Electric field gradient

<h2 id="definition">Definition</h2>
A given charge distribution of electrons and nuclei, ρ(r), generates an <a href="/facts/Electrostatic_potential/Nmqm6hm5">electrostatic potential</a> V(r). The derivative of this potential is the negative of the <a href="/facts/Electric_field/tt4MooBs">electric field</a> generated. The first derivatives of the field, or the second derivatives of the potential, is the electric field gradient. The nine components of the EFG are thus defined as the second partial derivatives of the electrostatic potential, evaluated at the position of a nucleus:

V
          
            i
            j
          
        
        =
        
          
            
              
                ∂
                
                  2
                
              
              V
            
            
              ∂
              
                x
                
                  i
                
              
              ∂
              
                x
                
                  j
                
              
            
          
        
        .
      
    
    {\displaystyle V_{ij}={\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}.}

For each nucleus, the components Vij are combined as a symmetric 3 × 3 matrix. Under the assumption that the charge distribution generating the electrostatic potential is external to the nucleus, the matrix is <a href="/facts/Traceless/dchFFvHt">traceless</a>, for in that situation <a href="/facts/Laplace%27s_equation/8n1YkmFS">Laplace's equation</a>, ∇2V(r) = 0, holds. Relaxing this assumption, a more general form of the EFG tensor which retains the symmetry and traceless character is

φ
          
            i
            j
          
        
        =
        
          V
          
            i
            j
          
        
        −
        
          
            1
            3
          
        
        
          δ
          
            i
            j
          
        
        
          ∇
          
            2
          
        
        V
        ,
      
    
    {\displaystyle \varphi _{ij}=V_{ij}-{\frac {1}{3}}\delta _{ij}\nabla ^{2}V,}

where ∇2V(r) is evaluated at a given nucleus.
As V (and φ) is symmetric, it can be <a href="/facts/Diagonalized/y3MkyTCy">diagonalized</a>. Different conventions exist for assigning the EFG tensor's principal components from the eigenvalues. In nuclear magnetic resonance spectroscopy, the Haeberlen convention is 
 
 
 
 
 |
 
 
 λ
 
 c
 
 
 
 |
 
 >
 
 |
 
 
 λ
 
 a
 
 
 
 |
 
 >
 
 |
 
 
 λ
 
 b
 
 
 
 |
 
 
 
 {\displaystyle |\lambda _{c}|>|\lambda _{a}|>|\lambda _{b}|}
 
, in order to maintain consistency with the convention for the nuclear shielding tensor. In other fields, however, they are assigned 
 
 
 
 
 |
 
 
 λ
 
 c
 
 
 
 |
 
 >
 
 |
 
 
 λ
 
 b
 
 
 
 |
 
 >
 
 |
 
 
 λ
 
 a
 
 
 
 |
 
 
 
 {\displaystyle |\lambda _{c}|>|\lambda _{b}|>|\lambda _{a}|}
 
, more usually denoted as 
 
 
 
 |
 
 V
 
 z
 z
 
 
 |
 ≥
 |
 
 V
 
 y
 y
 
 
 |
 ≥
 |
 
 V
 
 x
 x
 
 
 |
 
 
 {\displaystyle \vert V_{zz}\vert \geq \vert V_{yy}\vert \geq \vert V_{xx}\vert }
 
, in order of decreasing <a href="/facts/Absolute_value/dwJBpEs5">modulus</a>. Given the traceless character, 
 
 
 
 
 λ
 
 a
 
 
 +
 
 λ
 
 b
 
 
 >
 
 λ
 
 c
 
 
 =
 0
 
 
 {\displaystyle \lambda _{a}+\lambda _{b}>\lambda _{c}=0}
 
, only two of the principal components are independent. Typically these are described by 
 
 
 
 
 λ
 
 c
 
 
 
 
 {\displaystyle \lambda _{c}}
 
 or Vzz and the biaxially parameter or asymmetry parameter, η, defined as

η
        =
        
          
            
              
                V
                
                  x
                  x
                
              
              −
              
                V
                
                  y
                  y
                
              
            
            
              V
              
                z
                z
              
            
          
        
        =
        
          
            
              
                λ
                
                  b
                
              
              −
              
                λ
                
                  a
                
              
            
            
              λ
              
                c
              
            
          
        
        .
      
    
    {\displaystyle \eta ={\frac {V_{xx}-V_{yy}}{V_{zz}}}={\frac {\lambda _{b}-\lambda _{a}}{\lambda _{c}}}.}

where 
 
 
 
 0
 ≤
 η
 ≤
 1
 
 
 {\displaystyle 0\leq \eta \leq 1}
 
.
Electric field gradient, as well as the biaxially parameter, can be evaluated numerically for large electric systems as shown in.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<ul><li><a href="/facts/Elton_N._Kaufmann/t5rNHPMa">Kaufmann, Elton N</a>; Reiner J. Vianden (1979). "The electric field gradient in noncubic metals". Reviews of Modern Physics. 51 (1): 161–214. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1979RvMP...51..161K">1979RvMP...51..161K</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FRevModPhys.51.161">10.1103/RevModPhys.51.161</a>.</li></ul>

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

<ol>
<li id="fn:1">Hernandez-Gomez, J J; Marquina, V; Gomez, R W (25 July 2013). "Algorithm to compute the electric field gradient tensor in ionic crystals". Rev. Mex. Fís. 58 (1): 13–18. arXiv:1107.0059. Bibcode:2011arXiv1107.0059H. Retrieved 23 April 2016. <a href="https://www.researchgate.net/publication/51914406" target="_blank">https://www.researchgate.net/publication/51914406</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Electric field gradient open-in-new

Electric field gradient