“Thomas–Fermi equation”的意思、由来-开放百科全书

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

approaches zero as

becomes large, this equation models the charge distribution of a neutral atom as a function of radius

. Solutions where

becomes zero at finite

model positive ions.^[3] For solutions where

becomes large and positive as

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

for which

.^[4]^[5]

Transformations

This equation is similar to Lane–Emden equation with polytropic index

except the sign difference.

The original equation is invariant under the transformation

. Hence, the equation can be made equidimensional by introducing

into the equation, leading to

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

, which satisfies the boundary condition that

, but not the boundary condition y(0)=1. This particular solution is

Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932.^[6] If the transformation

is introduced, the equation becomes

The particular solution in the transformed variable is then

. So one assumes a solution of the form

and if this is substituted in the above equation and the coefficients of

are equated, one obtains the value for

, which is given by the roots of the equation

. The two roots are

. Since this solution already satisfies the second boundary condition, to satisfy the first boundary condition one writes

The first boundary condition will be satisfied if

. This condition is satisfied if

and since

, Sommerfeld found the approximation as

. Therefore, the approximate solution is

This solution predicts the correct solution accurately for large

, but still fails near the origin.

Solution near origin

Enrico Fermi^[7] provided the solution for

and later extended by Baker.^[8] Hence for

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

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 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. ^{{Cite journal|last=Esposito|first=Salvatore|date=|year=2002|title=Majorana solution of the Thomas-Fermi equation|url=https://arxiv.org/abs/physics/0111167|journal=American Journal of Physics|volume=70|pages=852|via=}}