# (p.387) Appendix D The Differential Cross-Section of a Binary Collision

# (p.387) Appendix D The Differential Cross-Section of a Binary Collision

In the derivation of the Boltzmann equation in Chapter 7, we introduced the cross-section related to the binary collisions to transform Eq. (7.46) into Eq. (7.48). As underlined in the derivation, the collisions between two particles are computed neglecting the interaction with the other particles in the system, i.e. as if the two particles were isolated. In this Appendix, we explain in more details the binary collision process and the meaning of the relation in Eq. (7.47).

Let us start from the Hamiltonian of a system of two identical particles

where ${p}_{i}=|{\mathbf{\text{p}}}_{i}|$. Making the canonical transformation of coordinates defined in (Eqs. 7.35) and (7.36), here rewritten as

the Hamiltonian becomes

where $\mathbf{\text{P}}$ and $\mathbf{\text{p}}$ are the momenta canonically conjugated to $\mathbf{\text{Q}}$ and $\mathbf{\text{q}}$, respectively. We therefore have the free motion of a particle of mass $2m$, the centre of mass, plus the dynamics of a particle of mass $m/2$ (the reduced mass) subject to the potential *V*. Let us then concentrate on the motion of a particle of mass $\mu =m/2$ in a central potential. We know that such a motion develops in a plane. Using polar coordinates $(r,\mathrm{\theta})$ in this plane, the energy of this mechanical problem is

The angular momentum $\mathrm{\ell}=\mu r\dot{\mathrm{\theta}}$ is a constant of the motion. Substituting in the expression of the energy, we get (p.388)

showing that the problem is equivalent to a one-dimensional motion in the effective potential $V(r)+{\mathrm{\ell}}^{2}/(2\mu {r}^{2})$. We are interested in the case of open orbits, which occur when $E>0$. Assuming, without loss of generality, that the particle approaches the origin from $r=\mathrm{\infty}$ at the angle $\theta =0$, the general solution, expressing *θ* as a function of *r* during the approach, is given by (Goldstein, 1980)

The distance of closest approach, *r*_{m}, is the largest positive root of the expression under square root in the denominator in Eq. (D.7); if we denote by ${\theta}_{m}$ the corresponding value of *θ*, the angle of deflection of the particle, i.e. the angle between the direction of approach and the direction along which the particle goes back to $r=\mathrm{\infty}$, is simply related to ${\theta}_{m}$. Again, without loss of generality, we can assume that ℓ is positive; then *χ* is given by $\mathrm{\pi}-2{\theta}_{m}$, i.e.

Both the energy *E* and the angular momentum ℓ are a simple function of the impact parameter *b* and of the modulus *v* of the velocity of the incoming particle at infinity. We have

Substituting in Eq. (D.8), we get

This expression determines *χ* as a function of *b* and *v*; for most of the potentials $V(r)$ of interest, this function, for given *v*, is a monotonic decreasing function of *b*.

Let us now recall the definition of the differential cross-section $\text{d}\mathrm{\sigma}/\text{d}\mathrm{\Omega}$ in a scattering problem from a centre of force. If there is a flux *I* of impinging monoenergetic particles (of energy $\mathrm{\epsilon}$ determined by their velocity of incidence *v*) per unit area and per unit second, they will be scattered to different angles according to their impact parameter. The number $\mathcal{N}(\mathrm{\Omega})\text{d}\mathrm{\Omega}$ of incident particles scattered per unit second into the solid angle element $\text{d}\mathrm{\Omega}$ about the direction *Ω* defines the differential cross-section by
(p.389)

On the other hand, we have just seen that the angle of deflection is uniquely determined by the impact parameter *b* and by *v*. Considering the cylindrical symmetry of the problem, we then have

where *ϕ* is the azimuthal angle defining, together with *r* and *θ*, the cylindrical coordinates that describe the scattering. We obtain therefore Eq. (7.47):