In
mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
, named after
Llewellyn Thomas
Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician. He is best known for his contributions to atomic and molecular physics and solid-state physics. His key achievements include calcula ...
and
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" an ...
, which can be derived by applying the
Thomas–Fermi model
The Thomas–Fermi (TF) model,
named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equati ...
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 at finite
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
In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer La ...
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''(0)=1. This particular solution is
:
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretic ...
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
, where we need to take the positive root to avoid the singularity at the origin. This solution already satisfies the first boundary condition (
), so, to satisfy the second boundary condition, one writes to the same level of accuracy for an arbitrary
:
The second 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
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" an ...
provided the solution for
and later extended by Edward B. Baker. Hence for
,
:
where
.
It has been reported by Salvatore Esposito
that the Italian physicist
Ettore Majorana
Ettore Majorana (,, uploaded 19 April 2013, retrieved 14 December 2019 ; born on 5 August 1906 – possibly dying after 1959) was an Italian theoretical physicist who worked on neutrino masses. On 25 March 1938, he disappeared under mysteri ...
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
.
References
{{DEFAULTSORT:Thomas Fermi equation
Equations of physics
Ordinary differential equations