In the following we are going to use a classical description of trajectories. In the case of electronic excitation collisions, often it is possible to consider the trajectory classically and to deal with the electrons quantum mechanically. This is called the semi-classical approximation.
We are now going to see how we can obtain the scattering function J(b), giving the scattering angle J in a binary collision as a function of parameter b, for a given interatomic potential function, U(r). But first, we need to discuss interatomic potentials.
Interatomic Potentials
The potential energy between two atoms, U(r) is a complex function of interatomic distance r, due to the polarization effects, changes in the electron wavefunctions from centered in one atom (atomic) to centered around two atoms (molecular), plus the complexities due to the shell structure. In addition, the potential depends on collision velocity due to the inability of electrons to accommodate to the temporal variation of the interaction. In particular, electrons can be excited or ejected during the collision altering the screening of the nuclei. The full quantum-mechanical calculation has been done only for very few cases, and only in the limit of stationary atoms. More typically, approximations are made. At high collisional (CM) energies, one is often interested in the more violent collisions, those where the nuclei approach closely. In this case, one can safely neglect attractive forces and treat elastic scattering as due to a purely repulsive potential.
The simplest case of repulsive potential is the Coulomb potential between two nuclei of charge Z1 and Z2.
The simplest way to include the effect of the electrons around the nuclei is to consider the potential to be a product of a Coulomb potential times a screening function f (r).
For some types of functions f(r) one can calculate the scattering function J(b) analytically, and this was necessary some decades ago when computers where not available or convenient to use. The first screening function is that to Bohr:
where a is called the screening radius. The screening of this function is too strong at large values of r. A more accurate potential is obtained from the Thomas-Fermi statistical description of the atom
but the screening function fTF(r/a) is not analytical. This precludes calculating analytically processes due to multiple collisions, like particle penetration, and collision cascades in solids. To solve this problem, the Thomas-Fermi function can be approximated in different regions of r by power functions such that the potential becomes
with values of n and Cn obtained by fits to the Thomas-Fermi U(r) in the internuclear range important for the collisions being considered. The merit of this approach is that it allows obtaining analytically J(b) that can be used in a multiple-collision calculation, like those using the Boltzmann transport equation.
An approximation useful at relatively low collision energies, is the Born-Mayer potential:
where A and b are constants.
Note that the Thomas-Fermi potential is universal in the scaling unit a, but strictly it is not accurate for values of Z lower than about 6 (carbon). The screening radius a decreases with the atomic number Z of the projectile and target atoms. Lindhard gives it as:
where a0 = 0.5292 Å is the Bohr radius. Firsov, fit numerical calculations to this more accurate formula.
An expression used in computer simulations, that fits the ZBL potential (see below), is:
The graph below shows a comparison between the Thomas-Fermi electron densities (from which the potential can be derived) and more accurate quantum-mechanical Hartree densities, obtained in the independent-electron approximation. It can be seen that the statistical Thomas-Fermi model does not account for the density oscillations that are due to the shell structure of the atoms. Note also that the Thomas-Fermi model overestimates the electron density, and therefore interatomic potentials, at large distances.
The graph below shows different forms of potentials that have been used over the years.
In the last decade, the ZBL potential (Ziegler-Biersack-Littmark) has been preferred. This is a "universal" fit to reasonably accurate quantum calculations of interatomic potentials in the repulsive region and is a better description than other models. The ZBL potential is used in current TRIM Monte Carlo simulations.
Potentials suited for low energies
When the collision energy becomes comparable with binding energies between atoms (typically 1-10 eV), we cannot ignore the attractive part of the interaction in the potential. A popular potential with attractive and repulsive components is the Lennard-Jones or 6-12 potential:
At large distances, the second term dominates, the potential falls like r--6, which is the form for van der Wals forces. The exponent 12 in the repulsive part is just a convenient value for analytical calculations, and fairly realistic for non-polar molecules. This potential is reasonable adequate for the interaction between rare gas atoms but it is used commonly in computer simulations of arbitrary solids due to its simple form.
Many body potentials
In collisions with molecules, or in solids, a more accurate description is provided by many-body potentials. These recognize the fact that the potential between two atoms depends on the presence of other neighboring atoms. As an example, the potential between an O atom and the O on the SiO molecule will be different from that between two O atoms. It will also depend on the position of the Si. That is, the potential between O and the oxygen atom in SiO is not spherically symmetric.
When we discuss molecular dynamics simulations, we will introduce many-body potentials (e.g., embedded-atom, Tersoff, Brenner potentials).
Classical Calculation of the scattering angle
The figure below shows trajectories of particles in the CM system scattering due to a central potential with repulsive and attractive components. Note that scattering occurs in a plane (the scattering plane), which is perpendicular to the angular momentum. The two-body problem can be reduced to a one body problem (see McDaniel notes).
During the collision, total energy is conserved, so the kinetic energy is decreased or increased in the same amount of the change of the potential energy U(r).
Conservation of angular momentum then leads to:
A solution always exists for a repulsive potential, and the largest real root is the distance of closest approach ra. The scattering angle in the CM system is given by:
Therefore, particles incident at impact parameter b, in an annular ring of area 2pb db are scattered into angular element dQ between Q and Q+dQ. Then, the differential scattering cross section Is(Q) dWCM, the effective area for scattering into Q and Q+dQ is related to the are of the annular ring by:
Since the same number of projectiles must be scattered in the LAB and CM system, the differential scattering cross sections must be the same:
From the relationship we have seem between the angles in the LAB and CM systems,
valid for elastic scattering with stationary atoms, that is independent of azimuth j.
The Subroutine MAGIC
Jochen Biersack developed a fast subroutine called MAGIC to calculate scattering angles within his TRIM simulation (TRIM = Transport of Recoils and Ions in Matter). It makes use of reduced variables introduced by Jens Linhard decades ago:
e (eps) reduced energy
e = E (a/14.4eV) (m/(m+M)) 1/Z1Z2
b impact parameter in units of
Input: CM energy in units of epsilon (eps), impact parameter b
Output: c2 = cos2Q, s2 = sin2Q
subroutine magic(eps,b,c2,s2)
r=b
rr=-2.7*alog(eps*b)
if(rr.lt.b) goto 330
rr=-2.7*alog(eps*rr)
if(rr.lt.b)
goto 330
r=rr
330
ex1=0.18175*exp(-3.1998*r)
ex2=0.50986*exp(-0.94229*r)
ex3=0.28022*exp(-0.4029*r)
ex4=0.028171*exp(-0.20162*r)
v=(ex1+ex2+ex3+ex4)/r
v1=-(v+3.1998*ex1+0.94229*ex2+0.4029*ex3+0.20162*ex4)/r
fr=b*b/r+v*r/eps-r
fr1=-b*b/(r*r)+(v+v1*r)/eps-1.
q=fr/fr1
r=r-q
if(abs(q/r).gt.0.001) goto 330
roc=-2.*(eps-v)/v1
sqe=sqrt(eps)
cc=(0.011615+sqe)/(0.0071222+sqe)
aa=2.*eps*(1.+(0.99229/sqe))*b**cc
ff=(sqrt(aa**2+1.)-aa)*((9.3066+eps)/(14.813+eps))
delta=(r-b)*aa*ff/(ff+1.)
co=(b+delta+roc)/(r+roc)
c2=co*co
s2=1.-c2
return
end
Homework
1) Compare the three formulas for the radius a given, in a three dimensional plot with axes: Z1, Z2, a, with a in Å and both atomic numbers ranging from 1 to 90.
2) Find Biersack's original paper when he discusses the Magic formula and write comments to the subroutine Magic.
3) Use MAGIC to calculate the relationship b(J) for H-Cu,
Si-Si, and Pb-C pairs, where the first atom is the projectile. Make three graphs, each with the three pairs, for laboratory energies 0.1 keV, 50 keV, and 10 MeV.
4) For the three pairs of problem 3, graph the distance of closest approach vs. projectile energy and vs. CM energy, for energies between 100 eV and 100 keV. Calculate enough points to have a smooth curve.