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} . Solutions where y {\displaystyle y} becomes zero at finite x {\displaystyle x} model positive ions. For solutions where y {\displaystyle y} becomes large and positive as x {\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 x {\displaystyle x} for which d y / d x = y / x {\displaystyle dy/dx=y/x} .
Transformations Introducing the transformation z = y / x {\displaystyle z=y/x} converts the equation to
1 x 2 d d x ( x 2 d z d x ) − z 3 / 2 = 0 {\displaystyle {\frac {1}{x^{2}}}{\frac {d}{dx}}\left(x^{2}{\frac {dz}{dx}}\right)-z^{3/2}=0}
This equation is similar to Lane–Emden equation with polytropic index 3 / 2 {\displaystyle 3/2} except the sign difference. The original equation is invariant under the transformation x → c x , y → c − 3 y {\displaystyle x\rightarrow cx,\ y\rightarrow c^{-3}y} . Hence, the equation can be made equidimensional by introducing y = x − 3 u {\displaystyle y=x^{-3}u} into the equation, leading to
x 2 d 2 u d x 2 − 6 x d u d x + 12 u = u 3 / 2 {\displaystyle x^{2}{\frac {d^{2}u}{dx^{2}}}-6x{\frac {du}{dx}}+12u=u^{3/2}}
so that the substitution x = e t {\displaystyle x=e^{t}} reduces the equation to
d 2 u d t 2 − 7 d u d t + 12 u = u 3 / 2 . {\displaystyle {\frac {d^{2}u}{dt^{2}}}-7{\frac {du}{dt}}+12u=u^{3/2}.}
Treating w = d u / d t {\displaystyle w=du/dt} as the dependent variable and u {\displaystyle u} as the independent variable, we can reduce the above equation to
w d w d u − 7 w = u 3 / 2 − 12 u . {\displaystyle w{\frac {dw}{du}}-7w=u^{3/2}-12u.}
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 y p ( x ) {\displaystyle y_{p}(x)} , which satisfies the boundary condition that y → 0 {\displaystyle y\rightarrow 0} as x → ∞ {\displaystyle x\rightarrow \infty } , but not the boundary condition y(0)=1. This particular solution is
y p ( x ) = 144 x 3 . {\displaystyle y_{p}(x)={\frac {144}{x^{3}}}.}
Arnold Sommerfeld used this particular solution and provided an approximate solution which can satisfy the other boundary condition in 1932. If the transformation x = 1 / t , w = y t {\displaystyle x=1/t,\ w=yt} is introduced, the equation becomes
t 4 d 2 w d t 2 = w 3 / 2 , w ( 0 ) = 0 , w ( ∞ ) ∼ t . {\displaystyle t^{4}{\frac {d^{2}w}{dt^{2}}}=w^{3/2},\quad w(0)=0,\ w(\infty )\sim t.}
The particular solution in the transformed variable is then w p ( t ) = 144 t 4 {\displaystyle w_{p}(t)=144t^{4}} . So one assumes a solution of the form w = w p ( 1 + α t λ ) {\displaystyle w=w_{p}(1+\alpha t^{\lambda })} and if this is substituted in the above equation and the coefficients of α {\displaystyle \alpha } are equated, one obtains the value for λ {\displaystyle \lambda } , which is given by the roots of the equation λ 2 + 7 λ − 6 = 0 {\displaystyle \lambda ^{2}+7\lambda -6=0} . The two roots are λ 1 = 0.772 , λ 2 = − 7.772 {\displaystyle \lambda _{1}=0.772,\ \lambda _{2}=-7.772} , where we need to take the positive root to avoid the singularity at the origin. This solution already satisfies the first boundary condition ( w ( 0 ) = 0 {\displaystyle w(0)=0} ), so, to satisfy the second boundary condition, one writes to the same level of accuracy for an arbitrary n {\displaystyle n}
W = w p ( 1 + β t λ ) n = [ 144 t 3 ( 1 + β t λ ) n ] t . {\displaystyle W=w_{p}(1+\beta t^{\lambda })^{n}=[144t^{3}(1+\beta t^{\lambda })^{n}]t.}
The second boundary condition will be satisfied if 144 t 3 ( 1 + β t λ ) n = 144 t 3 β n t λ n ( 1 + β − 1 t − λ ) n ∼ 1 {\displaystyle 144t^{3}(1+\beta t^{\lambda })^{n}=144t^{3}\beta ^{n}t^{\lambda n}(1+\beta ^{-1}t^{-\lambda })^{n}\sim 1} as t → ∞ {\displaystyle t\rightarrow \infty } . This condition is satisfied if λ n + 3 = 0 , 144 β n = 1 {\displaystyle \lambda n+3=0,\ 144\beta ^{n}=1} and since λ 1 λ 2 = − 6 {\displaystyle \lambda _{1}\lambda _{2}=-6} , Sommerfeld found the approximation as λ = λ 1 , n = − 3 / λ 1 = λ 2 / 2 {\displaystyle \lambda =\lambda _{1},\ n=-3/\lambda _{1}=\lambda _{2}/2} . Therefore, the approximate solution is
y ( x ) = y p ( x ) { 1 + [ y p ( x ) ] λ 1 / 3 } λ 2 / 2 . {\displaystyle y(x)=y_{p}(x)\{1+[y_{p}(x)]^{\lambda _{1}/3}\}^{\lambda _{2}/2}.}
This solution predicts the correct solution accurately for large x {\displaystyle x} , but still fails near the origin.
Solution near origin Enrico Fermi provided the solution for x ≪ 1 {\displaystyle x\ll 1} and later extended by Edward B. Baker. Hence for x ≪ 1 {\displaystyle x\ll 1} ,
y ( x ) =
1 − B x + 1 3 x 3 − 2 B 15 x 4 + ⋯
⋯ + x 3 / 2 [ 4 3 − 2 B 5 x + 3 B 2 70 x 2 + ( 2 27 + B 3 252 ) x 3 + ⋯ ] {\displaystyle {\begin{aligned}y(x)={}&1-Bx+{\frac {1}{3}}x^{3}-{\frac {2B}{15}}x^{4}+\cdots {}\\[6pt]&\cdots +x^{3/2}\left[{\frac {4}{3}}-{\frac {2B}{5}}x+{\frac {3B^{2}}{70}}x^{2}+\left({\frac {2}{27}}+{\frac {B^{3}}{252}}\right)x^{3}+\cdots \right]\end{aligned}}}
where B ≈ 1.588071 {\displaystyle B\approx 1.588071} . It has been reported by Salvatore 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.
References