Talk:Unit cell

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Extra math expressions

A given real-space cubic lattice will have dimensions:

Such that the position of any particular cell within the infinite lattice is:

Where h, k, and l are indices. The corresponding inverse-space lattice would be:

In the case where :

Vectors

There are many equivalent ways to define/construct the Cartesian basis for the unit cell in real-space. The unit cell vectors can be written as:

According to this, the vectors can be written as:

which is mathematically equivalent.

According to:

The vectors are written as:

This is, again, an equivalent expression. The equivalence can be show by:

TBD: Reciprocal vector components

Calculate q_hkl generally

    def q_hkl(self, h, k, l):
        """Determines the position in reciprocal space for the given reflection."""
        
        # The 'unitcell' coordinate system assumes:
        #  a-axis lies along x-axis
        #  b-axis is in x-y plane
        #  c-axis is vertical (or at a tilt, depending on beta)
        
        # Convert from (unitcell) Cartesian to (unitcell) fractional coordinates
        reduced_volume = sqrt( 1 - (cos(self.alpha))**2 - (cos(self.beta))**2 - (cos(self.gamma))**2 + 2*cos(self.alpha)*cos(self.beta)*cos(self.gamma) )
        #volume = reduced_volume*self.lattice_spacing_a*self.lattice_spacing_b*self.lattice_spacing_c
        a = ( self.lattice_spacing_a , \
                0.0 , \
                0.0  )
        b = ( self.lattice_spacing_b*cos(self.gamma) , \
                self.lattice_spacing_b*sin(self.gamma) , \
                0.0 )
        c = ( self.lattice_spacing_c*cos(self.beta) , \
                self.lattice_spacing_c*( cos(self.alpha) - cos(self.beta)*cos(self.gamma) )/( sin(self.gamma) ) , \
                self.lattice_spacing_c*reduced_volume/( sin(self.gamma) ) )
        
        # Compute (unitcell) reciprocal-space lattice vectors
        volume = np.dot( a, np.cross(b,c) )
        u = np.cross( b, c ) / volume # Along qx
        v = np.cross( c, a ) / volume # Along qy
        w = np.cross( a, b ) / volume # Along qz
        
        qhkl_vector = 2*pi*( h*u + k*v + l*w )
        qhkl = sqrt( qhkl_vector[0]**2 + qhkl_vector[1]**2 + qhkl_vector[2]**2 )
        
        return (qhkl, qhkl_vector)

                
        
    def q_hkl_length(self, h, k, l):
        
        qhkl, qhkl_vector = self.q_hkl(h,k,l)
        #qhkl = sqrt( qhkl_vector[0]**2 + qhkl_vector[1]**2 + qhkl_vector[2]**2 )
        
        return qhkl