  Cs$_2$LiLaBr$_6$:Ce$^{3+}$ (CLLB) is an elpasolite scintillator that offers
excellent linearity and gamma-ray energy resolution and sensitivity to thermal
neutrons with the ability to perform pulse-shape discrimination (PSD) to
distinguish gammas and neutrons. Our investigation of CLLB has indicated the
presence of intrinsic radioactive alpha background that we have determined to
be from actinium contamination of the lanthanum component. We measured the
pulse shapes for gamma, thermal neutron, and alpha events and determined that
PSD can be performed to separate the alpha background with a moderate figure of
merit of 0.98. We also measured the electron-equivalent-energy of the alpha
particles in CLLB and simulated the intrinsic alpha background from $^{227}$Ac
to determine the quenching factor of the alphas. A linear quenching
relationship $L_{\alpha} = E_{\alpha} \times q + L_0$ was found at alpha
particle energies above 5 MeV, with a quenching factor $q = 0.71$ MeVee/MeV and
an offset $L_0 = - 1.19$ MeVee.

Instrumentation and Detectors