For high intensity x-ray sources, the investigations of the nonlinear x-ray effects become essential. The x-ray third-order nonlinear asymmetric transmission case dynamical diffraction is investigated theoretically in perfect crystals. The three essential input parameters are taken into account: the deviation from the Bragg exact orientation, the asymmetry angle of reflecting atomic planes and the intensity of incident wave. The third-order nonlinear equations for the asymmetrical dynamical diffraction and two integrals of motion are presented. The exact solutions for the wave amplitudes, presented via Jacobi elliptic functions, have been found. The solutions are periodic functions of the depth. It is shown that the behavior of the waves inside the crystal is determined by the sign of a combined parameter, which includes the three input parameters, mentioned above. The regions, where this parameter is positive, negative or zero, are found. For positive values, the energy is concentrated both in the transmitted and diffracted waves while for the negative values, the energy is concentrated mainly in the transmitted beam. It is found that when this combined parameter is zero, the amplitudes are elementary periodic or non-periodic functions. For the periodic solutions, the nonlinear Pendellsung effect takes place, i.e. the transmitted and diffracted waves periodically exchange with their energies. For the period, called an extinction length, an exact expression is found. It is shown that for large values of the asymmetry factor, the observation of the nonlinear diffraction effects can be realized for sufficiently low intensities of the incident wave. The obtained results can be used for experimental investigation of the nonlinear Bragg diffraction and for preparation of high intensity x-ray beams with given parameters. The expression for the nonlinear extinction length allows to estimate the third-order nonlinear susceptibilities of crystals by measuring the extinction length in experiments.

对于高强度 X 射线源，非线性 X 射线效应的研究变得必不可少。理论上研究了完美晶体中的 X 射线三阶非线性不对称透射壳动态衍射。考虑了三个基本输入参数：与布拉格精确取向的偏差、反射原子平面的不对称角和入射波的强度。给出了非对称动力学衍射和二次运动积分的三阶非线性方程。已经找到了通过雅可比椭圆函数呈现的波幅的精确解。解是深度的周期函数。结果表明，晶体内部波的行为由组合参数的符号决定，该组合参数包括三个输入参数，上文提到的。找到此参数为正、负或零的区域。对于正值，能量集中在透射波和衍射波中，而对于负值，能量主要集中在透射波中。发现当该组合参数为零时，幅度是基本周期或非周期函数。对于周期解，会发生非线性 Pendellsung 效应，即透射波和衍射波会周期性地交换能量。对于称为灭绝长度的周期，可以找到一个确切的表达式。结果表明，对于较大的不对称因子值，对于足够低的入射波强度，可以实现非线性衍射效应的观察。所得结果可用于非线性布拉格衍射的实验研究和具有给定参数的高强度 X 射线束的制备。非线性消光长度的表达式允许通过测量实验中的消光长度来估计晶体的三阶非线性磁化率。