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Engineering LibreTexts

20.2: Reciprocal Space

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  • 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*.


    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 nhkl 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

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