We studied experimentally and theoretically the perpendicular anisotropy and the stripe-domain structure in both ${\mathrm{Fe}}_{x}{\mathrm{Si}}_{1\ensuremath{-}x}$ thin films and ${\mathrm{Fe}}_{x}{\mathrm{Si}}_{1\ensuremath{-}x}/\mathrm{Si}$ multilayers, the latter being in the low-modulation-length regime $(0.4\mathrm{nm}<\mathrm{\ensuremath{\lambda}}<7\mathrm{}\mathrm{nm}).$ The experimental study was made by means of the transversely biased initial susceptibility ${\ensuremath{\chi}}_{t\ensuremath{\beta}}$ via the magneto-optic Kerr effect. The samples under study were prepared by dc triode sputtering at ${T}_{S}=300\mathrm{K}.$ It is found that the appearance of stripe domains is more pronounced for decreasing \ensuremath{\lambda} as $x$ remains constant and may be caused by both the increase in effective magnetic thickness and the reduction in effective magnetization as \ensuremath{\lambda} decreases. For multilayers with $\ensuremath{\lambda}=0.4\mathrm{nm},$ the observed field dependence of ${\ensuremath{\chi}}_{t\ensuremath{\beta}}^{\ensuremath{-}1}$ is similar to that found in homogeneous thin films when weak stripe-domain structures arise as a consequence of the existence of perpendicular anisotropy ${K}_{N}.$ We propose a quasistatic one-dimensional model to explain the behavior of ${\ensuremath{\chi}}_{t\ensuremath{\beta}}^{\ensuremath{-}1}$ when stripe domains are present, and we analyze the critical occurrence of stripe domains. We calculated the so-called pseudo-uniaxial anisotropy field ${H}_{\mathrm{Ks}},$ associated with the stripes, in two extreme cases: exchange-driven susceptibility or magnetic free poles (nonzero divergence in the bulk). The latter case agrees better with experiment. We found that perpendicular anisotropy is not exclusive of a well-defined multilayer structure; i.e., ${K}_{N}$ arises even when there are no interfaces in the volume. By setting the experimental saturation field ${H}_{s}$ (obtained by hysteresis loops) into our model, we obtain both the perpendicular anisotropy constant ${K}_{N}{=10}^{4}--{10}^{5}{\mathrm{J}/\mathrm{m}}^{3}$ and the critical thickness ${t}_{c}$ for the occurrence of a stripe-domain structure. Some possible sources of perpendicular anisotropy are discussed, for example, the associated isotropic compressive stress \ensuremath{\sigma}, whose contribution is found to be $|{K}_{N}{|}_{\mathrm{magnetoel}}\ensuremath{\approx}1.5--4.5\ifmmode\times\else\texttimes\fi{}{10}^{4}{\mathrm{J}/\mathrm{m}}^{3}.$
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