Fractional coordinates

<p>In <a href="/facts/Crystallography/9B5toeqG">crystallography</a>, a fractional coordinate system (crystal coordinate system) is a <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a> in which basis vectors used to describe the space are the lattice vectors of a crystal (periodic) pattern. The selection of an origin and a basis define a unit cell, a parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped (3D) in higher dimensions) defined by the lattice basis vectors 
  
    
      
        
          
            a
          
          
            1
          
        
        ,
        
          
            a
          
          
            2
          
        
        ,
        …
        ,
        
          
            a
          
          
            d
          
        
      
    
    {\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\dots ,\mathbf {a} _{d}}
  
 where 
  
    
      
        d
      
    
    {\displaystyle d}
  
 is the dimension of the space. These basis vectors are described by lattice parameters (lattice constants) consisting of the lengths of the lattice basis vectors  
  
    
      
        
          a
          
            1
          
        
        ,
        
          a
          
            2
          
        
        ,
        …
        ,
        
          a
          
            d
          
        
      
    
    {\displaystyle a_{1},a_{2},\dots ,a_{d}}
  
 and the angles between them 
  
    
      
        
          α
          
            1
          
        
        ,
        
          α
          
            2
          
        
        ,
        …
        ,
        
          α
          
            
              
                d
                (
                d
                −
                1
                )
              
              2
            
          
        
      
    
    {\displaystyle \alpha _{1},\alpha _{2},\dots ,\alpha _{\frac {d(d-1)}{2}}}
  
.  
</p><p>Most cases in crystallography involve two- or three-dimensional space.  In the three-dimensional case, the basis vectors 
  
    
      
        
          
            a
          
          
            1
          
        
        ,
        
          
            a
          
          
            2
          
        
        ,
        
          
            a
          
          
            3
          
        
      
    
    {\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}}
  
 are commonly displayed as 
  
    
      
        
          a
        
        ,
        
          b
        
        ,
        
          c
        
      
    
    {\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }
  
 with their lengths denoted by 
  
    
      
        a
        ,
        b
        ,
        c
      
    
    {\displaystyle a,b,c}
  
 respectively, and the angles denoted by 
  
    
      
        α
        ,
        β
        ,
        γ
      
    
    {\displaystyle \alpha ,\beta ,\gamma }
  
, where conventionally, 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is the angle between 
  
    
      
        
          b
        
      
    
    {\displaystyle \mathbf {b} }
  
 and 
  
    
      
        
          c
        
      
    
    {\displaystyle \mathbf {c} }
  
, 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 is the angle between 
  
    
      
        
          c
        
      
    
    {\displaystyle \mathbf {c} }
  
 and 
  
    
      
        
          a
        
      
    
    {\displaystyle \mathbf {a} }
  
, and 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is the angle between 
  
    
      
        
          a
        
      
    
    {\displaystyle \mathbf {a} }
  
 and 
  
    
      
        
          b
        
      
    
    {\displaystyle \mathbf {b} }
  
.
</p>

Fractional coordinates open-in-new

Fractional coordinates