{"title":"寻找相对群强度的费米函数积分","authors":"George P. Ford, Darleane C. Hoffman","doi":"10.1016/S0550-306X(65)80008-8","DOIUrl":null,"url":null,"abstract":"<div><p>Values are presented of the definite integral <em>I</em>(<em>Z, T</em><sub>max</sub>), which is related to the relative number, <em>N</em>, of electrons in a β<sup>−</sup>-group with end point energy <em>T</em><sub>max</sub> by the simple expression <em>N</em> = <em>a</em><sup>2</sup><em>T</em><sub>max</sub><sup>2</sup><em>I</em>(<em>Z</em>, <em>T</em><sub>max</sub>), where <em>a</em> is the <em>y</em>-intercept in the Fermi-Kurie plot, <span><span><span><math><mrow><mi>I</mi><mo>(</mo><mi>Z</mi><mo>,</mo><msub><mi>T</mi><mrow><mi>max</mi></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>T</mi><mo>max</mo></msub></mrow></msubsup><mi>f</mi><mo>(</mo><mi>Z</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>h</mi><mo>(</mo><mi>T</mi><mo>)</mo><mi>d</mi><mi>T</mi><mo>,</mo></mrow></math></span></span></span></p><p>where <em>f</em> is the Fermi function and <span><span><span><math><mrow><mi>h</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>T</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>T</mi><mo>/</mo><msub><mi>T</mi><mrow><mi>max</mi></mrow></msub><mo>)</mo></mrow><mstyle><mn>2</mn></mstyle></msup><msup><mrow><mo>[</mo><mi>T</mi><mo>(</mo><mi>T</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>]</mo></mrow><mrow><mstyle><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mstyle></mrow></msup><msubsup><mi>T</mi><mrow><mi>max</mi></mrow><mrow><mstyle><mo>−</mo><mn>2</mn></mstyle></mrow></msubsup><mo>.</mo></mrow></math></span></span></span></p><p>The <em>Z</em>-ranges is 10(1)109; the <em>T</em><sub>max</sub>-range is 0.1(0.05)0.4, 0.5(0.1)5.2.</p></div>","PeriodicalId":100967,"journal":{"name":"Nuclear Data Sheets. Section A","volume":"1 ","pages":"Pages 411-433"},"PeriodicalIF":0.0000,"publicationDate":"1965-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0550-306X(65)80008-8","citationCount":"0","resultStr":"{\"title\":\"Fermi-function integrals for finding relative beta-group intensities\",\"authors\":\"George P. Ford, Darleane C. Hoffman\",\"doi\":\"10.1016/S0550-306X(65)80008-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Values are presented of the definite integral <em>I</em>(<em>Z, T</em><sub>max</sub>), which is related to the relative number, <em>N</em>, of electrons in a β<sup>−</sup>-group with end point energy <em>T</em><sub>max</sub> by the simple expression <em>N</em> = <em>a</em><sup>2</sup><em>T</em><sub>max</sub><sup>2</sup><em>I</em>(<em>Z</em>, <em>T</em><sub>max</sub>), where <em>a</em> is the <em>y</em>-intercept in the Fermi-Kurie plot, <span><span><span><math><mrow><mi>I</mi><mo>(</mo><mi>Z</mi><mo>,</mo><msub><mi>T</mi><mrow><mi>max</mi></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><msub><mi>T</mi><mo>max</mo></msub></mrow></msubsup><mi>f</mi><mo>(</mo><mi>Z</mi><mo>,</mo><mi>T</mi><mo>)</mo><mi>h</mi><mo>(</mo><mi>T</mi><mo>)</mo><mi>d</mi><mi>T</mi><mo>,</mo></mrow></math></span></span></span></p><p>where <em>f</em> is the Fermi function and <span><span><span><math><mrow><mi>h</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>T</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>T</mi><mo>/</mo><msub><mi>T</mi><mrow><mi>max</mi></mrow></msub><mo>)</mo></mrow><mstyle><mn>2</mn></mstyle></msup><msup><mrow><mo>[</mo><mi>T</mi><mo>(</mo><mi>T</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>]</mo></mrow><mrow><mstyle><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mstyle></mrow></msup><msubsup><mi>T</mi><mrow><mi>max</mi></mrow><mrow><mstyle><mo>−</mo><mn>2</mn></mstyle></mrow></msubsup><mo>.</mo></mrow></math></span></span></span></p><p>The <em>Z</em>-ranges is 10(1)109; the <em>T</em><sub>max</sub>-range is 0.1(0.05)0.4, 0.5(0.1)5.2.</p></div>\",\"PeriodicalId\":100967,\"journal\":{\"name\":\"Nuclear Data Sheets. Section A\",\"volume\":\"1 \",\"pages\":\"Pages 411-433\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1965-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0550-306X(65)80008-8\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nuclear Data Sheets. Section A\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0550306X65800088\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nuclear Data Sheets. Section A","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0550306X65800088","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Fermi-function integrals for finding relative beta-group intensities
Values are presented of the definite integral I(Z, Tmax), which is related to the relative number, N, of electrons in a β−-group with end point energy Tmax by the simple expression N = a2Tmax2I(Z, Tmax), where a is the y-intercept in the Fermi-Kurie plot,
where f is the Fermi function and
The Z-ranges is 10(1)109; the Tmax-range is 0.1(0.05)0.4, 0.5(0.1)5.2.
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