false,"creators":["Sagalovsky, Leonid"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Physics","degree_department":null,"school":null,"contributors":["Chang, Shau-Jin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-18T17:15:59Z","date_published":"2011-05-18T17:15:59Z","updated_at":"2026-07-22T22:25:23Z","subjects":["third-order","charged particle","beam optics","power series expansion","transfer matrices"],"languages":["en"],"rights":["1989 Leonid Sagalovsky"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["3478193"],"render_values":[{"text":"3478193","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23945","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Chang, Shau-Jin"]},{"key":"dc:creator","label":"Author","values":["Sagalovsky, Leonid"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-18T17:15:59Z","10000-01-01","1989"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation / Thesis","text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["third-order","charged particle","beam optics","power series expansion","transfer matrices"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["1989 Leonid Sagalovsky"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["3478193","http://hdl.handle.net/2142/23945"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The motion of a charged particle through a magnetic field configuration can be described in terms of deviation from a certain ideal trajectory. One uses power series expansion of the phase-space coordinates to obtain the transfer matrices for a particular optical system. In this thesis we present a complete third-order theory of computing transfer matrices and apply it to magnetic elements in an accelerator beam-line. A particular attention is devoted to studying particles' orbits in an extended fringing field of a dipole magnet. Analytical solutions are obtained up to the third order in the formalism of the matrix theory. They contain form factors describing the fall-off pattern of the field. These form factors are dimensionless line integrals of the field strength and its derivative. There is one such integral in the first-order solution, two in the second, and nine in the third. An alternate way of describing charged particle optics is also presented. It is based on a Hamiltonian treatment and uses certain symplectic operators, which are defined in terms of Poisson brackets, to parametrize the transfer map of a system. We apply this approach to the fringing field problem and obtain a third-order solution. We furthermore show how to convert this solution into conventional transfer matrices by examining the connection between the non-canonical matrix theory and the Hamiltonian description.","Submitted by Carolyn Mead (cmead2@illinois.edu) on 2011-05-18T17:15:59Z No. of bitstreams: 1 1989_sagalovsky.pdf: 7134770 bytes, checksum: e117a389994ff1a52f47b03d21e2b9a7 (MD5)","Made available in DSpace on 2011-05-18T17:15:59Z (GMT). No. of bitstreams: 1 1989_sagalovsky.pdf: 7134770 bytes, checksum: e117a389994ff1a52f47b03d21e2b9a7 (MD5) Previous issue date: 1989","Restriction data tranferred 2014-07-01T11:14:44-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: Thesis","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Carolyn Mead (cmead2@illinois.edu) on 2011-05-18T17:16:00Z Item is restricted indefinitely.","Thesis","U of I Only"]},{"key":"dc:title","label":"Title","values":["Third-order charged particle beam optics"]}]}],"canonical_facts":{"dc:contributor":["Chang, Shau-Jin"],"dc:creator":["Sagalovsky, Leonid"],"dc:date":["2011-05-18T17:15:59Z","10000-01-01","1989"],"dc:description":["The motion of a charged particle through a magnetic field configuration can be described in terms of deviation from a certain ideal trajectory. One uses power series expansion of the phase-space coordinates to obtain the transfer matrices for a particular optical system. In this thesis we present a complete third-order theory of computing transfer matrices and apply it to magnetic elements in an accelerator beam-line. A particular attention is devoted to studying particles' orbits in an extended fringing field of a dipole magnet. Analytical solutions are obtained up to the third order in the formalism of the matrix theory. They contain form factors describing the fall-off pattern of the field. These form factors are dimensionless line integrals of the field strength and its derivative. There is one such integral in the first-order solution, two in the second, and nine in the third. An alternate way of describing charged particle optics is also presented. It is based on a Hamiltonian treatment and uses certain symplectic operators, which are defined in terms of Poisson brackets, to parametrize the transfer map of a system. We apply this approach to the fringing field problem and obtain a third-order solution. We furthermore show how to convert this solution into conventional transfer matrices by examining the connection between the non-canonical matrix theory and the Hamiltonian description.","Submitted by Carolyn Mead (cmead2@illinois.edu) on 2011-05-18T17:15:59Z No. of bitstreams: 1 1989_sagalovsky.pdf: 7134770 bytes, checksum: e117a389994ff1a52f47b03d21e2b9a7 (MD5)","Made available in DSpace on 2011-05-18T17:15:59Z (GMT). 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