A model is proposed for the observed combination of power-law and ther
mal emission of keV X-rays from rotationally powered pulsars, For gamm
a-ray pulsars with accelerators very many stellar radii above the neut
ron star surface, 100 MeV curvature gamma-rays from e(-) or e(+) flowi
ng starward out of such accelerators are converted to e(+/-) pairs on
closed field lines all around the star. These pairs strongly affect X-
ray emission from near the star in two ways. (1) The pairs are a sourc
e of synchrotron. emission immediately following their creation in reg
ions where B similar to 10(10) G. This emission, in the photon energy
range 0.1 keV less than or similar to E-X less than or similar to 5 Me
V, has a power-law spectrum with energy index 0.5 and X-ray luminosity
that depends on the backflow current and is typically similar to 10(3
3) ergs s(-1). (2) The pairs ultimately form a cyclotron resonance ''b
lanket'' surrounding the star except for two holes along the open fiel
d line bundles that pass through it. In such a blanket, the gravitatio
nal pull on e(+/-) pairs toward the star is balanced by the hugely amp
lified push of outflowing surface-emitted X-rays wherever cyclotron re
sonance occurs. Because of it, the neutron star is surrounded by a lea
ky ''hohlraum'' of hot blackbody radiation with two small holes, which
prevents direct X-ray observation of a heated polar cap of a gamma-ra
y pulsar. Weakly spin-modulated radiation from the blanket together wi
th more strongly spin-modulated radiation from the holes through it wo
uld then dominate observed low-energy (0.1-10 keV) emission. For non-g
amma-ray pulsars, in which no such accelerators with their accompanyin
g extreme relativistic backflow toward the star are expected, opticall
y thick e(+/-) resonance blankets should not form (except in special c
ases very close to the open field line bundle). From such pulsars, bla
ckbody radiation from both the warm stellar surface and the heated pol
ar caps should be directly observable. In these pulsars, details of th
e surface magnetic field evolution, especially of polar cap areas, bec
ome relevant to observations. The models are compared to X-ray data fr
om Geminga, PSR 1055-52, PSR 0656 + 14, PSR 1929 + 10, and PSR 0950 08.