Plasma parameter

<h2 id="coupling-parameter">Coupling parameter</h2>
<p>A closely related parameter is the plasma coupling 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
, defined as a ratio of the Coulomb energy to the thermal one:

Γ
        =
        
          
            
              E
              
                C
              
            
            
              
                k
                
                  B
                
              
              
                T
                
                  e
                
              
            
          
        
        .
      
    
    {\displaystyle \Gamma ={\frac {E_{\text{C}}}{k_{\text{B}}T_{\text{e}}}}.}

</p><p>The Coulomb energy (per particle) is

E
          
            C
          
        
        =
        
          
            
              q
              
                e
              
              
                2
              
            
            
              4
              π
              
                ε
                
                  0
                
              
              ⟨
              r
              ⟩
            
          
        
        ,
      
    
    {\displaystyle E_{\text{C}}={\frac {q_{\text{e}}^{2}}{4\pi \varepsilon _{0}\langle r\rangle }},}

where for the typical inter-particle distance 
  
    
      
        ⟨
        r
        ⟩
      
    
    {\displaystyle \langle r\rangle }
  
 usually is taken the <a href="/facts/Wigner%E2%80%93Seitz_radius/C7BQ1fY7">Wigner–Seitz radius</a>. Therefore,

Γ
        =
        
          
            
              q
              
                e
              
              
                2
              
            
            
              4
              π
              
                ε
                
                  0
                
              
              
                k
                
                  B
                
              
              
                T
                
                  e
                
              
            
          
        
        
          
            
              
                4
                π
                
                  n
                  
                    e
                  
                
              
              3
            
            
              3
            
          
        
        .
      
    
    {\displaystyle \Gamma ={\frac {q_{\text{e}}^{2}}{4\pi \varepsilon _{0}k_{\text{B}}T_{\text{e}}}}{\sqrt[{3}]{\frac {4\pi n_{\text{e}}}{3}}}.}

</p><p>Clearly, up to a numeric factor of the order of unity,

Γ
        ∼
        
          Λ
          
            −
            2
            
              /
            
            3
          
        
        .
      
    
    {\displaystyle \Gamma \sim \Lambda ^{-2/3}.}

</p><p>In general, for multicomponent plasmas one defines the coupling parameter for each species <i>s</i> separately:

Γ
          
            s
          
        
        =
        
          
            
              q
              
                s
              
              
                2
              
            
            
              4
              π
              
                ε
                
                  0
                
              
              
                k
                
                  B
                
              
              
                T
                
                  s
                
              
            
          
        
        
          
            
              
                4
                π
                
                  n
                  
                    s
                  
                
              
              3
            
            
              3
            
          
        
        .
      
    
    {\displaystyle \Gamma _{s}={\frac {q_{s}^{2}}{4\pi \varepsilon _{0}k_{\text{B}}T_{s}}}{\sqrt[{3}]{\frac {4\pi n_{s}}{3}}}.}

</p><p>Here, <i>s</i> stands for either electrons or (a type of) ions.
</p>
<h2 id="the-ideal-plasma-approximation">The ideal plasma approximation</h2>
<p>One of the criteria which determine whether a collection of charged particles can rigorously be termed an <a href="/facts/Ideal_plasma/1d3btMVU">ideal plasma</a> is that Λ ≫ 1. When this is the case, collective electrostatic interactions dominate over binary collisions, and the plasma particles can be treated as if they only interact with a smooth background field, rather than through pairwise interactions (collisions).<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> The <a href="/facts/Equation_of_state/tLILkepz">equation of state</a> of each species in an ideal plasma is that of an <a href="/facts/Ideal_gas/80EfYGFj">ideal gas</a>.
</p>
<h2 id="plasma-properties-and-">Plasma properties and Λ</h2>
<p>Depending on the magnitude of Λ, plasma properties can be characterized as following:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<table><tbody><tr><th rowspan="2">Description</th><th colspan="2">Plasma parameter magnitude</th></tr><tr><th>Λ ≪ 1 (Γ ≫ 1)</th><th>Λ ≫ 1 (Γ ≪ 1)</th></tr><tr><th>Coupling</th><td>Strongly coupled plasma</td><td>Weakly coupled plasma</td></tr><tr><th>Debye sphere</th><td>Sparsely populated</td><td>Densely populated</td></tr><tr><th>Electrostatic influence</th><td>Almost continuously</td><td>Occasional</td></tr><tr><th>Typical characteristic</th><td>Cold and dense</td><td>Hot and diffuse</td></tr><tr><th>Examples</th><td>Solid-density laser ablation plasmasVery "cold" "high pressure" arc dischargeInertial fusion experimentsStellar interiors</td><td>Ionospheric physicsMagnetic fusion devicesSpace plasma physicsPlasma ball</td></tr></tbody></table>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20090325194948/http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_07.pdf">NRL Plasma Formulary 2007 ed.</a></li></ul>

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

<ol>
<li id="fn:1"><p>Chen, Francis F. (2006). Introduction to Plasma Physics and Controlled Fusion. New York: Springer. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Miyamoto, K. (1997). Fundamentals of Plasma Physics and Controlled Fusion. Iwanami, Tokyo.{{cite book}}:  CS1 maint: location missing publisher (link) <a href="/wiki/Template:Cite_book" target="_blank">/wiki/Template:Cite_book</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>J.D. Callen, University of Wisconsin-Madison, Draft Material for Fundamentals of Plasma Physics book: Collective Plasma Phenomena PDF <a href="http://homepages.cae.wisc.edu/~callen/chap1.pdf" target="_blank">http://homepages.cae.wisc.edu/~callen/chap1.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>See The plasma parameter lecture notes from Richard Fitzpatrick <a href="http://farside.ph.utexas.edu/teaching/plasma/lectures/node8.html" target="_blank">http://farside.ph.utexas.edu/teaching/plasma/lectures/node8.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Plasma parameter open-in-new

Plasma parameter