Resolution: Difference between revisions
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The resolution of a reflection <math>(hkl)</math> is defined as the inverse of the reciprocal lattice vector, i.e. <math>\frac{1}{r^2} = \mathbf{d}^{*} \cdot \mathbf{d}^{*}</math> with <math> \mathbf{d}^{*} = h \mathbf{a}^{*} + k \mathbf{b}^{*} + l \mathbf{c}^{*}</math>. | The resolution of a reflection <math>(hkl)</math> is defined as the inverse of the reciprocal lattice vector, i.e. <math>\frac{1}{r^2} = \mathbf{d}^{*} \cdot \mathbf{d}^{*}</math> with <math> \mathbf{d}^{*} = h \mathbf{a}^{*} + k \mathbf{b}^{*} + l \mathbf{c}^{*}</math>. | ||
The formula to calculate the resolution from the unit cell dimensions <math>a, b, c, \alpha, \beta, \gamma</math> looks a little | The formula to calculate the resolution from the unit cell dimensions <math>a, b, c, \alpha, \beta, \gamma</math> looks a little appalling: | ||
<math>\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}</math>. | <math>\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}</math>. | ||
Latest revision as of 08:56, 7 December 2022
The resolution of a reflection [math]\displaystyle{ (hkl) }[/math] is defined as the inverse of the reciprocal lattice vector, i.e. [math]\displaystyle{ \frac{1}{r^2} = \mathbf{d}^{*} \cdot \mathbf{d}^{*} }[/math] with [math]\displaystyle{ \mathbf{d}^{*} = h \mathbf{a}^{*} + k \mathbf{b}^{*} + l \mathbf{c}^{*} }[/math].
The formula to calculate the resolution from the unit cell dimensions [math]\displaystyle{ a, b, c, \alpha, \beta, \gamma }[/math] looks a little appalling:
[math]\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} }[/math].
When the spacegroup is orthorombic, however, this formula simplifies to [math]\displaystyle{ \frac{1}{r^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2} }[/math].