.. _sphx_glr_api_gallery_crystal_crystal_systems:

Crystal systems
-----------------

Classifying a unit cell into one of the 7 crystal systems from its
lattice parameters and the general unit-cell-volume formula; Steno's
constant interfacial angles; Miller indices from Hauy's rational face
intercepts; and the 14 Bravais lattices.


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    <div class="sphx-glr-thumbcontainer" tooltip="Christian Samuel Weiss (1815) sorted crystals into systems by the lengths of, and angles between, their crystallographic axes -- the idea that survives as today&#x27;s 7 crystal systems. classify_crystal_system classifies a unit cell purely from the equalities/inequalities among its six lattice parameters (a,b,c,\alpha,\beta,\gamma), and unit_cell_volume gives the general-cell volume formula that reduces to familiar shortcuts (abc, \frac{\sqrt3}{2}a^2c) in special cases.">

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  .. image:: /api/gallery/crystal/crystal_systems/images/thumb/sphx_glr_plot_01_crystal_systems_thumb.png
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  :doc:`/api/gallery/crystal/crystal_systems/plot_01_crystal_systems`

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      <div class="sphx-glr-thumbnail-title">Weiss's crystal systems: classifying a unit cell by its axes</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Nicolas Steno (1669) found that quartz crystals, however unevenly grown, always meet at the same angles between corresponding faces. The angle between two faces depends only on their Miller indices -- not on how big the faces are -- which interplanar_angle_cubic computes for a cubic crystal from the face normals.">

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  .. image:: /api/gallery/crystal/crystal_systems/images/thumb/sphx_glr_plot_02_steno_interfacial_angles_thumb.png
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  :doc:`/api/gallery/crystal/crystal_systems/plot_02_steno_interfacial_angles`

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      <div class="sphx-glr-thumbnail-title">Steno's law: constancy of interfacial angles</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Hauy (1784-1801) argued that every natural crystal face cuts the three axes at intercepts in small whole-number ratios. miller_indices_from_intercepts turns those rational intercepts into the integer Miller indices (hkl) by taking reciprocals and clearing fractions; the (hkl) triple then fixes each face&#x27;s interplanar spacing.">

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  .. image:: /api/gallery/crystal/crystal_systems/images/thumb/sphx_glr_plot_03_hauy_rational_indices_thumb.png
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  :doc:`/api/gallery/crystal/crystal_systems/plot_03_hauy_rational_indices`

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      <div class="sphx-glr-thumbnail-title">Hauy's law of rational indices: Miller indices from face intercepts</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Bravais (1848) showed there are exactly 14 distinct three-dimensional lattices: the 7 crystal systems combined with the centerings each allows (~chemistrykit.crystal.systems.crystal_systems.BRAVAIS_LATTICES). The cubic system has three -- primitive (P), body-centered (I), and face-centered (F) -- drawn here with cubic_lattice_points.">

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  .. image:: /api/gallery/crystal/crystal_systems/images/thumb/sphx_glr_plot_04_bravais_cubic_lattices_thumb.png
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  :doc:`/api/gallery/crystal/crystal_systems/plot_04_bravais_cubic_lattices`

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      <div class="sphx-glr-thumbnail-title">Bravais lattices: the 14 lattices and the three cubic centerings</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/crystal/crystal_systems/plot_01_crystal_systems
   /api/gallery/crystal/crystal_systems/plot_02_steno_interfacial_angles
   /api/gallery/crystal/crystal_systems/plot_03_hauy_rational_indices
   /api/gallery/crystal/crystal_systems/plot_04_bravais_cubic_lattices