We formulate new expressions for the scattering potential applicable to the elastic scattering of electrons by atoms and surfaces . The derivation is based on inclusion three forms of exchange - correlation /polarization potentials (HIP-PZ , mT-'VWN and fflP-PW ) . The accuracys of the formulated potentials are tested by : i- Comparing the calculated values of differential cross sections (DCS) and phase shifts (PS) to the reported experimental and theoretical data at 10 eV and 20 eV energies and over the angular range 20 °-120 ° for Ne , Ar or»d Mg atoms . For K atoms at 3.3 eV energy and over the angular range 10 °- 80 ° . And for Si atoms at 100 eV energy and over the angular range 0 °-180 ° . ii - Calculating the spin polarization for elastic electron scaltering from \Vatoms at 100 eV for initially unpolarized incident electron beam using two sets of wave function for the solutions of Roothaan Hartree - Fock equation . The results are compared with the published theoretical data ; the comparison gives: * the best agreement for the first set of wave function ( errors ~ 14%) using Hara exchange potential with ionization potential and Perdew - Wang correlation potential Hl-PW. ** identical results for the both sets of wave functions using Slater exchange potential. iii - Comparing the calculated values of the scattering intensities of MgO (100) surface for the energy range 20-140 eV to the experimental data for (00) electron beam direction at off normel incidence. This comparison is presented for four relaxation of top layer (0 . ± 3% , - 6% ) and for the (10) electron beam direction at normal incidence . These calculations are carried out by using HIP - PW and Slater potentials . Relaxation value (- 3 %) is found to be optimum for HEP - PW potential. iv - Comparing the calculated values of spin polarization profiles versus energy of the 00 beam far the unpolarized primarily beam at 0 = 11° to the theoretical and experimental reported data for W(100) surface . This comparison is conducted for two values of relaxation of the top layer (0 - -12%), using HIP -PW potential and by using two sets of wave function . Relaxation value of (-10%) is found to be optimum for this type of potential. Finally , we present determinated spectrums of dielectric of property e* for Si(111)2xl , Si(111)7x7 , Si(l00) and Si(110) surfaces , obtained by using the published surface differential reflectivity data (SDR) and non - linear least squares (NL - LSQ) method . The computed values of e2 are compared with reported theoretical data in the range 0.36 - 3.3 ev for Si(111)2x1 surface .