Scattering Experiments and Classical Theory of Atom–Atom Scattering

1.1 Crossed Atomic and Molecular Beams

We wish to investigate how atoms and molecules interact with each other in the most fundamental manner. What happens when they closely approach each other or collide? As molecules are far too small to see and the time-scale of their collisions is very short, we resort to making them collide with each other under strictly controlled conditions and examining the results in as much detail as we can. Figure 1.1 shows a schematic of a crossed atomic beam apparatus.

To the left hand side of the diagram there is a sophisticated analyzer which detects the products of the collision. The "quadrapole mass filter" selects products of a defined mass and the chopper permits the determination as to when the products arrive at the detector. From this it will be possible to determine the speed and kinetic energy of the products. The detector assembly remains fixed as it is the most complex component of the apparatus. To the lower right hand side of the diagram are the two atomic or molecular beam sources. These are at some fixed angle, normally 90°, relative to each other and can be rotated with respect to the detector assembly. This permits the measurement of the angular variation of the reaction probability (i.e. the differential cross section).

The heart of the apparatus is the collision region where the two beams cross and the collisions take place. This region is maintained under the greatest attainable vacuum so as to exclude any unwanted molecules.

1.2 Classical Theory of Atom–Atom Elastic Scattering

1.2.1 Hard Sphere Collisions

Let us now consider what happens when two atoms collide. Figure 1.2 shows two atoms traveling towards each other. This is what a collision would look like in the center-of-mass reference frame. We will discuss the difference between the "laboratory frame" and the "center-of-mass" frame a little later. In the center-of-mass frame the collision partners travel directly towards each other. The distance "b" in the figure is called the impact parameter and is the hypothetical distance of what would be the closest approach of the centers of the two atoms if there were no interaction between them. If the atoms were hard spheres then they would either collide or miss each other entirely. They would collide if the impact parameter was smaller than the sum of the radii of the two spheres.

Figure 1.3 shows a collision in which the two atoms just graze each other. If the impact parameter were any larger, then the atoms would miss each other entirely and there would be no collision. The maximum value of the impact parameter for a collision to occur is the sum of the radii of the two atoms, b = r1 + r2. If the projected path of atom 1 lies anywhere within the area of the circle shown in Figure 1.3 then a collision will take place. The area of this circle is called the collision cross section, σ = πR2, where R = r1 + r2.

Let us now consider hard sphere collisions in greater detail. What is the angle of scattering of the atoms after the collision? Figure 1.4 illustrates what happens when two "hard sphere" atoms collide. When their surfaces touch they bounce off each other like two billiard balls. The angle that the direction of impact makes with the normal to the surface (α) is the same as that made by the recoil velocity direction to this normal. This is termed specular reflection. The "scattering angle" is shown as θ in Figure 1.4. From the figure we see that

b = R sin α, (1.1)

where R = r1 + r2, and also

θ + 2α = π, (1.2)

and therefore

b = R sin (π - θ/2) = R cos θ/2. (1.3)

All the atoms colliding with an impact parameter b to db are scattered into scattering angles θ to θ + dθ. The cross section for scattering into polar angles θ to θ + dθ is therefore

2πb|db| = 2πR (cos θ/2) | R (sin θ/2) dθ/2| = 1/2πR2 sin θ dθ. (1.4)

The absolute value signs have been introduced as dθ/db is negative and the cross sectional area is a positive quantity.

The "differential cross section" is defined as the flux of atoms scattered into a given solid angle (dΩ = sin θ dθ dΦ) divided by the incident flux of atoms. In the present case we have not considered the azimuthal angle, Φ. We have in effect integrated over this angle. The solid angle, integrated over all Φ, is 2π sin θ dθ. The differential cross section for scattering into a solid angle dΩ is therefore

dσ/dΩ = R2/4. (1.5)

The integral of the differential cross section over all angles must be equal to the total or integral cross section

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

1.2.2 Scattering Under the Influence of a Potential

Figure 1.2 shows a schematic of two atoms colliding. The position of each atom is described by a vector with three spatial components r1 and r2. Thus the system overall requires six position coordinates to fully describe it. As the forces determining the scattering of the atoms depend on the magnitude of the distance between the two atoms, r = |r2 - r1|, it seems reasonable to change coordinates to r, the position of atom 2 relative to atom 1, and R, the position of the center-of-mass

r = r2 - r1, R = m1r2 + m2r1/m1 + m2 (1.7)

Classical mechanics is based on Newton's laws of motion and these are generally expressed in terms of the Cartesian (i.e. x, y, z) coordinates of each of...