An eikonal expansion is developed in order to provide systematic corrections to the eikonal approximation through order 1/k^2, where k is the wave number. The expansion is applied to wave functions for the Klein-Gordon equation and for the Dirac equation with a Coulomb potential. Convergence is rapid at energies above about 250 MeV. Analytical results for the eikonal wave functions are obtained for a simple analytical form of the Coulomb potential of a nucleus. They are used to investigate distorted-wave matrix elements for quasi-elastic electron scattering from a nucleus. Focusing factors are shown to arise from the corrections to the eikonal approximation. A precise form of the effective-momentum approximation is developed by use of a momentum shift that depends on the electron's energy loss.
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机译：为了通过1 / k ^ 2阶对系统的近似值进行系统的校正，开发了系统的展开式，其中k是波数。该扩展适用于Klein-Gordon方程和具有库仑电势的Dirac方程的波动函数。能量在约250 MeV以上时会迅速收敛。对于原子核的库仑势的简单分析形式，可以获得固有波函数的分析结果。它们被用于研究畸变波矩阵元素，以用于从原子核的准弹性电子散射。聚焦因子被证明是从对真实近似值的校正中产生的。有效动量近似的一种精确形式是通过使用依赖于电子能量损失的动量位移来开发的。
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