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We investigate the electronic states around a single vacancy in siliconcrystal by using the Green's function approach. The triply degenerate vacancystates within the band gap are found to be extended over a large distance$\sim20 {\rm \AA}$ from the vacancy site and contribute to the reciprocaltemperature dependence of the quadrupole susceptibility resulting in theelastic softening at low temperture. The Curie constant of the quadrupolesusceptibility for the trigonal mode ($O_{yz},O_{zx},O_{xy}$) is largelyenhanced as compared to that for the tetragonal mode ($O_{2}^{0},O_{2}^{2}$).The obtained results are consistent with the recent ultrasonic experiments insilicon crystal down to 20 mK. We also calculate the dipole and octupolesusceptibilities and find that the octupole susceptibilities are extremelyenhannced for a specific mode.

Quadrupole Susceptibility and Elastic Softening due to a Vacancy in Silicon Crystal