Lower bound for the population of hyperfine component $\mu =0$ particles   in the ground state of spin-1 condensates
  An analytical expression for the lower bound of the average number of hyperfine component $\mu =0$ particles in the ground state of spin-1 condensates (denoted as $\overset{\_\_}{\rho _{0}}$) under a magnetic field has been derived. In the derivation the total magnetization $M$ is kept rigorously conserved. Numerical examples are given to show the applicability of the analytical expression. It was found that, in a broad domain of parameters specified in the paper, the lower bound is very close to the actual $\overset{\_\_}{\rho _{0}}$. Thereby, in this domain, $\overset{\_\_}{% \rho _{0}}$ can be directly evaluated simply by using the analytical expression. 
C.G. Bao
cond-mat.quant-gas physics.atom-ph