Boundary Depth Scans

Collective Oscillation Density-Mass Ratio Scan

Plasma frequency squared is scanned across exact density and mass ratios. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The checked oscillation square starts from a ratio

The diagram pins normalized constant twelve, density ratio three, and mass ratio six. The square frequency is therefore six.

ω2=12⋅36=6 s−2\omega^{2}=12\cdot\frac{3}{6}=6\ \text{s}^{-2}
Oscillation ratio sourceDensity, mass ratio, and omega squared are checked.electronEFx=2 mgradient=3 N/(C*m)E=6 N/Cq=-1 CF=-6 NK=12 1/s^2density=3massRatio=6omega2=6 1/s^2

Density and mass ratio pull opposite directions

The first row is the checked source. Doubling density doubles omega squared; halving mass ratio doubles it again.

nmω236666126324\begin{array}{c|c|c}n&m&\omega^{2}\\3&6&6\\6&6&12\\6&3&24\\\end{array}
Oscillation ratio scanThe first row is the checked source diagram.electronEFx=2 mgradient=3 N/(C*m)E=6 N/Cq=-1 CF=-6 NK=12 1/s^2density=3massRatio=6omega2=6 1/s^2

The square frequency follows density over mass

The lesson stays with omega squared so every row remains exact. That is enough to see the collective frequency trend.

ω2=Knmratio\omega^{2}=K\frac{n}{m_{\text{ratio}}}
Collective ratio boundaryThe restoring force diagram anchors the ratio scan.electronEFx=2 mgradient=3 N/(C*m)E=6 N/Cq=-1 CF=-6 NK=12 1/s^2density=3massRatio=6omega2=6 1/s^2