:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.crystal``: the 7 crystal systems and general unit-cell
volume; hard-sphere packing (packing fraction, coordination number) for
the SC/BCC/FCC/HCP lattices; ionic-crystal lattice energy via the
Born-Lande and Kapustinskii equations, backed by a genuinely converging
(Evjen-method) numerical Madelung constant; Bragg's law and powder-XRD
peak positions with structure factors and systematic absences, Laue
interference, and Scherrer crystallite sizes; Schottky/Frenkel
point-defect equilibrium; Miller indices, interfacial angles, and the 14
Bravais lattices; and Pauling's radius-ratio rule and Goldschmidt's
perovskite tolerance factor.

Each script in this gallery is self-contained and can be run directly
with ``python examples/crystal/<section>/<script>.py``. Every script
also carries an RST module docstring as its title/description and uses
``# %%`` markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source
of truth for what you see, not a copy of it.

Sections
--------

- **crystal_systems** -- classifying a unit cell into one of the 7
  crystal systems, the general lattice-parameter volume formula, Steno's
  interfacial angles, Miller indices from face intercepts, and the 14
  Bravais lattices.
- **packing** -- atomic packing factor, coordination number, and atoms
  per cell for SC, BCC, FCC, and ideal HCP.
- **lattice_energy** -- the Born-Lande and Kapustinskii equations, and
  Shannon's effective ionic radii.
- **madelung** -- Madelung's lattice sum shell by shell, and the NaCl
  Madelung constant from a genuinely converging
  (Evjen-method) lattice summation, contrasted with a naive truncated sum
  that does not converge.
- **xrd** -- Bragg's law, cubic d-spacings, and simulated powder-XRD
  patterns with systematic absences for SC/BCC/FCC, Laue interference,
  and Scherrer crystallite sizes.
- **defects** -- Frenkel and Schottky point-defect concentration vs.
  temperature.
- **crystal_chemistry** -- Pauling's radius-ratio rule and Goldschmidt's
  perovskite tolerance factor.


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Crystal chemistry
-------------------

Rules of thumb from ionic radii: Pauling's radius-ratio rule for
coordination numbers and Goldschmidt's tolerance factor for perovskites.


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    <div class="sphx-glr-thumbcontainer" tooltip="Pauling&#x27;s first rule (1929): each cation is surrounded by a polyhedron of anions whose size is fixed by the radius ratio r_+/r_-. A polyhedron is stable only while the anions still touch the cation, which sets the limits \sqrt2-1=0.414 (octahedral), \sqrt3-1=0.732 (cubic), and so on. radius_ratio_coordination applies the rule; here it is tested against observed structures.">

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  .. image:: /api/gallery/crystal/crystal_chemistry/images/thumb/sphx_glr_plot_01_pauling_radius_ratio_thumb.png
    :alt:

  :doc:`/api/gallery/crystal/crystal_chemistry/plot_01_pauling_radius_ratio`

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      <div class="sphx-glr-thumbnail-title">Pauling's radius-ratio rule: predicting coordination numbers</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Goldschmidt (1926) noted that in an ideal cubic ABX_3 perovskite the A-X distance equals \sqrt2 times the B-X distance, so hard spheres fit exactly when t=(r_A+r_X)/[\sqrt2(r_B+r_X)]=1 (~chemistrykit.crystal.systems.crystal_chemistry.goldschmidt_tolerance_factor). Titanates with A-cations of different sizes span the range from tilted to cubic to ferroelectrically distorted.">

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  .. image:: /api/gallery/crystal/crystal_chemistry/images/thumb/sphx_glr_plot_02_goldschmidt_tolerance_factor_thumb.png
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  :doc:`/api/gallery/crystal/crystal_chemistry/plot_02_goldschmidt_tolerance_factor`

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      <div class="sphx-glr-thumbnail-title">Goldschmidt's tolerance factor: which perovskites are cubic</div>
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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>
    </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>
    </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
    :alt:

  :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>
    </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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Point defects
---------------

Frenkel and Schottky point-defect concentrations vs. temperature, from
Boltzmann-factor equilibrium.


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    <div class="sphx-glr-thumbcontainer" tooltip="Frenkel (1926) argued that at any T&gt;0 some ions leave their sites for interstitial gaps, because the configurational entropy gained outweighs the enthalpy cost at low concentration. The equilibrium number is n_F=\sqrt{NN_i}\exp(-\Delta H_F/2k_BT) (~chemistrykit.crystal.systems.defects.frenkel_defect_concentration). AgCl, where small Ag+ ions readily occupy interstitial sites, is the textbook Frenkel-dominated solid (\Delta H_F\approx1.4 eV).">

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  .. image:: /api/gallery/crystal/defects/images/thumb/sphx_glr_plot_01_frenkel_defects_thumb.png
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  :doc:`/api/gallery/crystal/defects/plot_01_frenkel_defects`

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      <div class="sphx-glr-thumbnail-title">Frenkel defects: vacancy-interstitial pairs in silver chloride</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Wagner and Schottky (1930) identified the second point-defect mechanism of ionic crystals: a cation and an anion leave the bulk together, keeping the crystal neutral and leaving a vacancy pair behind. The equilibrium number is n_S=N\exp(-\Delta H_S/2k_BT) (~chemistrykit.crystal.systems.defects.schottky_defect_concentration). In NaCl, the classic Schottky-dominated solid, \Delta H_S\approx2.3 eV.">

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  .. image:: /api/gallery/crystal/defects/images/thumb/sphx_glr_plot_02_schottky_defects_thumb.png
    :alt:

  :doc:`/api/gallery/crystal/defects/plot_02_schottky_defects`

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      <div class="sphx-glr-thumbnail-title">Schottky defects: paired cation and anion vacancies in rock salt</div>
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Lattice energy
----------------

The Born-Lande and Kapustinskii equations for ionic-crystal lattice
energy, and the Shannon ionic radii the Kapustinskii equation uses.


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    <div class="sphx-glr-thumbcontainer" tooltip="Born and Lande (1918) balanced the Madelung attraction -A/r against a short-range repulsion B/r^n; setting dU/dr=0 at the observed spacing r_0 gives U=-\frac{N_AM|z_+z_-|e^2}{4\pi\varepsilon_0r_0}(1-\frac{1}{n}), which BornLandeLatticeEnergy evaluates. This example shows the energy curve for NaCl and compares the equation with Born-Haber values for six rock-salt halides.">

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  .. image:: /api/gallery/crystal/lattice_energy/images/thumb/sphx_glr_plot_01_born_lande_thumb.png
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  :doc:`/api/gallery/crystal/lattice_energy/plot_01_born_lande`

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      <div class="sphx-glr-thumbnail-title">Born-Lande equation: lattice energies of the alkali halides</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Kapustinskii (1956) replaced the structure-specific Madelung constant with a near-universal constant per ion, so KapustinskiiLatticeEnergy needs only the number of ions per formula unit, the charges, and the ionic radii. On rock-salt halides it tracks the structure-based Born-Lande value; it also handles salts of other structures (fluorite CaF2) where no Madelung constant is supplied at all.">

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  .. image:: /api/gallery/crystal/lattice_energy/images/thumb/sphx_glr_plot_02_kapustinskii_thumb.png
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  :doc:`/api/gallery/crystal/lattice_energy/plot_02_kapustinskii`

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      <div class="sphx-glr-thumbnail-title">Kapustinskii equation: lattice energy without a crystal structure</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Shannon (1976) fitted one self-consistent radius per ion (for a given coordination number) so that cation-anion distances in thousands of crystals are reproduced by simple sums r_++r_-. SHANNON_IONIC_RADII_PM holds the 6-coordinate values; here their sums are checked against measured rock-salt nearest-neighbor distances, and their periodic trends are plotted.">

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  .. image:: /api/gallery/crystal/lattice_energy/images/thumb/sphx_glr_plot_03_shannon_ionic_radii_thumb.png
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  :doc:`/api/gallery/crystal/lattice_energy/plot_03_shannon_ionic_radii`

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      <div class="sphx-glr-thumbnail-title">Shannon's effective ionic radii: additivity and periodic trends</div>
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Madelung constant
-------------------

Madelung's electrostatic lattice sum for NaCl, shell by shell, and
Evjen's genuinely converging summation of it, contrasted with a naive
truncated sum that does not converge.


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    <div class="sphx-glr-thumbcontainer" tooltip="The NaCl Madelung-constant lattice sum is only conditionally convergent: a naive truncated sum over a growing cube of ions does not settle down as the cutoff grows, it oscillates. madelung_constant_nacl uses Evjen&#x27;s method (fractional boundary-charge weighting) to build a genuinely converging summation instead, and this example demonstrates both behaviors side by side against the literature value.">

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  .. image:: /api/gallery/crystal/madelung/images/thumb/sphx_glr_plot_01_madelung_convergence_thumb.png
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  :doc:`/api/gallery/crystal/madelung/plot_01_madelung_convergence`

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      <div class="sphx-glr-thumbnail-title">Evjen's method: converging the NaCl Madelung constant</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Madelung (1918) asked for the electrostatic energy of one ion in the field of every other ion of an infinite crystal. For rock salt it is U=-Me^2/(4\pi\varepsilon_0r_0) per ion pair, with the Madelung constant M=\sum_j \pm r_0/r_j summed shell by shell: 6 opposite charges at r_0, 12 like charges at \sqrt2r_0, 8 opposite charges at \sqrt3r_0, and so on. madelung_constant_nacl returns the converged value.">

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  .. image:: /api/gallery/crystal/madelung/images/thumb/sphx_glr_plot_02_madelung_lattice_sum_thumb.png
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  :doc:`/api/gallery/crystal/madelung/plot_02_madelung_lattice_sum`

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      <div class="sphx-glr-thumbnail-title">Madelung's lattice sum: the electrostatic energy of rock salt</div>
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Hard-sphere packing
---------------------

Atomic packing factor, coordination number, and atoms per cell for the
simple cubic, body-centered cubic, face-centered cubic, and ideal
hexagonal close-packed lattices.


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    <div class="sphx-glr-thumbcontainer" tooltip="Kepler (1611) guessed that no stacking of equal spheres fills space more densely than the layered (face-centered cubic / hexagonal close) packing, with fraction \pi/\sqrt{18}\approx0.7405. Comparing the four textbook lattices shows the close-packed pair at exactly that bound.">

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  .. image:: /api/gallery/crystal/packing/images/thumb/sphx_glr_plot_01_packing_fractions_thumb.png
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  :doc:`/api/gallery/crystal/packing/plot_01_packing_fractions`

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      <div class="sphx-glr-thumbnail-title">Kepler's conjecture: packing efficiency of SC, BCC, FCC, and HCP</div>
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Powder XRD
------------

Laue interference from a periodic array, Bragg's law and simulated
powder-XRD patterns with structure factors and systematic absences for
SC, BCC, and FCC, and Scherrer crystallite sizes from line broadening.


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    <div class="sphx-glr-thumbcontainer" tooltip="Each line of a diffraction pattern is a family of (hkl) planes that satisfies Bragg&#x27;s law n\lambda=2d\sin\theta. powder_xrd_peaks computes each reflection&#x27;s structure factor (~chemistrykit.crystal.systems.xrd.structure_factor) directly from atomic positions and Miller indices, so BCC&#x27;s and FCC&#x27;s systematic absences (reflections with zero intensity by symmetry) fall out automatically rather than being hardcoded as a rule.">

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  .. image:: /api/gallery/crystal/xrd/images/thumb/sphx_glr_plot_01_powder_xrd_thumb.png
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  :doc:`/api/gallery/crystal/xrd/plot_01_powder_xrd`

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      <div class="sphx-glr-thumbnail-title">Bragg's law: indexing powder XRD peaks</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Von Laue (1912) predicted that a crystal, as a periodic array of scatterers, should diffract X-rays like a grating, and Friedrich and Knipping saw the discrete spots. Summing the waves from N periodically spaced scatterers with structure_factor gives the Laue interference function: intensity N^2 at integer h (the Laue condition) and almost nothing in between, ever sharper as N grows.">

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  .. image:: /api/gallery/crystal/xrd/images/thumb/sphx_glr_plot_02_laue_interference_thumb.png
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  :doc:`/api/gallery/crystal/xrd/plot_02_laue_interference`

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      <div class="sphx-glr-thumbnail-title">Laue's interference: sharp diffraction spots from a periodic array</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Scherrer (1918) showed that tiny crystallites give broadened powder lines, with width \beta=K\lambda/(\tau\cos\theta) for crystallite size \tau. scherrer_crystallite_size inverts this: here, copper powder patterns are broadened for three crystallite sizes, and each size is recovered from the measured width of the (111) line.">

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  .. image:: /api/gallery/crystal/xrd/images/thumb/sphx_glr_plot_03_scherrer_crystallite_size_thumb.png
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  :doc:`/api/gallery/crystal/xrd/plot_03_scherrer_crystallite_size`

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      <div class="sphx-glr-thumbnail-title">Scherrer equation: crystallite size from peak broadening</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/crystal/crystal_chemistry/index.rst
   /api/gallery/crystal/crystal_systems/index.rst
   /api/gallery/crystal/defects/index.rst
   /api/gallery/crystal/lattice_energy/index.rst
   /api/gallery/crystal/madelung/index.rst
   /api/gallery/crystal/packing/index.rst
   /api/gallery/crystal/xrd/index.rst



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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_