20.2: Reciprocal Space

The animation below shows the relationship between the real lattice and the reciprocal lattice. Note that this 2D representation omits the c* vector, but that it follows the same rules as a* and b*.

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The key things to note are that:

• The reciprocal lattice has reciprocal vectors a* and b*, separated by the angle γ*.
• a* is perpendicular to the (100) planes, and equal in magnitude to the inverse of d₁₀₀.
• Similarly, b* is perpendicular to the (010) planes and equal in magnitude to the inverse of d₀₁₀.
• γ and γ* will sum to 180º.

Due to the linear relationship between planes (for example, d₂₀₀ = ½ d₁₀₀ ), a periodic lattice is generated. In general, the periodicity in the reciprocal lattice is given by

$$\rho_{h k l}^{*}=\frac{1}{d_{h k l}}$$ 1dhkl

In vector form, the general reciprocal lattice vector for the (h k l) plane is given by

$$s_{h k l}=\frac{\mathrm{n}_{h k l}}{d_{h k l}}$$

where n_hkl is the unit vector normal to the (h k l) planes.

This concept can be applied to crystals, to generate a reciprocal lattice of the crystal lattice. The units in reciprocal space are Å^-1 or nm^-1