Schottky anomaly

<h2 id="details">Details</h2>
<p>In a system where particles can have either a state of energy 0 or 
  
    
      
        ϵ
      
    
    {\displaystyle \epsilon }
  
, the <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of the energy of a particle in the <a href="/facts/Canonical_ensemble/GIhyzkb8">canonical ensemble</a> is:
</p><p>
  
    
      
        ⟨
        ϵ
        ⟩
        =
        ϵ
        ⋅
        
          
            
              e
              
                −
                β
                ϵ
              
            
            
              1
              +
              
                e
                
                  −
                  β
                  ϵ
                
              
            
          
        
        =
        
          
            ϵ
            
              
                e
                
                  +
                  β
                  ϵ
                
              
              +
              1
            
          
        
      
    
    {\displaystyle \langle \epsilon \rangle =\epsilon \cdot {\frac {e^{-\beta \epsilon }}{1+e^{-\beta \epsilon }}}={\frac {\epsilon }{e^{+\beta \epsilon }+1}}}

</p><p>with the inverse temperature 
  
    
      
        β
        =
        
          
            1
            
              
                k
                
                  
                    B
                  
                
              
              T
            
          
        
      
    
    {\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }T}}}
  
 and the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a> 
  
    
      
        
          k
          
            
              B
            
          
        
      
    
    {\displaystyle k_{\mathrm {B} }}
  
.
</p><p>The total energy of 
  
    
      
        N
      
    
    {\displaystyle N}
  
 independent particles is thus:
</p><p>
  
    
      
        U
        =
        N
        ⟨
        ϵ
        ⟩
        =
        
          
            
              N
              ϵ
            
            
              
                e
                
                  +
                  β
                  ϵ
                
              
              +
              1
            
          
        
      
    
    {\displaystyle U=N\langle \epsilon \rangle ={\frac {N\epsilon }{e^{+\beta \epsilon }+1}}}

</p><p>The heat capacity is therefore: 
</p><p>
  
    
      
        C
        =
        
          
            (
            
              
                
                  ∂
                  U
                
                
                  ∂
                  T
                
              
            
            )
          
          
            ϵ
          
        
        =
        −
        
          
            1
            
              
                k
                
                  
                    B
                  
                
              
              
                T
                
                  2
                
              
            
          
        
        
          
            
              ∂
              U
            
            
              ∂
              β
            
          
        
        =
        N
        
          k
          
            
              B
            
          
        
        
          
            (
            
              
                ϵ
                
                  
                    k
                    
                      
                        B
                      
                    
                  
                  T
                
              
            
            )
          
          
            2
          
        
        
          
            
              e
              
                +
                
                  
                    ϵ
                    
                      
                        k
                        
                          
                            B
                          
                        
                      
                      T
                    
                  
                
              
            
            
              
                (
                
                  
                    e
                    
                      +
                      
                        
                          ϵ
                          
                            
                              k
                              
                                
                                  B
                                
                              
                            
                            T
                          
                        
                      
                    
                  
                  +
                  1
                
                )
              
              
                2
              
            
          
        
      
    
    {\displaystyle C=\left({\frac {\partial U}{\partial T}}\right)_{\epsilon }=-{\frac {1}{k_{\mathrm {B} }T^{2}}}{\frac {\partial U}{\partial \beta }}=Nk_{\mathrm {B} }\left({\frac {\epsilon }{k_{\mathrm {B} }T}}\right)^{2}{\frac {e^{+{\frac {\epsilon }{k_{\mathrm {B} }T}}}}{\left(e^{+{\frac {\epsilon }{k_{\mathrm {B} }T}}}+1\right)^{2}}}}

</p><p>Plotting 
  
    
      
        C
      
    
    {\displaystyle C}
  
 as a function of temperature, a peak can be seen at 
  
    
      
        
          k
          
            
              B
            
          
        
        T
        ≈
        0.417
        ϵ
      
    
    {\displaystyle k_{\mathrm {B} }T\approx 0.417\epsilon }
  
. In this section

ϵ
            
              k
              
                
                  B
                
              
            
          
        
        =
        Δ
      
    
    {\displaystyle {\frac {\epsilon }{k_{\mathrm {B} }}}=\Delta }

for the 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
 in the introductory section.
</p>

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

<ol>
<li id="fn:1"><p>Tari, A: The Specific Heat of Matter at Low Temperatures, page 250. Imperial College Press, 2003. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Schottky anomaly open-in-new

Schottky anomaly