= Finite-acceptance dilution of beta asymmetry
{title2=$k=\langle\cos\theta\rangle_{\rm acceptance}$}
A detector collects a range of <solid angles>, not a single direction. If its nonnegative angular response is $w(\Omega)$, define
$$
G=\int w(\Omega)\,d\Omega,\qquad
k=\frac1G\int w(\Omega)\cos\theta\,d\Omega.
$$
Integrating the <polarized beta-decay angular distribution> gives a count proportional to $G(1+ak)$. For a uniform forward cone with half-angle $\theta_0$, put $c_0=\cos\theta_0$. Direct integration gives
$$
k=\frac{\int_{c_0}^{1}u\,du}{\int_{c_0}^{1}du}=\frac{1+c_0}{2}.
$$
The opposite cone has mean cosine $-k$. A hemisphere gives $k=1/2$; a narrow cone tends to $k=1$. Hence a <spin-reversal cross-ratio for beta decay> measures $ak$, and an acceptance correction is needed to obtain $a$.
With unequal angular responses, let the two laboratory mean cosines be $k_+$ and $k_-$, including their signs. Then the actual expectation is
$$
R=\frac{(1+ak_+)(1-ak_-)}{(1+ak_-)(1-ak_+)},
$$
which reduces to the simple square only when $k_-=-k_+$. The cancellation of stable multiplicative <detection efficiencies> survives, but their angular shapes still matter. An <energy>-integrated response likewise averages $A(E)P\beta(E)\cos\theta$ with the accepted spectrum; a mean speed cannot in general be substituted independently of acceptance.
Additive backgrounds, unequal polarization magnitudes or <detection efficiencies> that change on reversal do not have the assumed factorization. For example, a reversal-dependent efficiency cross-ratio multiplies $R$ by $\eta_+^{\,+}\eta_-^{\,-}/(\eta_-^{\,+}\eta_+^{\,-})$. <Background subtraction> also creates statistical complications: positive expected signal counts are assumed in this derivation, whereas finite samples with <Poisson distributions> can contain zero counts. Taking square roots of negative background-subtracted counts is not a valid estimator. These limits explain why reversal and independent preparation checks belong together.
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