= Solution
{disambiguate=magnetoresistance}
Combining the two results gives the <GMR ratio for identical thickness-averaged layers>:
$$
R_P=\frac{ab}{2D^2(a+b)},\qquad R_{AP}=\frac{a+b}{8D^2},\qquad
R_{AP}-R_P=\frac{(a-b)^2}{8D^2(a+b)}\ge0.
$$
Thus a field that aligns initially antiparallel moments produces \b[a resistance decrease]. One spin channel conducts well throughout the aligned stack; in the antiparallel stack both channels sample strong scattering. For the low-resistance-state normalization,
$$
\boxed{\frac{R_{AP}-R_P}{R_P}=\frac{(a-b)^2}{4ab}.}
$$
Normalizing the field-induced drop by $R_{AP}$ instead gives $(a-b)^2/(a+b)^2$. Large channel contrast can produce a large <giant magnetoresistance>; equal channels give none, and common nonmagnetic scattering dilutes the contrast. This is the spin-dependent alignment effect, distinct from <anisotropic magnetoresistance> of an individual magnetic layer.
\Image[/past-exam-of-the-advanced-quantum-condensed-matter-physics-course-of-the-university-of-cambridge/2013/paper-1-aqc2013-spin-channel-conductance.png]
{title=Parallel layer moments preserve a favourable spin channel; antiparallel moments equalize the two averaged channels}
{source=paper-1-aqc2013-spin-channel-conductance.py}
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