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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/scattering/plot_born_approximation_rutherford.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

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

.. _sphx_glr_api_gallery_quantum_scattering_plot_born_approximation_rutherford.py:


The Born approximation: Rutherford scattering from quantum mechanics
====================================================================

Max Born's 1926 collision theory gives the scattering amplitude, to first
order in the potential, as the Fourier transform of :math:`V(r)` at the
momentum transfer :math:`q = 2k\sin(\theta/2)`. For a screened Coulomb
(Yukawa) potential :math:`V = g\,e^{-\mu r}/r` it is
:math:`f_B = -2mg/(\mu^2 + q^2)`. Removing the screening gives
Rutherford's classical 1911 cross section exactly. Here the numerical
:func:`~physicskit.quantum.chapters.scattering.born_amplitude` is compared
with the closed form and with
:func:`~physicskit.particle.scattering.rutherford_dsigma_domega`.

.. GENERATED FROM PYTHON SOURCE LINES 15-27

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.particle.scattering import rutherford_dsigma_domega
    from physicskit.quantum.chapters.scattering import born_amplitude, momentum_transfer, yukawa_born_amplitude

    g, mass, k = 1.0, 1.0, 2.0  # coupling Z1 Z2 alpha, reduced mass, wavenumber
    theta = np.linspace(0.05, np.pi, 300)
    q = momentum_transfer(k, theta)
    E_kin = k**2 / (2 * mass)








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Screening length controls the forward peak
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-44

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(9, 5))
    for mu in (2.0, 0.5, 0.1):
        ax.semilogy(np.degrees(theta), yukawa_born_amplitude(q, g, mu, mass) ** 2, label=rf"Yukawa, $\mu = {mu}$")
    theta_num = np.linspace(0.1, np.pi, 12)
    f_num = born_amplitude(lambda r: g * np.exp(-0.5 * r) / r, momentum_transfer(k, theta_num), mass)
    ax.semilogy(np.degrees(theta_num), f_num**2, "o", color="C1", label=r"numerical Born integral, $\mu = 0.5$")
    ax.semilogy(np.degrees(theta), rutherford_dsigma_domega(theta, 1, 1, E_kin, alpha=g), "k--", label="Rutherford (1911)")
    ax.set_xlabel(r"scattering angle $\theta$ (degrees)")
    ax.set_ylabel(r"$d\sigma/d\Omega = |f_B|^2$")
    ax.set_title(r"Born approximation for $V = g\,e^{-\mu r}/r$: $\mu \to 0$ gives Rutherford")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/scattering/images/sphx_glr_plot_born_approximation_rutherford_001.png
   :alt: Born approximation for $V = g\,e^{-\mu r}/r$: $\mu \to 0$ gives Rutherford
   :srcset: /api/gallery/quantum/scattering/images/sphx_glr_plot_born_approximation_rutherford_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 45-51

The Born amplitude is the Fourier transform of the potential
-------------------------------------------------------------

Different potentials with the same strength scatter differently because
their Fourier transforms differ: a Gaussian well cuts off large momentum
transfers exponentially, a Yukawa potential only as a power law.

.. GENERATED FROM PYTHON SOURCE LINES 51-67

.. code-block:: Python


    q_grid = np.linspace(0, 6, 60)
    potentials = {
        r"Yukawa $e^{-r}/r$": lambda r: -np.exp(-r) / r,
        r"Gaussian $e^{-r^2}$": lambda r: -np.exp(-(r**2)),
        r"exponential $e^{-r}$": lambda r: -np.exp(-r),
    }
    fig2, ax2 = plt.subplots(figsize=(8, 4.5))
    for name, V in potentials.items():
        ax2.semilogy(q_grid, np.abs(born_amplitude(V, q_grid)), label=name)
    ax2.set_xlabel("momentum transfer q")
    ax2.set_ylabel(r"$|f_B(q)|$")
    ax2.set_title("Born amplitudes of three attractive potentials")
    ax2.legend()
    fig2.tight_layout()




.. image-sg:: /api/gallery/quantum/scattering/images/sphx_glr_plot_born_approximation_rutherford_002.png
   :alt: Born amplitudes of three attractive potentials
   :srcset: /api/gallery/quantum/scattering/images/sphx_glr_plot_born_approximation_rutherford_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 68-72

Check
-----
The Yukawa Born amplitude is -2 m g / (q^2 + mu^2); the numerical Born
integral reproduces it, and as mu -> 0 |f|^2 becomes Rutherford's formula.

.. GENERATED FROM PYTHON SOURCE LINES 72-74

.. code-block:: Python

    np.testing.assert_allclose(f_num, yukawa_born_amplitude(momentum_transfer(k, theta_num), g, 0.5, mass), rtol=1e-4)
    np.testing.assert_allclose(yukawa_born_amplitude(q, g, 1e-8, mass) ** 2, rutherford_dsigma_domega(theta, 1, 1, E_kin, alpha=g), rtol=1e-8)








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