Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases
This work explores the evaluation of hydrogenic matrix elements for non-relativistic and relativistic cases under the Screened Hydrogenic Model (SHM). It focuses on scenarios where the initial and final states have different effective charges <inline-formula><math xmlns="http://www.w3....
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2025-06-01
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| author | Héctor O. Di Rocco Julio C. Aguiar |
| author_facet | Héctor O. Di Rocco Julio C. Aguiar |
| author_sort | Héctor O. Di Rocco |
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| description | This work explores the evaluation of hydrogenic matrix elements for non-relativistic and relativistic cases under the Screened Hydrogenic Model (SHM). It focuses on scenarios where the initial and final states have different effective charges <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>≠</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>, deriving closed-form solutions for particular cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In addition, analytical expressions for radial matrix elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><msup><mi>n</mi><mo>′</mo></msup><msup><mi>l</mi><mo>′</mo></msup><mo>|</mo><msup><mi>r</mi><mi>β</mi></msup><mo>|</mo><mi>n</mi><mi>l</mi><mo>⟩</mo></mrow></semantics></math></inline-formula> and their relativistic counterparts are presented. These are applicable for discrete–discrete transitions and allow simplifications for specific configurations using Laplace transforms. The study discusses generalizations of SHM for calculating cross-sections in hot and dense plasmas, employing the Plane Wave Born Approximation (PWBA). It also addresses the transition from LS to jj coupling for matrix elements, providing rules for such transformations. |
| format | Article |
| id | doaj-art-cfbf01da0c3246cf8dd4048a2e9316fa |
| institution | Kabale University |
| issn | 2218-2004 |
| language | English |
| publishDate | 2025-06-01 |
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| spelling | doaj-art-cfbf01da0c3246cf8dd4048a2e9316fa2025-08-20T03:58:26ZengMDPI AGAtoms2218-20042025-06-011376010.3390/atoms13070060Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic CasesHéctor O. Di Rocco0Julio C. Aguiar1Instituto de Física “Arroyo Seco”, Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, Tandil B7000GHG, ArgentinaAutoridad Regulatoria Nuclear, Av. Del Libertador 8250, Buenos Aires C1429BNP, ArgentinaThis work explores the evaluation of hydrogenic matrix elements for non-relativistic and relativistic cases under the Screened Hydrogenic Model (SHM). It focuses on scenarios where the initial and final states have different effective charges <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>≠</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>, deriving closed-form solutions for particular cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In addition, analytical expressions for radial matrix elements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><msup><mi>n</mi><mo>′</mo></msup><msup><mi>l</mi><mo>′</mo></msup><mo>|</mo><msup><mi>r</mi><mi>β</mi></msup><mo>|</mo><mi>n</mi><mi>l</mi><mo>⟩</mo></mrow></semantics></math></inline-formula> and their relativistic counterparts are presented. These are applicable for discrete–discrete transitions and allow simplifications for specific configurations using Laplace transforms. The study discusses generalizations of SHM for calculating cross-sections in hot and dense plasmas, employing the Plane Wave Born Approximation (PWBA). It also addresses the transition from LS to jj coupling for matrix elements, providing rules for such transformations.https://www.mdpi.com/2218-2004/13/7/60hydrogenic matrix elementsScreened Hydrogenic Modelhot and dense plasmasLS to jj coupling |
| spellingShingle | Héctor O. Di Rocco Julio C. Aguiar Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases Atoms hydrogenic matrix elements Screened Hydrogenic Model hot and dense plasmas LS to jj coupling |
| title | Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases |
| title_full | Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases |
| title_fullStr | Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases |
| title_full_unstemmed | Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases |
| title_short | Hydrogenic Matrix Elements with Different Effective Charges: Non-Relativistic and Relativistic Cases |
| title_sort | hydrogenic matrix elements with different effective charges non relativistic and relativistic cases |
| topic | hydrogenic matrix elements Screened Hydrogenic Model hot and dense plasmas LS to jj coupling |
| url | https://www.mdpi.com/2218-2004/13/7/60 |
| work_keys_str_mv | AT hectorodirocco hydrogenicmatrixelementswithdifferenteffectivechargesnonrelativisticandrelativisticcases AT juliocaguiar hydrogenicmatrixelementswithdifferenteffectivechargesnonrelativisticandrelativisticcases |