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13.4: Deriving the Thomas-Fermi Equation

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    19023
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    As the electron around an atom moves, energy is converted between energy of the Coulomb interaction and kinetic energy of the electron. The action is

    \[\mathbb{S} = \left|\int\limits_{r_1}^{r_2} \mathcal{L}dr\right|. \nonumber \]

    The path found in nature minimizes the action.

    \[\delta = \left|\int\limits_{r_1}^{r_2} \mathcal{L}dr\right| = 0 \nonumber \]

    The integral is over position, not time. In Chapter 11, we called this idea the Principle of Least Action. Reference [136, p. 52] calls this idea in this context the Second Hohenberg-Kohn Theorem. To find the path, we set up and solve the Euler-Lagrange equation. The Euler-Lagrange equation in the case where the independent variable is a vector of the form \(\overrightarrow{r} = r \hat{a}_r\) instead of a scalar (with no \(\theta\) or \(\phi\) dependence anywhere) is given by

    \[\frac{\partial \mathcal{L}}{\partial (\text{path})} - \overrightarrow{\nabla} \cdot \left( \frac{\partial \mathcal{L}}{\partial \left(\frac{d(\text{path})}{dr}\right)}\right) \hat{a}_r = 0 \label{13.4.3} \]

    As described above, generalized path is voltage \(V = V (r)\), and generalized potential is charge density \(\rho_{ch} = \rho_{ch}(r)\). As discussed in Chapter 12, we could have made the opposite choice. In fact, the opposite choice may seem more logical because the words voltage and potential are often used synonymously. The same result is obtained regardless of the choice. However, the algebra is less messy with this choice, and this choice is more consistent with the literature.

    Next, evaluate the Euler-Lagrange equation, Equation \ref{13.4.3}, using the Lagrangian of Equation 13.3.51. The resulting equation is the equation of motion. Consider some of the pieces needed. The derivative of the Lagrangian with respect to the path is

    \[\frac{\partial \mathcal{L}}{\partial V} = \frac{5}{2}c_0V^{3/2}. \nonumber \]

    In Chapter 11, this quantity was defined as the generalized potential. Above, we defined \(\rho_{ch}\) as the generalized potential. Both \(\frac{\partial \mathcal{L}}{\partial V}\) and \(\rho_{ch}\) have units \(\frac{C}{m^3}\). According to Equation 13.3.44, \(\frac{\partial \mathcal{L}}{\partial V}\) is \(\rho_{ch}\) multiplied by a constant, and that constant is close to one. Since \(\frac{\partial \mathcal{L}}{\partial V}\) isn't equal to \(\rho_{ch}\), our equations are not completely consistent. However, the difference is small given the extreme assumptions made elsewhere. We also need the generalized momentum.

    \[\frac{\partial \mathcal{L}}{\partial \left(\frac{dV}{dr}\right)} = \epsilon \frac{dV}{dr}. \nonumber \]

    \[\frac{\partial \mathcal{L}}{\partial \left(\frac{dV}{dr}\right)} \hat{a}_r = \epsilon \overrightarrow{\nabla}V \nonumber \]

    Next, put these pieces into the Euler-Lagrange equation.

    \[\frac{5}{2}c_0V^{3/2} - \overrightarrow{\nabla} \cdot \left( \epsilon \overrightarrow{\nabla}V \right) = 0 \nonumber \]

    Use Equation 13.2.6.

    \[\frac{5}{2}c_0V^{3/2} - \epsilon \nabla^2 V = 0 \nonumber \]

    \[ \nabla^2 V = \frac{5}{2\epsilon}c_0V^{3/2} \label{13.4.9} \]

    Next, following Fermi's original work [177], change variables

    \[V = \frac{-y}{r} \label{13.4.10} \]

    where \(y\) has the units \(V \cdot m\). The Laplacian term on the left can be simplified using Equation 13.2.5.

    \[ \nabla^2 V =\nabla^2 \left(\frac{-y}{r}\right) \nonumber \]

    \[ \nabla^2 V = \frac{1}{r^2} \frac{\partial}{\partial r} \left[r^2 \frac{\partial}{\partial r} \left(\frac{-y}{r}\right)\right] \nonumber \]

    \[ \nabla^2 V = \frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \left( \frac{y}{r^2} - \frac{1}{r}\frac{\partial y}{\partial r} \right)\right] \nonumber \]

    \[ \nabla^2 V = \frac{1}{r^2} \frac{\partial}{\partial r} \left( y - r\frac{\partial y}{\partial r} \right) \nonumber \]

    \[ \nabla^2 V = \frac{1}{r^2} \left( \frac{\partial y}{\partial r} - \frac{\partial y}{\partial r} - r^2\frac{\partial^2 y}{\partial r^2}\right) \nonumber \]

    \[ \nabla^2 V = -\frac{1}{r^2} \frac{\partial^2 y}{\partial r^2} \nonumber \]

    Equation \ref{13.4.9} now simplifies.

    \[-\frac{1}{r} \frac{\partial^2 y}{\partial r^2} = \frac{-5}{2\epsilon}c_0\left(\frac{-y}{r}\right)^{3/2} \nonumber \]

    \[\frac{-1}{r}\frac{d^2y}{dr^2} = \frac{-5}{2\epsilon}c_0 (-1)^{1/2}\left(\frac{y}{r}\right)^{3/2} \nonumber \]

    \[\frac{d^2y}{dr^2} = c_1 r^{-1/2}y^{3/2} \label{13.4.19} \]

    In the equation above, the constant is

    \[c_1 = -\frac{5}{2\epsilon}c_0 (-1)^{1/2}. \nonumber \]

    \[c_1 = \frac{-5}{2\epsilon}\left[\left(\frac{-5mq}{3\hbar^2}\right)^{3/2}\left(\frac{-q}{3\pi^2}\right)\right](-1)^{1/2} \nonumber \]

    \[c_1 = \frac{5}{2\epsilon}\left[\left(\frac{5mq}{3\hbar^2}\right)^{3/2}\frac{q}{3\pi^2}\right] \nonumber \]

    To clean Equation \ref{13.4.19} up further, choose

    \[\mathrm{t} = c_1^{-2/3}r. \nonumber \]

    The variable t here is the name of the independent variable, and it does not represent time. It is a scaled version of the position \(r\).

    \[\frac{d^2y}{d\mathrm{t}^2} = \mathrm{t}^{-1/2}y^{3/2} \label{13.4.24} \]

    Equation Equation \ref{13.4.24} is called the Thomas-Fermi equation. We have finished the derivation. The Thomas Fermi equation along with appropriate boundary conditions can be solved for \(y(t)\). Since the equation is nonlinear, numerical techniques are likely used to solve it. Once \(y(t)\) is found, Equations 13.3.40 and Equation \ref{13.4.10} can be used to find \(V (r)\) and \(\rho_{ch}(r)\). From this equation of motion, we can find \(\rho_{ch}(r)\), where, on average, the electrons are likely to be found as a function of distance from the nucleus in spherical coordinates.


    This page titled 13.4: Deriving the Thomas-Fermi Equation is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.