A design of a cryogenic detector for a WIMP dark matter search based o
n a single crystal absorber using Series Array of Superconducting Tunn
el Junctions (SASTJs) as readout sensors has been presented. In order
to investigate the behaviour of such a detector the response of two sp
atially separated Series Arrays of Superconducting Tunnel Junctions (S
ASTJs) to the non-equilibrium phonons generated in a single crystal of
BaF2, by external alpha-particles has been measured. A strong positio
n dependence in pulse height and pulse shape of the detector response
has been observed. The fast time response of the SASTJs permits time r
esolved measurements of the spatially separated particle interactions
in the crystal.