By the generalized linear quantum transformation theory, we concisely derive the analytic expressions of partition function for general quadratic systems of multi-mode boson and fermion Fock space without any information for the energy spectrum. Under a general condition, we firstly acquire the exact expressions of energy spectra for these systems.PACS numbers: 03.65. Fd,

The temperature dependences of acceptor concentration n A , conductivity mobility C , and resistivity of gallium (Ga)-doped Czochralski (CZ)-silicon (Si) crystals were studied in the temperature range from 220 to 360 K (À53 to 87 C). Crystals with Ga concentration N A from 6:1 Â 10 14 to 2:0 Â 10 18 atoms/cm 3 were analyzed by Hall-effect measurements using the van der Pauw method in the temperature range from 80 to 360 K. The temperature dependences of n A and of crystals with less than 10 16 atoms/cm 3 showed the same trends as those of B-doped p-type Si crystals, while those of crystals with more than 10 16 atoms/cm 3 differed from those of B-doped crystals, mainly because the temperature dependence of n A is not in the saturation range and the degree of ionization n A =N A decreases with decreasing temperature from 360 to 220 K. #