Geometrical Corrections in Angular Correlation Measurements

It is shown that given a point-point angular correlation of the form $W\left(\theta \right)=\Sigma l^{}{a}_l{P}_l\left(cos\theta \right)$, the experimentally measured angular correlation using finite-sized detectors of arbitrary shape and efficiency distributions is given by $W\left(\theta \right)=\Sigma {l{l}^{\prime }m{m}^{\prime }}^{}{a}_l{b}_{\mathrm{lm}}{c}_{l{m}^{\prime }}{g_{m{m}^{\prime }}}^{l{l}^{\prime }}{P}_{l^{\prime }}\left(cos\theta \right)$, where $b_{\mathrm{lm}}$ and $c_{\mathrm{lm}}$ are the Legendre coefficients describing the efficiency functions of the detectors, and ${g_{m{m}^{\prime }}}^{l{l}^{\prime }}$ are numerical coefficients. An extensive table of these latter coefficients, sufficient for most applications, is included. The manner in which detector symmetries of various types affect the form of the measured angular correlation is discussed; in particular it is shown that if the efficiency function of both counters is invariant to reflection about both horizontal and vertical axes, the measured angular correlation will contain no $P_l\text{'}\mathrm{s}$ of higher order than the $P_l$ of highest order appearing in the point-point correlation. The above formula for the measured angular correlation is also shown to apply if an axially-extended source instead of a point source is used, the detector coefficients simply being replaced by a new set of suitably averaged coefficients. Tables of correction factors to fourth order in detector and axial source size are included for the special cases of rectangular and circular detectors of constant efficiency.