Sigma Program - Lennard-Jones Potential
Experiments have shown that the collision cross section depends fairly strongly on temperature even for a rigid structure like fullerene C60. This phenomenon is due to the nature of the ion-helium interaction, which is attractive at long distances and repulsive at very short distances. The model potential used in the sigma program (in Lennard-Jones mode) to describe the interaction between a specific atom in the polyatomic ion and a buffer gas helium atom (separated by a distance R) is described by the equation below.
V(R) = A [ B(r*/R)n - C(r*/R)6 - D(r*/R)4]
A = n E* / [n(3+γ)-12(1+γ)]
B = 12(1+γ)/n
C = 4γ
D = 3(1- γ)
In this (n,6,4)
potential, E* and r* are the depth and position of the potential
well, respectively, n is the exponent describing the ion-neutral
repulsion, and γ is a dimensionless parameter defining the
relative contributions of the R-6 and R-4
V4(R) = - (q2α) / (2R4)
The remaining part, Vn,6(R) = V(R) - V4(R), has the form of an (n,6) Lennard-Jones (LJ) potential and can be expressed by using the LJ parameters r and E (LJ well position and depth) as shown below.
Vn,6(R) = (nE)/(n-6) [ (6/n) (r/R)n - (r/R)6 ]
parameters n, r, E, and q, the potential V(R) is
completely defined, and the parameters γ ,
r*, and E* can be determined.
r(N) = r(60) (0.86882 - 0.99427 N + N 0.99913)
This equation was obtained by a three-parameter fit to hundreds of experimental data points for ions with 11 to 170 atoms. Presently (since 1999) we are using the ion-helium LJ parameters r(60) and E shown in the following table optimized for a (12,6,4) potential. In addition, the table has values for r(20) and r(160) obtained by using the formula for r(N) above.