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Based on the perturbation theory up to the second order, analyticalasymptotic expressions for the variation of the population of hyperfinecomponent $\mu=0 $ particles in the ground state of spin-1 condensates againsta magnetic field $B$ has been derived. The ranges of $B$ in which theasymptotic expressions are applicable have been clarified via a comparison ofthe numerical results from the analytical expressions and from adiagonalization of the Hamiltonian in a complete spin-space. It was found that,For $^{87}$Rb, the two analytical expressions, one for a weak and the other onefor a strong field, together cover the whole range of $B$ from 0 to infinite.For Na, the analytical expressions are valid only if $B$ is very weak orsufficiently strong.

Asymptotic expressions for the hyperfine populations in the ground state of spin-1 condensates against a magnetic field