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Argon–Krypton Electron Gain Enthalpy Investigation

Atomic Physics • Electron Scattering • Physical Chemistry

Mathematical Notes

This notebook contains the technical development of the investigation.

Unlike the Research Journal, which records the chronological evolution of the project, this notebook focuses exclusively on the mathematics. Every derivation is reconstructed independently wherever possible before being compared against the published literature.

Contents

  1. Polarization Potential
  2. Zero-Energy Radial Equation
  3. Exact Zero-Energy Solution
  4. Jost Transformation
  5. Volterra Integral Equation
  6. Perturbation Expansion
  7. Scattering Length
  8. Comparison with O'Malley (1961)

Polarization Potential

Starting Point

V(r)=−C4r4V(r)=-\frac{C_4}{r^4}

For convenience,

β2=2mC4ℏ2\beta^2=\frac{2mC_4}{\hbar^2}

giving

V(r)=−ℏ2β22mr4.V(r)=-\frac{\hbar^2\beta^2}{2mr^4}.

Physical Meaning

The long-range interaction between an incident electron and a neutral noble-gas atom is dominated by the induced polarization potential.

...

Zero-Energy Radial Equation

Starting from

u′′+[k2−2mℏ2V(r)]u=0u''+\left[k^2-\frac{2m}{\hbar^2}V(r)\right]u=0

the zero-energy limit gives

u′′+β2r4u=0.u''+\frac{\beta^2}{r^4}u=0.

...

Exact Zero-Energy Solution

Introduce

x=βr.x=\frac{\beta}{r}.

After substitution,

...

Eventually,

u1=rsin⁡βr,u_1=r\sin\frac{\beta}{r},
u2=rcos⁡βr.u_2=r\cos\frac{\beta}{r}.

Boundary conditions select the physical solution.

...

Jost Transformation

...

Volterra Integral Equation

...

Perturbation Expansion

Current status:

🟡 Under revision

Current issue:

The perturbation expansion originally began around

m(r)=1m(r)=1

rather than

m0(r)=cos⁡βr.m_0(r)=\cos\frac{\beta}{r}.

The latter corresponds to the exact zero-energy solution and produces the appropriate distorted-wave expansion discussed by O'Malley.

Comparison with O'Malley (1961)

Topic Independent Derivation O'Malley
Starting point ... ...
Exact solution ... ...
Matching ... ...
Special functions None Modified Mathieu
Current agreement ... ...