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 subject to the boundary conditions[3] If

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

model positive ions.

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

[5][6] Introducing the transformation

The original equation is invariant under the transformation

Hence, the equation can be made equidimensional by introducing

into the equation, leading to so that the substitution

reduces the equation to Treating

as the independent variable, we can reduce the above equation to But this first order equation has no known explicit solution, hence, the approach turns to either numerical or approximate methods.

The equation has a particular solution

, which satisfies the boundary condition that

This particular solution is Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932.

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

, where we need to take the positive root to avoid the singularity at the origin.

This solution already satisfies the first boundary condition (

), so, to satisfy the second boundary condition, one writes to the same level of accuracy for an arbitrary

The second boundary condition will be satisfied if

, 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.

Enrico Fermi[8] provided the solution for

[10][11] It has been reported by Salvatore Esposito[12] 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in 1928 a semi-analytical series solution to the Thomas–Fermi equation for the neutral atom, which however remained unpublished until 2001.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