Lower bound for the variation of the hyperfine populations in the ground state of spin-1 condensates against a magnetic field

A simple and analytical expression for the variation of the lower bound and upper bound of the population density ρ0 of hyperfine component μ=0 particles in the ground state of spin-1 condensates against a magnetic field B has been derived. The lower bound has a distinguished feature, namely, it will tend to the actual ρ0 when B tends to zero and infinite. This feature assures that, in the whole range of B, the lower bound can provide an effective constraint. Numerical examples are given to demonstrate the applicability of the bound.