Thomas–Fermi equation

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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

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

Llewellyn Hilleth Thomas was a British physicist and applied mathematician. He is best known for his contributions to atomic physics and solid-state physics, in particular:

Contents

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

subject to the boundary conditions

${\displaystyle y(0)=1\quad ;\quad y(+\infty )=0}$

If ${\displaystyle y}$ approaches zero as ${\displaystyle x}$ becomes large, this equation models the charge distribution of a neutral atom as a function of radius ${\displaystyle x}$. Solutions where ${\displaystyle y}$ becomes zero at finite ${\displaystyle x}$ model positive ions. [3] For solutions where ${\displaystyle y}$ becomes large and positive as ${\displaystyle x}$ 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 ${\displaystyle x}$ for which ${\displaystyle dy/dx=y/x}$. [4] [5]

Transformations

Introducing the transformation ${\displaystyle z=y/x}$ converts the equation to

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

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

In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden. The equation reads

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

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

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

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

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

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

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

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

Arnold Johannes Wilhelm Sommerfeld, was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored a large number of students for the new era of theoretical physics. He served as doctoral supervisor for many Nobel Prize winners in physics and chemistry.

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

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

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

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

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

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

Solution near origin

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

Enrico Fermi was an Italian and naturalized-American physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi held several patents related to the use of nuclear power, and was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. He made significant contributions to the development of statistical mechanics, quantum theory, and nuclear and particle physics.

{\displaystyle {\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 ${\displaystyle B\approx 1.588071}$. [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 ${\displaystyle B=1.588071022611375312718684509423950109452746621674825616765677418166551961154309262332033970138428665}$.

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References

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. 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.
4. March 1983, p. 10, Figure 1.
5. p. 1562, R. P. Feynman, N. Metropolis, and E. Teller. "Equations of State of Elements Based on the Generalized Thomas-Fermi Theory". Physical Review75, #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 A372, 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 Physics20 (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. arXiv:. doi:10.1119/1.1484144.