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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Impact parameter</span></span>
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<p>In <a href="Physics" title="Physics">physics</a>, the <b>impact parameter</b> <span class="texhtml mvar" style="font-style:italic;">b</span> is defined as the perpendicular distance between the <a href="Trajectory" title="Trajectory">path</a> of a <a href="Projectile" title="Projectile">projectile</a> and the center of a potential <a href="Field_(physics)" title="Field (physics)">field</a> <span class="texhtml"><i>U</i>(<i>r</i>)</span> created by an object that the projectile is approaching (see diagram). It is often referred to in <a href="Nuclear_physics" title="Nuclear physics">nuclear physics</a> (see <a href="Rutherford_scattering" class="mw-redirect" title="Rutherford scattering">Rutherford scattering</a>) and in <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>.
</p><p>The impact parameter is related to the <a href="Scattering" title="Scattering">scattering</a> angle <span class="texhtml mvar" style="font-style:italic;">θ</span> by<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi -2b\int _{r_{\text{min}}}^{\infty }{\frac {dr}{r^{2}{\sqrt {1-(b/r)^{2}-2U/(mv_{\infty }^{2})}}}},}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>θ<!-- θ --></mi>
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<mi>r</mi>
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<mi>d</mi>
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<mi>r</mi>
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<mn>2</mn>
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<mn>1</mn>
<mo>−<!-- − --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \theta =\pi -2b\int _{r_{\text{min}}}^{\infty }{\frac {dr}{r^{2}{\sqrt {1-(b/r)^{2}-2U/(mv_{\infty }^{2})}}}},}</annotation>
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</math></span><img src="./ecd95d79c40e9fb89e3f9e3df0e959ba7468e350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:45.736ex; height:8.176ex;" alt="{\displaystyle \theta =\pi -2b\int _{r_{\text{min}}}^{\infty }{\frac {dr}{r^{2}{\sqrt {1-(b/r)^{2}-2U/(mv_{\infty }^{2})}}}},}" loading="lazy"></span></dd></dl>
<p>where <span class="texhtml mvar" style="font-style:italic;">v<sub>∞</sub></span> is the velocity of the projectile when it is far from the center, and <span class="texhtml"><i>r</i><sub>min</sub></span> is its closest distance from the center.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
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<div class="mw-heading mw-heading2"><h2 id="Scattering_from_a_hard_sphere">Scattering from a hard sphere</h2></div>
<p>The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. Here, the object that the projectile is approaching is a hard sphere with radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>. In the case of a hard sphere, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(r)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(r)=0}</annotation>
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</math></span><img src="./c5c0e2b512e04e1bf1f031ab0e7f8de261d3fe55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.901ex; height:2.843ex;" alt="{\displaystyle U(r)=0}" loading="lazy"></span> when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>></mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r>R}</annotation>
</semantics>
</math></span><img src="./8971c9610113faec012a76ec2d47fa6235e16d2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r>R}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(r)=\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U(r)=\infty }</annotation>
</semantics>
</math></span><img src="./a19130ece63097200769dadf3dc889ab58283ea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.063ex; height:2.843ex;" alt="{\displaystyle U(r)=\infty }" loading="lazy"></span> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\leq R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>≤<!-- ≤ --></mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\leq R}</annotation>
</semantics>
</math></span><img src="./e3b3e9b51ba7a6dba138c1dc33c0324609fa43ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.911ex; height:2.343ex;" alt="{\displaystyle r\leq R}" loading="lazy"></span>. When <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>></mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b>R}</annotation>
</semantics>
</math></span><img src="./3e9736103b61b44e0493d381bc3fbc417b62ed09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.86ex; height:2.176ex;" alt="{\displaystyle b>R}" loading="lazy"></span>, the projectile misses the hard sphere. We immediately see that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation>
</semantics>
</math></span><img src="./2a7bc6e34b53e0e8a8815159c356b1acccf7ea24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}" loading="lazy"></span>. When <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>≤<!-- ≤ --></mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b\leq R}</annotation>
</semantics>
</math></span><img src="./4adb663a68b1f7c68f633599e0308f3448229f76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.86ex; height:2.343ex;" alt="{\displaystyle b\leq R}" loading="lazy"></span>, we find that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=R\cos {\tfrac {\theta }{2}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>=</mo>
<mi>R</mi>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>θ<!-- θ --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b=R\cos {\tfrac {\theta }{2}}.}</annotation>
</semantics>
</math></span><img src="./99a2195a206fe900264ca7fd795250e18bd77e1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.05ex; height:3.676ex;" alt="{\displaystyle b=R\cos {\tfrac {\theta }{2}}.}" loading="lazy"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Collision_centrality">Collision centrality</h2></div>
<p>In <a href="Particle_physics" title="Particle physics">high-energy nuclear physics</a> — specifically, in <a href="Collider" title="Collider">colliding-beam experiments</a> — collisions may be classified according to their impact parameter. Central collisions have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\approx 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>≈<!-- ≈ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b\approx 0}</annotation>
</semantics>
</math></span><img src="./a5db79dbb027653ec86d710804c3b1e8322a8859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b\approx 0}" loading="lazy"></span>, peripheral collisions have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<b<2R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo><</mo>
<mi>b</mi>
<mo><</mo>
<mn>2</mn>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0<b<2R}</annotation>
</semantics>
</math></span><img src="./61f38f41d6afa3d5333859a0995f2afcab076e54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.283ex; height:2.176ex;" alt="{\displaystyle 0<b<2R}" loading="lazy"></span>, and ultraperipheral collisions (UPCs)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>2R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>></mo>
<mn>2</mn>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b>2R}</annotation>
</semantics>
</math></span><img src="./8fe6295acc77c521343d7c3b333a74745d1c9d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.022ex; height:2.176ex;" alt="{\displaystyle b>2R}" loading="lazy"></span>, where the colliding <a href="Atomic_nucleus" title="Atomic nucleus">nuclei</a> are viewed as <a href="Hard_spheres" title="Hard spheres">hard spheres</a> with radius <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>.
</p><p>Because the <a href="Strong_interaction" title="Strong interaction">color force</a> has an extremely short range, it cannot couple quarks that are separated by much more than one <a href="Nucleon" title="Nucleon">nucleon</a>'s radius; hence, strong interactions are suppressed in peripheral and ultraperipheral collisions. This means that final-state particle multiplicity (the total number of particles resulting from the collision), is typically greatest in the most central collisions, due to the <a href="Parton_(particle_physics)" title="Parton (particle physics)">partons</a> involved having the greatest probability of interacting in some way. This has led to <a href="Charged_particle" title="Charged particle">charged particle</a> multiplicity being used as a common measure of collision centrality, as charged particles are much easier to detect than uncharged particles.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p><p>Because strong interactions are effectively impossible in ultraperipheral collisions, they may be used to study electromagnetic interactions — i.e. <a href="Two-photon_physics" title="Two-photon physics">photon–photon</a>, photon–nucleon, or photon–nucleus interactions — with low background contamination. Because UPCs typically produce only two to four final-state particles, they are also relatively "clean" when compared to central collisions, which may produce hundreds of particles per <a href="Event_(particle_physics)" title="Event (particle physics)">event</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Distance_of_closest_approach" title="Distance of closest approach">Distance of closest approach</a></li>
<li><a href="Hyperbolic_trajectory#Impact_parameter" title="Hyperbolic trajectory">Hyperbolic trajectory § Impact parameter</a></li>
<li><a href="Schwarzschild_geodesics#Bending_of_light_by_gravity" title="Schwarzschild geodesics">Schwarzschild geodesics § Bending of light by gravity</a></li>
<li><a href="Tests_of_general_relativity" title="Tests of general relativity">Tests of general relativity</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Landau L. D. and Lifshitz E. M. (1976) <i>Mechanics</i>, 3rd. ed., Pergamon Press. <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-08-021022-8</bdi> (hardcover) and <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-08-029141-4</bdi> (softcover).</span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFmitopercourseware2021" class="citation web cs1">mitopercourseware, MIT (3 September 2021). <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/nuclear-engineering/22-105-electromagnetic-interactions-fall-2005/readings/chap6.pdf">"Notes"</a> <span class="cs1-format">(PDF)</span>.</cite></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/impar.html">"Impact Parameter for Nuclear Scattering"</a>. <i>hyperphysics.phy-astr.gsu.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-09-03</span></span>.</cite></span>
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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">* Relativistic Heavy Ion Physics without Nuclear Contact, C.A. Bertulani and G. Baur, Physics Today, March 1994, pg. 22.</span>
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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFDrozhzhova2017" class="citation journal cs1">Drozhzhova, Tatiana (January 2017). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F798%2F1%2F012061">"Centrality and collision event-plane determination in ALICE at the LHC"</a>. <i>Journal of Physics: Conference Series</i>. <b>798</b> (1): 012061. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017JPhCS.798a2061D">2017JPhCS.798a2061D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-6596%2F798%2F1%2F012061">10.1088/1742-6596/798/1/012061</a></span>.</cite></span>
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<ul><li><a rel="nofollow" class="external free" href="http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/rutsca2.html">http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/rutsca2.html</a></li></ul>
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