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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/quantum_statistics/plot_01_thermal_de_broglie_wavelength.py"
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.. only:: html

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

.. _sphx_glr_api_gallery_statmech_quantum_statistics_plot_01_thermal_de_broglie_wavelength.py:


De Broglie's matter waves: the thermal wavelength of a gas
=============================================================

De Broglie's relation :math:`\lambda=h/p`, averaged over a gas's thermal
spread of momenta, gives the thermal de Broglie wavelength

.. math::

    \Lambda=\frac{h}{\sqrt{2\pi mk_BT}},

computed by :func:`~chemistrykit.statmech.thermal_de_broglie_wavelength`.
Translational motion can be treated classically when :math:`\Lambda` is
much smaller than the mean spacing between molecules,
:math:`(V/N)^{1/3}=(k_BT/P)^{1/3}`. The plot compares the two for several
gases at 1 bar: only the lightest particles at the lowest temperatures
come close.

.. GENERATED FROM PYTHON SOURCE LINES 21-50

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.constants as sc

    from chemistrykit.constants import K_B
    from chemistrykit.statmech import thermal_de_broglie_wavelength

    species = {
        "electron": (sc.m_e, "black"),
        "He": (4.0026 * sc.atomic_mass, "crimson"),
        "H2": (2.016 * sc.atomic_mass, "darkorange"),
        "N2": (28.014 * sc.atomic_mass, "seagreen"),
        "Ar": (39.948 * sc.atomic_mass, "steelblue"),
    }

    T = np.logspace(0.0, 3.5, 300)
    P = 1.0e5
    spacing = (K_B * T / P) ** (1.0 / 3.0)

    fig, ax = plt.subplots(figsize=(7, 5))
    for name, (mass, color) in species.items():
        ax.loglog(T, thermal_de_broglie_wavelength(mass, T), color=color, label=rf"$\Lambda$ ({name})")
    ax.loglog(T, spacing, color="gray", linestyle="--", linewidth=2, label=r"mean spacing $(k_BT/P)^{1/3}$ at 1 bar")
    ax.set_xlabel("T (K)")
    ax.set_ylabel("length (m)")
    ax.set_title("Thermal de Broglie wavelength vs. intermolecular spacing")
    ax.legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/statmech/quantum_statistics/images/sphx_glr_plot_01_thermal_de_broglie_wavelength_001.png
   :alt: Thermal de Broglie wavelength vs. intermolecular spacing
   :srcset: /api/gallery/statmech/quantum_statistics/images/sphx_glr_plot_01_thermal_de_broglie_wavelength_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 51-56

At room temperature Lambda is a small fraction of an angstrom for any
molecule, some 100-1000 times smaller than the ~34 angstrom spacing of a
gas at 1 bar -- the quantitative reason classical translational
statistics work. An electron's wavelength, by contrast, is comparable to
that spacing, and much larger than the spacing in a metal.

.. GENERATED FROM PYTHON SOURCE LINES 56-64

.. code-block:: Python


    T_room = 298.15
    d = (K_B * T_room / P) ** (1.0 / 3.0)
    for name, (mass, _) in species.items():
        lam = thermal_de_broglie_wavelength(mass, T_room)
        print(f"{name:8s}: Lambda = {lam * 1e10:8.3f} angstrom, spacing / Lambda = {d / lam:8.1f}")

    plt.show()




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

 .. code-block:: none

    electron: Lambda =   43.168 angstrom, spacing / Lambda =      0.8
    He      : Lambda =    0.505 angstrom, spacing / Lambda =     68.3
    H2      : Lambda =    0.712 angstrom, spacing / Lambda =     48.5
    N2      : Lambda =    0.191 angstrom, spacing / Lambda =    180.7
    Ar      : Lambda =    0.160 angstrom, spacing / Lambda =    215.8





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   **Total running time of the script:** (0 minutes 0.092 seconds)


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