  This study presents the development of hydrophobic silica aerogel for use as
a radiator in threshold-type Cherenkov counters. These counters are to be used
for separating positrons and positive muons produced by kaon decay in the
J-PARC TREK/E36 experiment. We chose to employ aerogel with a refractive index
of 1.08 to identify charged particles with momenta of approximately 240
MeV/$c$, and the radiator block shape was designed with a trapezoidal
cross-section to fit the barrel region surrounding the kaon stopping target in
the center of the TREK/E36 detector system. Including spares, we obtained 30
crack-free aerogel blocks segmented into two layers, each layer having a
thickness of 2 cm and a length of 18 cm, to fill 12 counter modules. Optical
measurements showed that the produced aerogel tiles had the required refractive
indices and transparency.
