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Experimentally observed ground state band based on the $1/2^{-}[521]$ Nilssonstate and the first exited band based on the $7/2^{-}[514]$ Nilsson state inthe odd-$Z$ nucleus $^{255}$Lr are studied by the cranked shell model (CSM)with the paring correlations treated by the particle-number-conserving (PNC)method. This is the first time the detailed theoretical investigations beingperformed on these rotational bands. Both the experimental kinematic anddynamic moment of inertia ($\mathcal{J}^{(1)}$ and $\mathcal{J}^{(2)}$) versusrotational frequency are reproduced quite well by the PNC-CSM calculations. Bycomparing the theoretical kinematic moment of inertia $\mathcal{J}^{(1)}$ withthe experimental ones extracted from different spin assignments, the spin$17/2^{-}\rightarrow13/2^{-}$ is assigned to the lowest-lying $196.6(5)$ keVtransition of the $1/2^{-}[521]$ band, and $15/2^{-}\rightarrow11/2^{-}$ to the$189(1)$ keV transition of the $7/2^{-}[514]$ band, respectively. The proton$N=7$ major shell is included in the calculations. The intruder of the high$-j$low$-\Omega$ orbitals $1j_{15/2}$ $ (1/2^{-}[770])$ at the high spin leads tothe band-crossing at $\hbar\omega\approx0.20$ ($\hbar\omega\approx0.25$) MeVfor the $7/2^{-}[514]$ $\alpha=-1/2$ ($\alpha=+1/2$) band, and at$\hbar\omega\approx0.175$ MeV for the $1/2^{-}[521]$ $\alpha=-1/2$ band,respectively. Further investigations show that the band-crossing frequenciesare quadrupole deformation dependent.

Rotational Properties of the Odd-$Z$ Transfermium Nucleus $^{255}$Lr by a Particle-number-conserving Method in the Cranked Shell Model