<section id="vavref"> <stitle>Vavilov theory</stitle> <par>Vavilov<cite refid="bib-VAVI"/> derived a more accurate straggling distribution by introducing the kinematic limit on the maximum transferable energy in a single collision, rather than using <inlinemath><math><msub><mi>E</mi><mrow><mtext>max</mtext></mrow></msub> <mo>=</mo><mi>∞</mi></math></inlinemath>. Now we can write<cite refid="bib-SCH1"/>: <eqnarray ><subeqn><math><mi>f</mi> <mfenced open='(' close=')'> <mi>ε</mi><mo>,</mo><mi>δ</mi><mi>s</mi></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow> <mrow><mi>ξ</mi></mrow> </mfrac><msub><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced open='(' close=')'> <msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo> <mi>κ</mi><mo>,</mo><msup><mi>β</mi><mrow><mn>2</mn></mrow> </msup></mfenced></math></subeqn></eqnarray> where <eqnarray><subeqn><math><msub><mi>φ</mi><mrow><mi>v</mi></mrow></msub> <mfenced open='(' close=')'> <msub><mi>λ</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo> <mi>κ</mi><mo>,</mo> <msup><mi>β</mi><mrow><mn>2</mn></mrow></msup></mfenced> <mo>=</mo> <mfrac><mrow><mn>1</mn></mrow> <mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow> </mfrac> <msubsup><mo>∫</mo> <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>∞</mi></mrow> <mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>∞</mi></mrow></msubsup> <mi>φ</mi><mfenced open='(' close=')'><mi>s</mi></mfenced> <msup><mi>e</mi><mrow><mi>λ</mi><mi>s</mi></mrow></msup> <mi>d</mi><mi>s</mi><mspace width='2cm'/><mi>c</mi><mo>≥</mo><mn>0</mn> </math></subeqn> <subeqn><math><mi>φ</mi><mfenced open='(' close=')'><mi>s</mi></mfenced> <mo>=</mo><mo>exp</mo><mfenced open='[' close=']'><mi>κ</mi> <mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi> <mrow><mn>2</mn></mrow></msup><mi>γ</mi><mo>)</mo></mrow> </mfenced><mo>exp</mo><mfenced open='[' close=']'><mi>ψ</mi> <mfenced open='(' close=')'><mi>s</mi></mfenced></mfenced> <mo>,</mo> </math></subeqn>
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