Thomas–Fermi equation

In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi,^[1]^[2] which can be derived by applying the Thomas–Fermi model to atoms. The equation reads

{\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}

subject to the boundary conditions

y(0)=1\quad ;\quad y(+\infty )=0

If $y$ approaches zero as $x$ becomes large, this equation models the charge distribution of a neutral atom as a function of radius $x$ . Solutions where $y$ becomes zero at finite $x$ model positive ions.^[3] For solutions where $y$ becomes large and positive as $x$ becomes large, it can be interpreted as a model of a compressed atom, where the charge is squeezed into a smaller space. In this case the atom ends at the value of $x$ for which $dy/dx=y/x$ .^[4]^[5]

Transformations

Introducing the transformation $z=y/x$ converts the equation to

{\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dz}{dx}}\right)-z^{3/2}=0

This equation is similar to Lane–Emden equation with polytropic index $3/2$ except the sign difference. The original equation is invariant under the transformation $x\rightarrow cx,\ y\rightarrow c^{-3}y$ . Hence, the equation can be made equidimensional by introducing $y=x^{-3}u$ into the equation, leading to

x^{2}{\frac {d^{2}u}{dx^{2}}}-6x{\frac {du}{dx}}+12u=u^{3/2}

so that the substitution $u=e^{t}$ reduces the equation to

{\frac {d^{2}u}{dt^{2}}}-7{\frac {du}{dt}}+12u=u^{3/2}.

If $w(u)={\frac {du}{dt}}$ then the above equation becomes

w{\frac {dw}{du}}-7w+12u=u^{3/2}.

But this first order equation has no known explicit solution, hence, the approach turns to either numerical or approximate methods.

Sommerfeld's approximation

The equation has a particular solution $y_{p}(x)$ , which satisfies the boundary condition that $y\rightarrow 0$ as $x\rightarrow\infty$ , but not the boundary condition y(0)=1. This particular solution is

y_{p}(x)={\frac {144}{x^{3}}}.

Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932.^[6] If the transformation $x=1/t,\ w=yt$ is introduced, the equation becomes

t^{4}{\frac {d^{2}w}{dt^{2}}}=w^{3/2},\quad w(0)=0,\ w(\infty )\sim t.

The particular solution in the transformed variable is then $w_{p}(t)=144t^{4}$ . So one assumes a solution of the form $w=w_{p}(1+\alpha t^{\lambda })$ and if this is substituted in the above equation and the coefficients of $\alpha$ are equated, one obtains the value for $\lambda$ , which is given by the roots of the equation $\lambda ^{2}+7\lambda -6=0$ . The two roots are $\lambda _{1}=0.772,\ \lambda _{2}=-7.772$ . Since this solution already satisfies the second boundary condition, to satisfy the first boundary condition one writes

W=w_{p}(1+\beta t^{\lambda })^{n}=[144t^{3}(1+\beta t^{\lambda })]t.

The first boundary condition will be satisfied if $144t^{3}(1+\beta t^{\lambda })^{n}=144t^{3}\beta ^{n}t^{\lambda n}(1+\beta ^{-1}t^{-\lambda })^{n}\sim 1$ as $t\rightarrow \infty$ . This condition is satisfied if $\lambda n+3=0,\ 144\beta ^{n}=1$ and since $\lambda _{1}\lambda _{2}=-6$ , Sommerfeld found the approximation as $\lambda =\lambda _{1},\ n=-3/\lambda _{1}=\lambda _{2}/2$ . Therefore, the approximate solution is

y(x)=y_{p}(x)\{1+[y_{p}(x)]^{\lambda _{1}/3}\}^{\lambda _{2}/2}.

This solution predicts the correct solution accurately for large $x$ , but still fails near the origin.

Solution near origin

Enrico Fermi^[7] provided the solution for $x\ll 1$ and later extended by Baker.^[8] Hence for $x\ll 1$ ,

{\begin{aligned}y(x)={}&1-Bx+{\frac {1}{3}}x^{3}-{\frac {2B}{15}}x^{4}+\cdots {}\\[6pt]&\cdots +x^{3/2}\left[{\frac {4}{3}}-{\frac {2B}{5}}x+{\frac {3B^{2}}{70}}x^{2}+\left({\frac {2}{27}}+{\frac {B^{3}}{252}}\right)x^{3}+\cdots \right]\end{aligned}}

where $B\approx 1.588071$ .^[9]^[10]

Approach by Majorana

It has been reported by Esposito^[11] that the Italian physicist Ettore Majorana found in 1928 a semi-analytical series solution to the Thomas-Fermi equation for the neutral atom, which however remained unpublished until 2001.

Using this approach it is possible to compute the constant B mentioned above to practically arbitrarily high accuracy; for example, its value to 100 digits is $B=1.588071022611375312718684509423950109452746621674825616765677418166551961154309262332033970138428665$ .

References

↑ Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.
↑ Bender, Carl M., and Steven A. Orszag. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 2013.
↑ pp. 9-12, N. H. March (1983). "1. Origins – The Thomas–Fermi Theory". In S. Lundqvist and N. H. March. Theory of The Inhomogeneous Electron Gas. Plenum Press. ISBN 978-0-306-41207-3.
↑ March 1983, p. 10, Figure 1.
↑ p. 1562, R. P. Feynman, N. Metropolis, and E. Teller. "Equations of State of Elements Based on the Generalized Thomas-Fermi Theory". Physical Review 75, #10 (May 15, 1949), pp. 1561-1573.
↑ Sommerfeld, A. "Integrazione asintotica dell’equazione differenziale di Thomas–Fermi." Rend. R. Accademia dei Lincei 15 (1932): 293.
↑ Fermi, Enrico. "Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente." Zeitschrift für Physik A Hadrons and Nuclei 48.1 (1928): 73–79.
↑ Baker, Edward B. "The application of the Fermi–Thomas statistical model to the calculation of potential distribution in positive ions." Physical Review 36.4 (1930): 630.
↑ Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111], Francisco M.Fernández, Physics Letters A 372, 28 July 2008, 5258-5260, doi:10.1016/j.physleta.2008.05.071.
↑ The analytical solution of the Thomas-Fermi equation for a neutral atom, G I Plindov and S K Pogrebnya, Journal of Physics B: Atomic and Molecular Physics 20 (1987), L547, doi:10.1088/0022-3700/20/17/001.
↑ Esposito, Salvatore (2002). "Majorana solution of the Thomas-Fermi equation". American Journal of Physics. 70: 852.

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[1] Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.

[2] Bender, Carl M., and Steven A. Orszag. Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory. Springer Science & Business Media, 2013.

[3] . 9-12, N. H. March (1983). "1. Origins – The Thomas–Fermi Theory". In S. Lundqvist and N. H. March. Theory of The Inhomogeneous Electron Gas. Plenum Press. ISBN 978-0-306-41207-3.

[4] March 1983, p. 10, Figure 1.

[5] . 1562, R. P. Feynman, N. Metropolis, and E. Teller. "Equations of State of Elements Based on the Generalized Thomas-Fermi Theory". Physical Review 75, #10 (May 15, 1949), pp. 1561-1573.

[6] Sommerfeld, A. "Integrazione asintotica dell’equazione differenziale di Thomas–Fermi." Rend. R. Accademia dei Lincei 15 (1932): 293.

[7] Fermi, Enrico. "Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente." Zeitschrift für Physik A Hadrons and Nuclei 48.1 (1928): 73–79.

[8] Baker, Edward B. "The application of the Fermi–Thomas statistical model to the calculation of potential distribution in positive ions." Physical Review 36.4 (1930): 630.

[9] Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111], Francisco M.Fernández, Physics Letters A 372, 28 July 2008, 5258-5260, doi:10.1016/j.physleta.2008.05.071.

[10] The analytical solution of the Thomas-Fermi equation for a neutral atom, G I Plindov and S K Pogrebnya, Journal of Physics B: Atomic and Molecular Physics 20 (1987), L547, doi:10.1088/0022-3700/20/17/001.

[11] Esposito, Salvatore (2002). "Majorana solution of the Thomas-Fermi equation". American Journal of Physics. 70: 852.