Nuclear cross section

<h2 id="microscopic-and-macroscopic-cross-section">Microscopic and macroscopic cross section</h2>
Nuclear cross sections are used in determining the <a href="/facts/Nuclear_reaction/202yClUr">nuclear reaction</a> rate, and are governed by the reaction rate equation for a particular set of particles (usually viewed as a "beam and target" thought experiment where one particle or nucleus is the "target", which is typically at rest, and the other is treated as a "beam", which is a projectile with a given energy).
For particle interactions incident upon a thin sheet of material (ideally made of a single <a href="/facts/Isotope/KD9B1y6o">isotope</a>), the nuclear reaction rate equation is written as:

r
          
            x
          
        
        =
        Φ
         
        
          σ
          
            x
          
        
         
        
          ρ
          
            A
          
        
        =
        Φ
        
          Σ
          
            x
          
        
      
    
    {\displaystyle r_{x}=\Phi \ \sigma _{x}\ \rho _{A}=\Phi \Sigma _{x}}

where:

<ul><li>
 
 
 
 
 r
 
 x
 
 
 
 
 {\displaystyle r_{x}}
 
 : number of reactions of type x, units: [1/time⋅volume]</li>
<li>
 
 
 
 Φ
 
 
 {\displaystyle \Phi }
 
 : beam flux, units: [1/area⋅time]</li>
<li>
 
 
 
 
 σ
 
 x
 
 
 
 
 {\displaystyle \sigma _{x}}
 
 : microscopic cross section for reaction 
 
 
 
 x
 
 
 {\displaystyle x}
 
, units: [area] (usually <a href="/facts/Barn_(unit)/0yCEtKJ9">barns</a> or cm2).</li>
<li>
 
 
 
 
 ρ
 
 A
 
 
 
 
 {\displaystyle \rho _{A}}
 
 : density of atoms in the target in units of [1/volume]</li>
<li>
 
 
 
 
 Σ
 
 x
 
 
 ≡
 
 σ
 
 x
 
 
  
 
 ρ
 
 A
 
 
 
 
 {\displaystyle \Sigma _{x}\equiv \sigma _{x}\ \rho _{A}}
 
: macroscopic cross-section [1/length]</li></ul>
Types of reactions frequently encountered are s: scattering, 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
: radiative capture, a: absorption (radiative capture belongs to this type), f: fission, the corresponding notation for cross-sections being: 
 
 
 
 
 σ
 
 s
 
 
 
 
 {\displaystyle \sigma _{s}}
 
, 
 
 
 
 
 σ
 
 γ
 
 
 
 
 {\displaystyle \sigma _{\gamma }}
 
, 
 
 
 
 
 σ
 
 a
 
 
 
 
 {\displaystyle \sigma _{a}}
 
, etc. A special case is the total cross-section 
 
 
 
 
 σ
 
 t
 
 
 
 
 {\displaystyle \sigma _{t}}
 
, which gives the probability of a neutron to undergo any sort of reaction (
 
 
 
 
 σ
 
 t
 
 
 =
 
 σ
 
 s
 
 
 +
 
 σ
 
 γ
 
 
 +
 
 σ
 
 f
 
 
 +
 …
 
 
 {\displaystyle \sigma _{t}=\sigma _{s}+\sigma _{\gamma }+\sigma _{f}+\ldots }
 
).
Formally, the equation above defines the macroscopic cross-section (for reaction x) as the proportionality constant between a particle flux incident on a (thin) piece of material and the number of reactions that occur (per unit volume) in that material. The distinction between macroscopic and microscopic cross-section is that the former is a property of a specific lump of material (with its density), while the latter is an intrinsic property of a type of nuclei.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Neutron_cross-section/gAkJmXXA">Neutron cross-section</a></li>
<li><a href="/facts/Scattering_cross-section/lo014h2D">Scattering cross-section</a></li></ul>

<ul><li>Nuclear Reactor Analysis by <a href="/facts/James_J._Duderstadt/dVfYnTWf">James J. Duderstadt</a> and Louis J. Hamilton - Published by John Wiley & Sons, Inc.</li>
<li>Perkins, Donald H. (1999). Introduction to High Energy Physics. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-62196-0.</li>
<li><a href="/facts/Samar_Mubarakmand/TPQ2M0T9">Mubarakmand, Samar</a>; <a href="/facts/Masud_Ahmad/Q738L4hG">Masud Ahmad</a>; M. Anwar; M.S. Chaudhry (1977). "Cross-section measurements with a neutron generator". The Nucleus. 42 (1–2). Nilore, Islamabad: PINSTECH: 115–185.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Younes, Walid; Loveland, Walter (2021). An Introduction to Nuclear Fission. Springer. pp. 10, 25–26, 56–58. ISBN 9783030845940. <a href="9783030845940" target="_blank">9783030845940</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Simon & Schuster Paperbacks. pp. 333–334, 282–287. ISBN 9781451677614. <a href="9781451677614" target="_blank">9781451677614</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Nuclear cross section open-in-new

Nuclear cross section