= Common-spacer dilution in the two-current model
{title2=$m=p^2/[(1+s)^2-p^2]$}
Define a dimensionless channel contrast $p=(b-a)/(a+b)$ and spacer ratio $s=r/(a+b)$. Positive $a,b$ give $|p|<1$, and substitution gives the displayed expression. Equal channel responses, $p=0$, eliminate the model's magnetic-configuration signal. Increasing the common spacer resistance at fixed $a,b$ increases the denominator and decreases $m$.
\Image[/spin-channel-resistor-model-and-spacer-dilution.png]
{title=Two spin channels turn magnetic alignment into a different terminal resistance}
{description=Original CPP circuit and chosen parameter curves, not fits to the historical CIP measurements. The example uses $a=1\,\Omega$, $b=4\,\Omega$, $r=1\,\Omega$. Curves vary the defined contrast $p$ at fixed spacer ratio $s$; this does not assert that changing a real chromium thickness leaves every other material and magnetic property fixed.}
{source=spin-channel-resistor-model-and-spacer-dilution.py}
This is a controlled comparison within the model. In an actual multilayer, changing a spacer thickness can change the magnetic coupling and the interfaces as well as electrical transport. <Spin-flip scattering> also mixes the channels that the calculation kept independent.
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