Debye Length Scale Proportionality Scan - Condensed Matter and Plasma Physics Step by Step

A Debye-length scale-proportionality scan varies temperature or density across rows, showing how lambda_D follows the square-root scale with T/n in the shielding ledger. The arithmetic is exact only for the stated densities, temperatures, charges, Debye lengths, and rows in the ideal Maxwellian plasma model. Real plasmas add collisions, nonuniform fields, distribution functions, boundary/sheath effects, calibration, finite precision, and measurement uncertainty.

Debye length squared is scanned across exact scale squares. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Scale square raises lambda square at fixed density

The checked source keeps density ratio at three. Scale squared nine therefore gives lambda squared three.

\lambda^{2}=\frac{9}{3}=3\ \text{m}^{2}

Changing scale alone changes lambda square linearly

Density stays fixed in every row. Raising scale squared from six to fifteen raises lambda squared from two to five.

\begin{array}{c|c|c}S&n&\lambda^{2}\\6&3&2\\9&3&3\\15&3&5\\\end{array}

Scale and density are separate knobs

This is the companion to the density scan. One knob multiplies the numerator; the other divides it.

S\uparrow\quad\Rightarrow\quad \lambda^{2}=\frac{S}{n}\uparrow