We describe the construction and operation of an x-ray beam size monitor (xBSM), a device measuring $e^+$ and $e^-$ beam sizes in the CESR-TA storage ring using synchrotron radiation. The device can measure vertical beam sizes of $10-100~mu$m on a turn-by-turn, bunch-by-bunch basis at $e^pm$ beam energies of $sim2~$GeV. At such beam energies the xBSM images x-rays of $epsilonapprox$1-10$~$keV ($lambdaapprox 0.1-1$ nm) that emerge from a hard-bend magnet through a single- or multiple-slit (coded aperture) optical element onto an array of 32 InGaAs photodiodes with 50$~mu$m pitch. Beamlines and detectors are entirely in-vacuum, enabling single-shot beam size measurement down to below 0.1$~$mA ($2.5times10^9$ particles) per bunch and inter-bunch spacing of as little as 4$~$ns. At $E_{rm b}=2.1 $GeV, systematic precision of $sim 1~mu$m is achieved for a beam size of $sim12~mu$m; this is expected to scale as $propto 1/sigma_{rm b}$ and $propto 1/E_{rm b}$. Achieving this precision requires comprehensive alignment and calibration of the detector, optical elements, and x-ray beam. Data from the xBSM have been used to extract characteristics of beam oscillations on long and short timescales, and to make detailed studies of low-emittance tuning, intra-beam scattering, electron cloud effects, and multi-bunch instabilities.
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