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

${\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}}$

subject to the boundary conditions

${\displaystyle y(0)=1,\quad \quad {\begin{cases}y(\infty )=0\quad {\text{for neutral atoms}}\\y(x_{0})=0\quad {\text{for positive ions}}\\y(x_{1})-x_{1}y'(x_{1})=0\quad {\text{for compressed neutral atoms}}\end{cases}}}$

If ${\displaystyle y}$ approaches zero as ${\displaystyle x}$ becomes large, this equation models the charge distribution of a neutral atom as a function of radius ${\displaystyle x}$. Solutions where ${\displaystyle y}$ becomes zero at finite ${\displaystyle x}$ model positive ions. For solutions where ${\displaystyle y}$ becomes large and positive as ${\displaystyle x}$ 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 ${\displaystyle x}$ for which ${\displaystyle dy/dx=y/x}$.