Neutron–proton ratio

<h2 id="semi-empirical-description">Semi-empirical description</h2>
<p>For stable nuclei, the neutron-proton ratio is such that the <a href="/facts/Nuclear_binding_energy/PU8bREHW">binding energy</a> is at a <a href="/facts/Local_minimum/ADlgnyzV">local minimum</a> or close to a minimum.
</p><p>From the liquid drop model, this bonding energy is approximated by empirical <a href="/facts/Bethe%25E2%2580%2593Weizs%25C3%25A4cker_formula/XQhTYz7e">Bethe–Weizsäcker formula</a>
</p>

E
          
            B
          
        
        =
        
          a
          
            V
          
        
        A
        −
        
          a
          
            S
          
        
        
          A
          
            2
            
              /
            
            3
          
        
        −
        
          a
          
            C
          
        
        
          
            
              Z
              
                2
              
            
            
              A
              
                1
                
                  /
                
                3
              
            
          
        
        −
        
          a
          
            A
          
        
        
          
            
              (
              A
              −
              2
              Z
              
                )
                
                  2
                
              
            
            A
          
        
        ±
        δ
        (
        A
        ,
        Z
        )
        .
      
    
    {\displaystyle E_{B}=a_{V}A-a_{S}A^{2/3}-a_{C}{\frac {Z^{2}}{A^{1/3}}}-a_{A}{\frac {(A-2Z)^{2}}{A}}\pm \delta (A,Z).}

<p>Given a value of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and ignoring the contributions of nucleon spin pairing (i.e. ignoring the 
  
    
      
        ±
        δ
        (
        A
        ,
        Z
        )
      
    
    {\displaystyle \pm \delta (A,Z)}
  
 term), the binding energy is a quadratic expression in 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 that is minimized when the neutron-proton ratio is 
  
    
      
        N
        
          /
        
        Z
        ≈
        1
        +
        
          
            
              a
              
                C
              
            
            
              2
              
                a
                
                  A
                
              
            
          
        
        
          A
          
            2
            
              /
            
            3
          
        
      
    
    {\displaystyle N/Z\approx 1+{\frac {a_{C}}{2a_{A}}}A^{2/3}}
  
.
</p>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Isotope/KD9B1y6o">Isotope § Variation in properties between isotopes</a></li>
<li><a href="/facts/Nuclear_fission/LfkIylvm">Nuclear fission</a></li>
<li><a href="/facts/Nuclear_drip_line/k3PfJUoy">Nuclear drip line</a></li></ul>

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

<ol>
<li id="fn:1"><p>"21.2: Patterns of Nuclear Stability". Chemistry LibreTexts. 2014-11-18. Retrieved 2019-04-10. <a href="https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/21%3A_Nuclear_Chemistry/21.2%3A_Patterns_of_Nuclear_Stability" target="_blank">https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/21%3A_Nuclear_Chemistry/21.2%3A_Patterns_of_Nuclear_Stability</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Radioactive Decay". chemed.chem.purdue.edu. Retrieved 2019-04-09. <a href="http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch23/modes.php" target="_blank">http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch23/modes.php</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Neutron–proton ratio open-in-new

Neutron–proton ratio