Ion Implantation and Synthesis of Materials - Studium

2.6 Screening Functions 17−3.8860.772⎧⎪⎛ r ⎞ ⎫⎪χ ( r) = ⎨1 + ⎜.2/3 ⎟ ⎬⎪⎩⎝12⎠ ⎪⎭(2.9)Lindhard and coworkers proposed two somewhat simpler and more approximativeThomas–Fermi screening functions given byrχ ( r) = 1 −(3 + r )2 1/2(2.10)andχ ( r) = 1−1/2.r(2.11)The Moliere form of the screening function is composed of three exponentials:χ ( r) = 7 pexp( − qr) + 11pexp( − 4 qr) + 2 pexp( −20 qr).(2.12)The screening function for a Bohr atom with x = r/a TF isχ ( x ) = exp( −x),Bohr(2.13)and the screening function for a Lenz–Jensen atom isχ ( r) = [0.7466 exp( − 1.038 r)] + [0.2433exp( −0.3876 r)]LJ+ [0.01018 exp( −0.206 r)].(2.14)While classical models can be used to provide basic insight into screened interatomicpotentials, the use of quantum mechanically derived charge distributionsbased on the Hartree–Fock atomic model create significant changes in the detailsof the interaction potential. In an attempt to find an analytic function that accuratelypredicts the interatomic potential between atoms, Ziegler, Biersack, andLittmark (ZBL; 1985) extended an earlier study made by Wilson et al. (1977), performingdetailed calculations of solid-state interatomic potentials for 261 atompairs. The details of the calculations can be found in Ziegler et al. (1985). The calculatedtotal interaction potential was used together with the screening function,(2.8), to produce the universal screening functionχU= 0.1818 exp( − 3.2 x) + 0.5099 exp( −0.9423 x)+ 0.2802 exp( − 0.4028 x) + 0.02817 exp( −0.2016),(2.15)