The equation of state for ultra-dense matter can be tested from observations of the ratio of mass to radius of neutron stars .
This could be measured precisely from the redshift of a narrow line produced on the surface .
X-rays bursts have been intensively searched for such features , but so far without detection .
Here instead we search for redshifted lines in the persistent emission , where the accretion flow dominates over the surface emission .
We discuss the requirements for narrow lines to be produced , and show that narrow absorption lines from highly ionized iron can potentially be observable in accreting low mass X-ray binaries ( low B field ) which have either low spin or low inclination so that Doppler broadening is small .
This selects Serpens X-1 as the only potential candidate persistent LMXB due to its low inclination .
Including surface models in the broad band accretion flow model predicts that the absorption line from He-like iron at 6.7 keV should be redshifted to \sim 5.1 ~ { } \mathchar 45 ~ { } 5.7 keV ( 10 - 15 km for 1.4 \mathrm { M _ { \odot } } ) and have an equivalent width of 0.8 ~ { } \mathchar 45 ~ { } 8 eV for surface temperatures of 7 ~ { } \mathchar 45 ~ { } 10 \times 10 ^ { 6 } K. We use the high resolution Chandra grating data to give a firm upper limit of 2 ~ { } \mathchar 45 ~ { } 3 eV for an absorption line at \sim 5 keV .
We discuss possible reasons for this lack of detection ( the surface temperature and the geometry of the boundary layer etc . ) .
Future instruments with better sensitivity are required in order to explore the existence of such features .