Measurement of the beam-normal single-spin asymmetry for elastic electron scattering from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi mathvariant="normal">C</mml:mi><mml:mprescripts/><mml:none/><mml:mn>12</mml:mn></mml:mmultiscripts></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Al</mml:mi><mml:mprescripts/><mml:none/><mml:mn>27</mml:mn></mml:mmultiscripts></mml:math>

We report measurements of the parity-conserving beam-normal single-spin elastic scattering asymmetries ${B}_{n}$ on $^{12}\mathrm{C}$ and $^{27}\mathrm{Al}$, obtained with an electron beam polarized transverse to its momentum direction. These measurements add an additional kinematic point to a series of previous measurements of ${B}_{n}$ on $^{12}\mathrm{C}$ and provide a first measurement on $^{27}\mathrm{Al}$. The experiment utilized the ${\mathrm{Q}}_{\mathrm{weak}}$ apparatus at Jefferson Lab with a beam energy of 1.158 GeV. The average laboratory scattering angle for both targets was $7.{7}^{\ensuremath{\circ}}$, and the average ${Q}^{2}$ for both targets was 0.024 37 ${\mathrm{GeV}}^{2}$ ($Q=0.1561$ GeV). The asymmetries are ${B}_{n}=\ensuremath{-}10.68\ifmmode\pm\else\textpm\fi{}0.90\phantom{\rule{0.222222em}{0ex}}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.57\phantom{\rule{0.222222em}{0ex}}(\mathrm{syst})$ ppm for $^{12}\mathrm{C}$ and ${B}_{n}=\ensuremath{-}12.16\ifmmode\pm\else\textpm\fi{}0.58\phantom{\rule{0.222222em}{0ex}}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.62\phantom{\rule{0.222222em}{0ex}}(\mathrm{syst})$ ppm for $^{27}\mathrm{Al}$. The results are consistent with theoretical predictions, and are compared to existing data. When scaled by $Z/A$, the $Q$ dependence of all the far-forward angle ($\ensuremath{\theta}&lt;{10}^{\ensuremath{\circ}}$) data from $^{1}\mathrm{H}$ to $^{27}\mathrm{Al}$ can be described by the same slope out to $Q\ensuremath{\approx}0.35$ GeV. Larger-angle data from other experiments in the same $Q$ range are consistent with a slope about twice as steep.