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, which can be derived by applying the Thomas–Fermi model to atoms. The equation reads subject to the boundary conditions If formula_3 approaches zero as formula_4 becomes large, this equation models the charge distribution of a neutral atom as a function of radius formula_4. Solutions where formula_3 becomes zero at finite formula_4 model positive ions. For solutions where formula_3 becomes large and positive as formula_4 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 formula_4 for which formula_11. Transformations. Introducing the transformation formula_12 converts the equation to This equation is similar to Lane–Emden equation with polytropic index formula_14 except the sign difference. The original equation is invariant under the transformation formula_15. Hence, the equation can be made equidimensional by introducing formula_16 into the equation, leading to so that the substitution formula_18 reduces the equation to If formula_20 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 formula_22, which satisfies the boundary condition that formula_23 as formula_24, 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. If the transformation formula_26 is introduced, the equation becomes The particular solution in the transformed variable is then formula_28. So one assumes a solution of the form formula_29 and if this is substituted in the above  equation and the coefficients of formula_30 are equated, one obtains the value for formula_31, which is given by the roots of the equation formula_32. The two roots are formula_33. 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 formula_35 as formula_36. This condition is satisfied if formula_37 and since formula_38, Sommerfeld found the approximation as formula_39. Therefore, the approximate solution is This solution predicts the correct solution accurately for large formula_4, but still fails near the origin. Solution near origin. Enrico Fermi provided the  solution for formula_42 and later extended by Baker. Hence for formula_42, where formula_45. 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 formula_46.