# Resolution

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The resolution of a reflection $\displaystyle{ (hkl) }$ is defined as the inverse of the reciprocal lattice vector, i.e. $\displaystyle{ \frac{1}{r^2} = \mathbf{d}^{*} \cdot \mathbf{d}^{*} }$ with $\displaystyle{ \mathbf{d}^{*} = h \mathbf{a}^{*} + k \mathbf{b}^{*} + l \mathbf{c}^{*} }$.
The formula to calculate the resolution from the unit cell dimensions $\displaystyle{ a, b, c, \alpha, \beta, \gamma }$ looks a little appaling:
$\displaystyle{ \frac{1}{r^2} \frac{1}{\sin^2\beta - \sin^2\alpha} \left( \frac{l}{c} - \frac{k\cos \alpha}{b} - \frac{h\cos \beta}{a}\right)^2 + \left( \frac{ak-bh\cos\gamma}{ab\sin\gamma}\right)^2 + \frac{h^2}{a^2} }$.
When the spacegroup is orthorombic, however, this formula simplifies to $\displaystyle{ \frac{1}{r^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2} }$.