The theory of resonant scattering of synchrotron radiation by vibrating nuclei in a crystal lattice with absorption or emission of phonons is developed under the conditions of incoherent scattering of x-rays over various nuclei. Consistent with previous works, we use the traditional approach based on Maxwell's equations and time-dependent quantum mechanics. A concept of coherence in scattering processes, properties of synchrotron radiation and conditions of measurements are analysed. The

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... ns for energy dependence and time evolution of scattered radiation are derived in detail in the case of the nonsplit nuclear levels. The effect of hyperfine splitting on time evolution is also discussed. The time evolution of the scattered radiation soon after the primary flash depends on the temporal properties of the monochromator and has a more complicated form than the exponential nuclear decay. Later, the monochromator does not influence the time evolution. The energy dependence of space-integrated nuclear inelastic scattering is analogous to that of nuclear inelastic absorption.