.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/solids/plot_01_debye_t_cubed_law.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_statmech_solids_plot_01_debye_t_cubed_law.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_statmech_solids_plot_01_debye_t_cubed_law.py:


Debye's T-cubed law for the heat capacity of a solid
=======================================================

Debye (1912) replaced Einstein's single vibrational frequency with the
full spectrum of lattice sound waves, :math:`g(\nu)\propto\nu^2` up to a
cutoff :math:`\nu_D`. :class:`~chemistrykit.statmech.DebyeSolid` gives

.. math::

    C_V=9R\left(\frac{T}{\Theta_D}\right)^3\int_0^{\Theta_D/T}
    \frac{x^4e^x}{(e^x-1)^2}\,dx
    \;\xrightarrow{T\to0}\;\frac{12\pi^4}{5}R\left(\frac{T}{\Theta_D}\right)^3

per mole of atoms. The low-frequency acoustic modes stay thermally active
at low temperature, so :math:`C_V` falls as :math:`T^3`. An Einstein solid
with a comparable frequency falls exponentially, much faster than
measured low-temperature heat capacities do.

.. GENERATED FROM PYTHON SOURCE LINES 22-58

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import K_B, H, R
    from chemistrykit.statmech import DebyeSolid, VibrationalPartitionFunctionHarmonic

    theta_D = 343.0  # copper, K (Kittel, Introduction to Solid State Physics, Table 5.1)
    copper = DebyeSolid(debye_temperature=theta_D)

    # The Einstein temperature that best matches Debye at high T is Theta_E = sqrt(3/5) Theta_D.
    einstein_mode = VibrationalPartitionFunctionHarmonic(frequency=np.sqrt(3.0 / 5.0) * theta_D * K_B / H)

    T = np.linspace(1.0, 600.0, 400)
    Cv_debye = copper.heat_capacity_v(T)
    Cv_einstein = 3.0 * einstein_mode.heat_capacity_v(T)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].plot(T / theta_D, Cv_debye / (3 * R), color="steelblue", label="Debye")
    axes[0].plot(T / theta_D, Cv_einstein / (3 * R), color="crimson", linestyle="--", label="Einstein")
    axes[0].axhline(1.0, color="gray", linestyle=":", linewidth=0.8, label="Dulong-Petit 3R")
    axes[0].set_xlabel(r"$T / \Theta_D$")
    axes[0].set_ylabel(r"$C_V / 3R$")
    axes[0].set_title("Debye vs. Einstein solid")
    axes[0].legend()

    T_low = np.logspace(-0.5, 2.0, 200)
    axes[1].loglog(T_low, copper.heat_capacity_v(T_low), color="steelblue", label="Debye")
    axes[1].loglog(T_low, copper.low_temperature_heat_capacity(T_low), color="black", linestyle=":", label=r"$T^3$ law")
    axes[1].loglog(T_low, 3.0 * einstein_mode.heat_capacity_v(T_low), color="crimson", linestyle="--", label="Einstein")
    axes[1].set_ylim(1e-6, 30.0)
    axes[1].set_xlabel("T (K)")
    axes[1].set_ylabel(r"$C_V$ (J mol$^{-1}$ K$^{-1}$)")
    axes[1].set_title(r"Copper ($\Theta_D$ = 343 K): the $T^3$ law")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/statmech/solids/images/sphx_glr_plot_01_debye_t_cubed_law_001.png
   :alt: Debye vs. Einstein solid, Copper ($\Theta_D$ = 343 K): the $T^3$ law
   :srcset: /api/gallery/statmech/solids/images/sphx_glr_plot_01_debye_t_cubed_law_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 59-61

Below about Theta_D / 50 the full Debye result and the T^3 law agree to
better than 0.1 %:

.. GENERATED FROM PYTHON SOURCE LINES 61-67

.. code-block:: Python


    for t in (2.0, 5.0, 20.0, 100.0):
        ratio = copper.heat_capacity_v(t) / copper.low_temperature_heat_capacity(t)
        print(f"T = {t:5.1f} K: Cv = {copper.heat_capacity_v(t):.4e} J/(mol K), Cv / T^3-law = {ratio:.4f}")

    plt.show()




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    T =   2.0 K: Cv = 3.8535e-04 J/(mol K), Cv / T^3-law = 1.0000
    T =   5.0 K: Cv = 6.0211e-03 J/(mol K), Cv / T^3-law = 1.0000
    T =  20.0 K: Cv = 3.8529e-01 J/(mol K), Cv / T^3-law = 0.9998
    T = 100.0 K: Cv = 1.4766e+01 J/(mol K), Cv / T^3-law = 0.3066





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.138 seconds)


.. _sphx_glr_download_api_gallery_statmech_solids_plot_01_debye_t_cubed_law.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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