A model for odd-$A$ nuclei is described in which the nucleon is coupled to a possible asymmetric, rotating core. The Hamiltonian consists of two parts: $H_R$, the Hamiltonian for a rotating core of fixed shape with quadrupole moments of inertia and $H_P$, the Hamiltonian for the single particle moving in an asymmetric oscillator potential with l·s and ${\mathbf{l}}^2$ terms. The energy eigenvalues were obtained by an exact diagonalization of this Hamiltonian using an appropriate core-particle basis. To account for the filling of a shell, a basis truncation procedure has been developed based, in general, upon the well-known Nilsson single-particle levels. The model has been applied to the $2s-1d$ shell and assignments of the three model parameters ($P$, a core strength parameter, and the usual $\beta$ and $\gamma$ parameters) have been made to all odd-$A$ nuclei in the shell having at least three levels with known spin and positive parity. The state functions so obtained were used to calculate the static moments and reduced matrix elements of various electromagnetic transitions. This survey shows that these nuclei are generally quite well represented by the model with the result that $\gamma$ is either zero for $A<\mathrm{}25\mathrm{}$ or about 30°, representing maximum asymmetry for $A\ge 25$. The truncations and eigenvalues for each nucleus fitted are appended.

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