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A method is developed to derive expressions for n-photon interband transition rates with arbitrary n, in which the real crystal energy spectrum is taken into account. For particular cases the results of the theory under consideration coincide both with the perturbation theory and Semiclassical treatment [13 to 161. The effect of the radiation coherent properties on the multi-photon processes is discussed. Contrary to previous calculations it is possible to give a qualitative explanation of the experimental data [a, 61. Es wird eine IntroductionNumerous experimental observation of interband multi-photon transitions in solids (see e.g. [l to 61) became possible due to the progress in laser technique. There are some reasons for the intensive investigation of multi-photon processes. The space pumping in solid state lasers is one of them. The second reason is connected with the study ofthe crystal energy structure. When the single-photon transition into an energy band is forbidden or this transition can be observed only in the far uv spectral region, multi-photon optics provides a useful tool for investigating this band. Furthermore, multi-photon processes have an essential influence on the kinetic properties of crystals in the presence of the strong laser radiation.Two different theoretical methods are usually applied to calculate multiphoton interband transition probabilities. The first one based on the usual perturbation theory was developed by Braunstein and Ockman and others. One can derive the n-photon transition probability in the n-th order perturbation theory. If, however, the number of photons involved is larger than 3, the number ofintermediate states being taken into account becomes enormously large making the probability evaluation practically impossible. Furthermore, the perturbation theory begins to break down when the radiation intensity is too large or when the radiation frequency is close to one of the crystal eigenfrequencies.The second methodis connected with the use of semiclassical and (or) adiabatic approximations, where a part of the electron-photon interaction is included into the zeroth-order Hamiltonian. Hence one can derive equations for n-photon transition probabilities with arbitrary n in the first order of the time-dependent

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