Calculation Of The Electron Density On The Surface Of Atoms And Research Of Internal Stress In Films Based On Cheng's Theory

Cheng Kaijia's improved Thomas-Fermi-Dirac theory (TFDC) is another profound exploration of electron theory since the appearance of density functional theory (DFT) and Yu's theory (EET). TFDC argued that the continuity of electronic density at atomic interface was a quantum mechanics condition of continuous wave function. It has been applied preliminarily to the interpretation of the origin of interfacial stress in films and the calculation of the critical grain size with stable dislocations in nano-crystal. TFDC originated from Thomas-Fermi-Dirac (TFD) model. Numerical solutions of TFD equation based on the TFD model and electron density of TFD equation under boundary condition of non-isolated neutral atoms were research focuses for Cheng's theory researchers. Therefore, constructing numerical solutions of TFD equation and solving the electron density on the surface of atoms will be of great significance to promote development and application of TFDC theory.In this paper, based on the Feynman et al.′s calculated method, the classical fourth order Runge-Kutta method was applied to solve TFD equation. The algorithm has high precision, easy convergence, stability, and conducive to good programming advantages. Using the corresponding Matlab program, the numerical solutions of the TFD equation for Cu, Fe, C and other twenty-four elements were solved. Based on the above numerical solution of the TFD equation, the boundary condition of non-isolated neutral atoms was quoted to calculate these twenty-four elements′electron density values on the surface of atoms. The solving process of electron density on the surface of atoms in the compound and solid solution was constructed and the electron density on the surface of atoms in Fe3C was calculated.Cu films were prepared on the surface of Ni substrate by using DC electro- deposition method. Deflections of cantilever consisted of Cu film and Ni substrate were measured in situ by using cantilever beam measuring device. Based on measured deflection and formula of Berry, average internal stresses (experiment values) in Cu film were calculated. The distribution of internal stresses was also analyzed according to the average internal stress in Cu films. The average internal stresses (theoretical values) resulted from interface stresses were evaluated based on TFDC theory and elastic mechanics principle. The results showed that theoretical values agreed well with experiment values (Both were compressive stresses and the values were close to each other). The usefulness of internal stress calculation model according to TFDC theory was preliminarily validated.