  We study the synchrotron radiation as the observed non-thermal X-ray emission
from old pulsars ($\gtrsim1-10$Myr) to investigate the particle acceleration in
their magnetospheres. We assume that the power-law component of the observed
X-ray spectra is caused by the synchrotron radiation from electrons and
positrons in the magnetosphere. We consider two pair production mechanisms of
X-ray emitting particles, the magnetic and the photon-photon pair productions.
High-energy photons, which ignite the pair production, are emitted via the
curvature radiation of the accelerated particles. We use the analytical
description for the radiative transfer and estimate the luminosity of the
synchrotron radiation. We find that for pulsars with the spin-down luminosity
$L_{\rm sd}\lesssim10^{33}$ erg s$^{-1}$, the locations of the particle
acceleration and the non-thermal X-ray emission are within $\lesssim10^7$cm
from the centre of the neutron star, where the magnetic pair production occurs.
For pulsars with the spin-down luminosity $L_{\rm sd}\lesssim10^{31}$ erg
s$^{-1}$ such as J0108-1431, the synchrotron radiation is difficult to explain
the observed non-thermal component even if we consider the existence of the
strong and small-scale surface magnetic field structures.
