Thomas-Fermi equation revisited: A variation on a theme by Majorana

This paper revisits Majorana's method of converting the second-order Thomas-Fermi equation into a first-order one, applying it to both neutral and weakly ionized atoms to efficiently recalculate and validate key atomic physics quantities against previous numerical results.

The Gist

Imagine you are trying to bake the perfect cake for a giant crowd (an atom with many electrons). In 1927, two scientists, Thomas and Fermi, came up with a recipe. However, their recipe was written in a very complicated language: a "second-order differential equation."

Think of a second-order equation like a recipe that tells you not just how much flour to add, but how the rate at which you add flour is changing based on how fast you are stirring. It's a double-layered puzzle. To solve it, you usually have to do a lot of heavy lifting, like climbing a steep, winding mountain to get a view from the top.

Enter Ettore Majorana, a brilliant physicist from the 1930s who was also a bit of a mystery (he disappeared without a trace). Majorana looked at this complicated mountain and said, "Wait a minute. There's a hidden elevator."

The "Hidden Elevator" (Scaling)

Majorana realized that the recipe for the atom has a special property called scaling. Imagine if you could shrink or stretch your cake pan, and the recipe would still work perfectly, just with different numbers. Majorana found a way to use this "stretching" trick to turn the complicated two-step mountain climb into a simple, one-step elevator ride.

He converted the difficult second-order equation into a much simpler first-order equation.

• Old Way: "How is the slope changing, and how is the height changing?" (Hard to solve).
• Majorana's Way: "If I know the current height and the slope, I can predict the next step directly." (Much easier).

The Two Types of Atoms

The paper focuses on two specific types of "cakes":

1. Neutral Atoms: These are atoms with a perfect balance of protons and electrons. This is the "standard" cake everyone studied for decades.
2. Weakly Ionized Atoms: These are atoms that have lost a tiny bit of their electron "frosting." They are slightly different, and until now, applying Majorana's elevator trick to them was a bit of a mystery.

The author of this paper, Berthold-Georg Englert, says, "Let's take Majorana's elevator and see if it works for the second type of cake, too."

The Magic Map (The Power Series)

Once you are in the elevator (solving the first-order equation), you still need to know exactly where you are going. In the 1980s, scientists calculated the answers by doing millions of tiny, tedious steps on a computer. It was like measuring the distance to the moon by walking step-by-step with a ruler.

Englert uses a Power Series. Think of this as a magic map or a GPS.

• Instead of walking step-by-step, the map gives you a formula that tells you exactly where you are at any point.
• The author rediscovered a "map" (a mathematical series) that Majorana's method revealed.
• He used this map to recalculate the numbers for the atom's energy and structure.

The Results: Faster and Smarter

The paper is essentially a "re-check" of old work.

• The Old Way (1980s): Took a long time, used heavy numerical methods, and was prone to small errors because it was so tedious.
• The New Way (Englert's Method): Uses Majorana's "elevator" and the "magic map." It is much faster, cleaner, and confirms the old numbers with high precision.

It's like realizing that instead of digging a tunnel through a mountain with a spoon (the old 1980s method), you could have just used a tunnel boring machine (Majorana's method) all along.

Why Does This Matter?

Atoms are the building blocks of everything. Knowing exactly how they hold together (their "binding energy") helps us understand chemistry, materials science, and even how stars burn.

By refining these numbers, the author isn't just doing math for fun; he is polishing the lenses through which physicists view the universe. He shows that sometimes, looking back at the insights of a genius from 60 years ago (Majorana) can give us a shortcut to better answers today.

In a Nutshell

• The Problem: Calculating the structure of atoms was like solving a double-layered puzzle.
• The Hero: Ettore Majorana found a trick to turn it into a single-layer puzzle.
• The Mission: The author applied this trick to a specific type of atom (ionized) that hadn't been fully explored with this method.
• The Outcome: They used a "magic map" (power series) to calculate the answers quickly and accurately, confirming old data and making the process much simpler for future scientists.

It's a story about how a clever shortcut discovered decades ago is still saving us time and effort in the lab today.

Technical Summary

1. Problem Statement

The paper addresses the Thomas–Fermi (TF) equation, a second-order non-linear differential equation central to the statistical model of many-electron atoms. While the equation was solved numerically in the 1980s (specifically in the monograph by Englert and Schwinger, 1988), those calculations were computationally tedious and relied on power series with limited, non-overlapping intervals of convergence.

The author seeks to:

1. Revisit and elaborate on a method discovered by Ettore Majorana in the 1930s (published posthumously) that exploits the scaling properties of the TF equation to convert the second-order differential equation into an equivalent first-order differential equation.
2. Apply this method to two specific physical cases:
   • Case (i): Neutral atoms ($f(0)=1, f(\infty)=0$).
   • Case (ii): Weakly ionized atoms ($f(1)=0, f'(1)=-\Lambda^2$).
3. Recalculate key physical constants and integrals with high precision to verify and improve upon previous numerical results.

2. Methodology

A. Scaling Transformation and Scale-Invariant Functions

The TF equation is given by $f''(x) = x^{-1/2}f(x)^{3/2}$. The equation possesses a scaling symmetry: if $f(x)$ is a solution, then $f_\lambda(x) = \lambda^3 f(\lambda x)$ is also a solution.

Majorana's insight involves constructing scale-invariant combinations of the solution $f(x)$ and its derivatives. Englert defines three such functions:

• $P(x) = x^{3/2}f(x)^{1/2}$
• $Q(x) = -x^4 f'(x)$
• $R(x) = -f(x)^{-4/3}f'(x)$

These functions satisfy specific differential relationships. By choosing the appropriate pair based on boundary conditions, the second-order problem is reduced to a first-order ordinary differential equation (ODE).

B. Case (i): Neutral Atoms

For neutral atoms, the boundary conditions are $f(0)=1$ and $f(\infty)=0$.

• Transformation: The author seeks $R$ as a function of $P$. Through variable substitution ($P = 12t^3$ and $R = (3/16)^{1/3}u(t)$), the system reduces to Majorana's equation:
  $$\frac{du}{dt} = -\frac{81 - t u(t)^2}{1 - t^2 u(t)}$$
  with boundary conditions $u(1)=1$ and $u(0) = (16/3)^{1/3}B$, where $B$ is the slope of the TF function at the origin.
• Reconstruction: The original TF function $F(x)$ and its derivative are recovered via integrals involving $u(t)$. Crucially, the variable $t$ ranges from $0$ to $1$, covering the entire physical domain $x \in [0, \infty)$.

C. Case (ii): Weakly Ionized Atoms

For weakly ionized atoms, the solution vanishes at a finite radius $x=1$.

• Transformation: The author seeks $Q$ as a function of $P$. Using substitutions ($P = 12s$ and $Q = 432v(s)$), a similar first-order equation is derived:
  $$\frac{dv}{ds} = -\frac{8}{3} \frac{sv(s) - s^4}{v(s) - s^2}$$
  with boundary conditions $v(1)=1$ and $v(0) = 1/(432\Lambda^2)$.

D. Numerical Implementation via Power Series

The primary advantage of this method is the convergence properties of the solutions $u(t)$ and $v(s)$.

• Series Expansion: The author expands $u(t)$ and $v(s)$ in powers of $(1-t)$ and $(1-s)$, respectively.
• Convergence: Unlike the direct power series for $F(x)$, which have finite, non-overlapping intervals of convergence (requiring matching at intermediate points), the series for $u(t)$ and $v(s)$ converge globally for the entire range $0 \le t, s \le 1$.
• Recurrence Relations: The coefficients of these series are determined by recurrence relations derived from the differential equations. This allows for the calculation of high-precision values for physical constants.

3. Key Contributions

1. Derivation of the Ionized Case: While Majorana's method was known for neutral atoms, Englert explicitly derives and analyzes the analogous first-order equation for weakly ionized atoms (Case ii), a solution relevant for finite systems.
2. Global Convergence: The paper demonstrates that the Majorana transformation converts the problem into one solvable by power series that converge over the entire physical domain, eliminating the need for the "tedious" matching procedures used in previous numerical studies.
3. Recalculation of Constants: The author recalculates fundamental constants with high precision using the new series method:
   • Slope $B$ for neutral atoms: $1.58807102261...$
   • Parameter $\Lambda$ for ionized atoms: $32.729416116173...$
   • Asymptotic parameters $\beta$ and $\alpha$.
4. Evaluation of Integrals: The method facilitates the calculation of various integrals involving powers of $F(x)$ and $x$, which are essential for energy formulas. These include integrals of the form $\int x^k F(x)^l dx$.

4. Results

• Verification of Previous Values: The values obtained for $B$, $\Lambda$, $\beta$, and $\alpha$ match the digits found in the 1980s literature, confirming the accuracy of the older, more computationally expensive methods.
• New Precision: The author provides extended decimal expansions for these constants and derived quantities (e.g., binding energy coefficients).
• Integral Values: Table 4 in the paper lists recalculated integrals (e.g., $\int F(x) dx$, $\int F(x)^2 dx$). The results confirm previous values but with higher precision and without the numerical instability associated with the older methods.
• Energy Formulas: The recalculated integrals are used to refine the coefficients for:
  • Binding Energy: $E \approx -c_7 Z^{7/3} + \frac{1}{2}Z^2 - c_5 Z^{5/3}$.
  • Ionization Energy: The leading term for weakly ionized atoms is calculated as $I \approx 3.151526$ eV.
  • Relativistic Corrections: Coefficients for relativistic corrections ($c_{rel}$) are updated using the new integral values.

5. Significance

• Mathematical Elegance: The paper highlights the profound insight of Ettore Majorana, showing how scaling symmetries can drastically simplify non-linear differential equations in physics.
• Computational Efficiency: The method provides a "simple way" to compute physical numbers that previously required complex numerical integration and series matching. The global convergence of the Majorana series makes the algorithm robust and easy to implement.
• Pedagogical Value: The paper serves as a comprehensive guide to applying Majorana's technique, extending it to a new physical regime (ionized atoms) and providing explicit recurrence relations for practical computation.
• Future Applications: The author suggests that this transformation technique could be applied to other non-linear equations of the form $f''(x) = x^a f(x)^b$, such as the Lane-Emden equation for polytropic gas balls, and to more complex ionization boundary conditions.

In summary, Englert's paper revitalizes a historical mathematical technique, demonstrating its superior utility for modern high-precision calculations in atomic physics, specifically for the Thomas–Fermi model of both neutral and weakly ionized atoms.