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 d 2 y d x 2 = 1 x y 3 / 2 {\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}} subject to the boundary conditions y ( 0 ) = 1 , { y ( ∞ ) = 0 for neutral atoms y ( x 0 ) = 0 for positive ions y ( x 1 ) − x 1 y ′ ( x 1 ) = 0 for compressed neutral atoms {\displaystyle y(0)=1,\quad \quad {\begin{cases}y(\infty )=0\quad {\text{for neutral atoms}}\\y(x_{0})=0\quad {\text{for positive ions}}\\y(x_{1})-x_{1}y'(x_{1})=0\quad {\text{for compressed neutral atoms}}\end{cases}}} If y {\displaystyle y} approaches zero as x {\displaystyle x} becomes large, this equation models the charge distribution of a neutral atom as a function of radius x {\displaystyle x} .