Talk:Unit cell

Extra math expressions

A given real-space cubic lattice will have dimensions:

$\left(a,b,c\right)$

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

$\mathbf {r} _{hkl}=\left\langle ah,bk,cl\right\rangle$

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

$\mathbf {q} _{hkl}=2\pi \left\langle {\frac {h}{a}},{\frac {k}{b}},{\frac {l}{c}}\right\rangle$

$q_{hkl}=2\pi {\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}}}$

In the case where $a=b=c$:

${\begin{alignedat}{2}q_{hkl}&=2\pi {\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{a}}\right)^{2}+\left({\frac {l}{a}}\right)^{2}}}\\&={\frac {2\pi }{a}}{\sqrt {h^{2}+k^{2}+l^{2}}}\end{alignedat}}$

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:

${\begin{array}{l}\mathbf {a} ={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} ={\begin{bmatrix}b\cos {\gamma }\\b\sin {\gamma }\\0\end{bmatrix}}\\\mathbf {c} ={\begin{bmatrix}c\sin {\theta _{c}}\cos {\phi _{c}}\\c\sin {\theta _{c}}\sin {\phi _{c}}\\c\cos {\theta _{c}}\end{bmatrix}}={\begin{bmatrix}c\cos {\beta }\\c{\frac {\cos {\alpha }-\cos {\beta }\cos {\gamma }}{\sin {\gamma }}}\\c{\sqrt {1-\cos ^{2}{\beta }-\left({\frac {\cos {\alpha }-\cos {\beta }\cos {\gamma }}{\sin {\gamma }}}\right)^{2}}}\end{bmatrix}}\end{array}}$

According to this, the vectors can be written as:

${\begin{array}{l}\mathbf {a} ={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} ={\begin{bmatrix}b\cos {\gamma }\\b\sin {\gamma }\\0\end{bmatrix}}\\\mathbf {c} ={\begin{bmatrix}c\cos {\beta }\\c_{y}\\c_{z}\\\end{bmatrix}}={\begin{bmatrix}c\cos {\beta }\\{\frac {c}{\sin \gamma }}\left(\cos \alpha -\cos \beta \cos \gamma \right)\\{\frac {c}{\sin \gamma }}{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}\\\end{bmatrix}}\end{array}}$

which is mathematically equivalent.

According to:

• K. N. Trueblood, H.-B. Bürgi, H. Burzlaff, J. D. Dunitz, C. M. Gramaccioli, H. H. Schulz, U. Shmueli and S. C. Abrahams Atomic Dispacement Parameter Nomenclature. Report of a Subcommittee on Atomic Displacement Parameter Nomenclature Acta Cryst 1996, A52, 770-781. doi: 10.1107/S0108767396005697

The vectors are written as:

${\begin{array}{l}\mathbf {a} ={\begin{bmatrix}a\\0\\0\end{bmatrix}}\\\mathbf {b} ={\begin{bmatrix}b\cos {\gamma }\\b\sin {\gamma }\\0\end{bmatrix}}\\\mathbf {c} ={\begin{bmatrix}c\cos {\beta }\\-c\sin \beta \cos \alpha ^{*}\\{\frac {1}{c^{*}}}\\\end{bmatrix}}={\begin{bmatrix}c\cos {\beta }\\c{\frac {\cos \alpha -\cos \beta \cos \gamma }{\sin \gamma }}\\{\frac {c}{\sin \gamma }}{\sqrt {1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma }}\\\end{bmatrix}}\end{array}}$

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

${\begin{alignedat}{2}c_{z}&=1/c^{*}\\&={\frac {c}{\sin \gamma }}{\sqrt {1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma }}\\&=c{\sqrt {\frac {1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma }{\sin ^{2}\gamma }}}\\&=c{\sqrt {{\frac {1}{\sin ^{2}\gamma }}+{\frac {2\cos \alpha \cos \beta \cos \gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\alpha }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\beta }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\gamma }{\sin ^{2}\gamma }}}}\\&=c{\sqrt {{\frac {1-\cos ^{2}\gamma }{\sin ^{2}\gamma }}+{\frac {2\cos \alpha \cos \beta \cos \gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\alpha }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\beta }{\sin ^{2}\gamma }}}}\\&=c{\sqrt {{\frac {\sin ^{2}\gamma }{\sin ^{2}\gamma }}+{\frac {2\cos \alpha \cos \beta \cos \gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\alpha }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\beta (\sin ^{2}\gamma +\cos ^{2}\gamma )}{\sin ^{2}\gamma }}}}\\&=c{\sqrt {1+{\frac {2\cos \alpha \cos \beta \cos \gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\alpha }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\beta \sin ^{2}\gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\beta \cos ^{2}\gamma }{\sin ^{2}\gamma }}}}\\&=c{\sqrt {1-\cos ^{2}\beta {\frac {\sin ^{2}\gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\alpha }{\sin ^{2}\gamma }}+{\frac {2\cos \alpha \cos \beta \cos \gamma }{\sin ^{2}\gamma }}-{\frac {\cos ^{2}\beta \cos ^{2}\gamma }{\sin ^{2}\gamma }}}}\\&=c{\sqrt {1-\cos ^{2}\beta -{\frac {1}{\sin ^{2}\gamma }}\left(\cos ^{2}\alpha -2\cos \alpha \cos \beta \cos \gamma +\cos ^{2}\beta \cos ^{2}\gamma \right)}}\\&=c{\sqrt {1-\cos ^{2}\beta -{\frac {1}{\sin ^{2}\gamma }}\left(\cos \alpha -\cos \beta \cos \gamma \right)\left(\cos \alpha -\cos \beta \cos \gamma \right)}}\\&=c{\sqrt {1-\cos ^{2}\beta -\left({\frac {\cos \alpha -\cos \beta \cos \gamma }{\sin \gamma }}\right)^{2}}}\\\end{alignedat}}$

TBD: Reciprocal vector components

${\begin{alignedat}{2}\mathbf {u} &=...\\\mathbf {v} &=...\\\mathbf {w} &={\frac {\mathbf {a} \times \mathbf {b} }{\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}\\&={\frac {1}{V}}\mathbf {a} \times \mathbf {b} \\&={\frac {1}{V}}{\begin{bmatrix}0\\0\\ab\sin {\gamma }\end{bmatrix}}\\\end{alignedat}}$

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