A new way for the development of an x-ray quarter-wave plate is suggested. The idea is based on an effect of total external reflection from an anisotropic crystal in an anomalous dispersion region. It is found that in the case of reflection of plane polarized synchrotron radiation from a uniaxial crystal near the total external reflection angle the reflected radiation can be circularly polarized (right or left handed). The reflected beam polarization is calculated for a hexagonal BN crystal near the boron K absorption edge and for a graphite crystal near carbon K absorption edge. It is shown that the method developed gives a simple way to construct an effective x-ray quarter-wave plate. [S0031-9007(98)05347-2] PACS numbers: 78.70.Ck, 07.85.Fv, 78.20.Ci, 78.70.DmExperiments on polarization dependence of x-ray reflectivity fine structure (XRFS) in hexagonal boron nitride (h-BN) crystal [1,2] and on orientational dependence of x-ray absorption near edge structure (XANES) in graphite crystal [3] prove that an appreciable anisotropy of permittivity tensor (PT) in soft x-ray region exists in lowsymmetry crystals when the photon energy is close to the absorption edge. In this case PT depends appreciably on the chemical state and environment of the scattering atoms. Earlier we have shown [4,5] that this is caused by scattering of the photo or virtual electrons from the neighboring atoms (in the case of the x-ray elastic scattering process photo electrons do not appear in the final state, but the virtual electrons exist in the intermediate state). Because of photo and virtual electron interactions with surrounding atoms the PT becomes anisotropic. This anisotropy has a sharp photon energy dependence in the anomalous dispersion region. Our previous calculations of both and XANES in graphite crystal [6] appear to be in reasonable agreement with experimental spectra.Such an anisotropy of the PT leads not only to polarization and orientation dependencies of x-ray spectra but also to differences in fine structures of left and right-hand circularly polarized radiation in both reflectivity [5] and absorption [7] x-ray spectra. Moreover, this anisotropy in combination with the total external reflection gives an opportunity to create an x-ray polaroid device [8,9].In this paper we suggest to use the effect of total external reflection of x-ray radiation for development of the efficient x-ray quarter-wave plate. The idea of the quarter-wave plate is based, first, on the difference between the critical angles of total external reflection for two different polarizations of the incident radiation in low-symmetry crystals in the anomalous dispersion region and, second, on the sharp dependence of the phase difference between reflected and incident beam from the glancing angle near the total external reflection region.