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, which can be derived by applying the Thomas–Fermi model to atoms. The equation reads subject to the boundary conditions If approaches zero as becomes large, this equation models the charge distribution of a neutral atom as a function of radius. Solutions where becomes zero atfinite model positive ions. 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.
Transformations
Introducing the transformation converts the equation to 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 so that the substitution reduces the equation to If then the above equation becomes 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 as, but not the boundary condition y=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. 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 as. 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
provided the solution for and later extended by Baker. Hence for, where.
Approach by Majorana
It has been reported by Esposito 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.
