  We show the spectral properties of 15 spin-powered X-ray pulsars, and the
correlation between the average power-law photon index and spin-down rate.
Generation order parameters (GOPs) based on polar-cap models are introduced to
characterize the X-ray pulsars. We calculate three definitions of generation
order parameters due to the different effects of magnetic and electric fields
on photon absorption during cascade processes, and study the relations between
the GOPs and spectral properties of X-ray pulsars. There exists a possible
correlation between the photon index and GOP in our pulsar sample. Furthermore,
we present a method due to the concept of GOPs to estimate the non-thermal
X-ray luminosity for spin-powered pulsars. Then X-ray luminosity is calculated
in the context of our polar-cap accelerator model which is well consistent with
the most observed X-ray pulsar data. The ratio between X-ray luminosity
estimated by our method and the pulsar's spin-down power is well consistent
with the $L_{\rm X}\sim 10^{-3}L_{\rm sd}$ feature.

Astrophysics