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Kinetic Theory and Transport Phenomena$

Rodrigo Soto

Print publication date: 2016

Print ISBN-13: 9780198716051

Published to Oxford Scholarship Online: June 2016

DOI: 10.1093/acprof:oso/9780198716051.001.0001

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(p.236) C Scattering processes

(p.236) C Scattering processes

Source:
Kinetic Theory and Transport Phenomena
Author(s):

Rodrigo Soto

Publisher:
Oxford University Press

C.1 Classical mechanics

C.1.1 Kinematics of binary collisions

Consider classical particles that interact through pairwise conservative forces that decay sufficiently rapidly at large distances. We are interested in describing the collision of a pair of particles that come from infinity with velocities c1 and c2. By energy conservation it is not possible for the particles to become captured in bound states. Therefore, the interaction lasts for a finite time, after which the particles escape again to infinity with velocities c1 and c2. For simplicity the case of particles of equal mass is considered; the extension to dissimilar masses is direct. Energy and momentum conservation imply that

(C.1)
c12+c22=c12+c22,c1+c2=c1+c2.

Defining the pre- and postcollisional relative velocities as g=c2c1 and g=c2c1, respectively, the general solution of eqns (C.1) is[1]

(C.2)
g=g(gnˆ)nˆ,

where nˆ is a unit vector that will be determined later in terms of the interparticle potential and the relative velocity, but up to now is arbitrary. If the particles interact through a central potential, nˆ points to the centre of mass. Noting that gnˆ=gnˆ, one can directly invert the relation (C.2) as

(C.3)
g=g(gnˆ)nˆ,

where we note that the same expression is recovered with the roles of g and g exchanged.

Following (Cercignani, 2013), the transformations gg [eqn (C.2)] and gg [eqn (C.3)] are characterised by the same Jacobian Jnˆ, which depends on the unit vector. The Jacobian of the combined transformation is hence Jnˆ2. However, as the combined transformation yields the initial relative velocity, the Jacobian must be one, from which we obtain that |Jnˆ|=1. Finally, using that the centre-of-mass velocity is conserved, we get that the Jacobian of the transformation from the initial to the final velocities is one; that is,

(C.4)
d3c1d3c2=d3c1d3c2,

an expression that is used in the derivation of the Lorentz and Boltzmann equations in Chapters 3 and 4.

(p.237) C.1.2 Geometrical parameterisation

The unit vector nˆ is determined by the geometry of the collision. In the centre-of-mass reference frame, the impact parameter b is defined as the distance from the projected unscattered trajectory to the origin. By conservation of energy, |g|=|g|, therefore angular momentum conservation implies that, for the postcollisional state, the impact parameter is also b. The deflection angle χ is related to ϕ0, which points to the point of minimum distance, by χ=|π2ϕ0| (Fig. C.1). Furthermore, ϕ0 is obtained by integrating the motion under the influence of the interparticle potential U (Landau and Lifshitz, 1976),

(C.5)
ϕ0=rmin(b/r2)dr1(b/r)2(2U/μg2),

where μ=m1m2/(m1+m2) is the reduced mass for particles of masses m1 and m2, which equals μ=m/2 for particles of equal mass.

C Scattering processes

Fig. C.1 Geometrical parameterisation of the scattering process in the centre-of-mass reference frame. The precollisional relative velocity g and the postcollisional one, g depend on the impact parameter b. The deflection angle is χ, and the unit vector nˆ points from the point of minimal distance to the origin, forming an angle ϕ0 with the horizontal axis.

If there is a beam of particles that approach the centre, it is useful to define the differential cross section dσ as the number of particles that are scattered between χ and χ+dχ divided by the particle flux n, which is assumed to be uniform. If there is a one-to-one relation between b and χ, the particles that will be deflected in the referred range of angles have impact parameters between b(χ) and b(χ)+db(χ). Now, by simple geometry, dσ=2πbdb, which using the chain rule gives dσ=2πb(χ)b(χ)χdχ. Finally, instead of the differential of the deflection angle, the cross section is usually referred to the element of solid angle, dΩ=2πsinχdχ, as

(C.6)
dσ=b(χ)sinχb(χ)χdΩ.

C.1.3 Scattering for hard sphere, Coulomb, and gravitational potentials

In the case of hard spheres of diameter D, the impact parameter and the deflection angle are related by b=Dsinϕ0=Dsin((πχ)/2). Replacing this expression in (C.6) gives

(C.7)
dσhs=(D2/4)dΩ;

that is, after the collision, particles are scattered isotropically. This result is used in the statistical modelling of hard sphere collisions, where the postcollisional relative velocity is selected from an isotropic distribution.

Coulomb and self-gravitating systems are characterised by a potential U=α/r, where the sign of α depends on the signs of the interacting charges for the Coulomb case and is always negative for the gravitational interaction. Using eqn (C.5), one gets, after some trigonometric transformations, ϕ0=arctanμg2b/α. The deflection angle therefore satisfies

(C.8)
tanχ2=αμg2b,

(p.238) which allows one to compute the differential cross section as

(C.9)
dσ=α2μg2sin2χ22dΩ.

Notably, this does not depend on the sign of α. This implies that, for Coulomb interactions, the statistical scattering properties do not depend on the signs of the charges.

For large separations, the deflection angle for gravitational interactions is χ2α/(μg2b). After the collision, the relative velocity acquires a transversal component, δg=gsinχ4Gmgb, where for particles of equal mass we have used that α=Gmm and μ=m/2. Finally, each of the colliding particles gets a transversal velocity component of

(C.10)
δc=gsinχ22Gm/gb,

where we have used that c2/1=V±g/2 and that the velocity of the centre of mass V is unchanged. This result is used to derive the dynamical friction properties for self-gravitating systems in Chapter 6.

C.2 Quantum mechanics

C.2.1 Time-dependent perturbation theory

The method used in this book to study transitions or scattering processes between quantum states is time-dependent perturbation theory (Messiah, 2014, Cohen-Tannoudji et al., 1992). Consider a system described by an unperturbed Hamiltonian H0 whose eigenstates |n and energies εn are known,

(C.11)
H0|n=εn|n.

Initially, the system is prepared in the pure state |i, and at time t=0 a perturbation in the Hamiltonian is switched on, such that it now reads,

(C.12)
H=H0+V(t),

where we write V(t)=ϵVˆ(t) (ϵ1) to indicate that the perturbation, which may depend on time, is small.

Our aim is to determine how the system evolves and the probability that, after some time, the system will be in a state |k, different from the initial one. We assume that the perturbation is weak enough that the Hilbert space is unchanged and that the state of the system at any time |ψ(t) can be written in terms of the unperturbed states,

(C.13)
|ψ(t)=jcj(t)|j.

Inserting this expansion into the Schrödinger equation, iddt|ψ=H|ψ, results in

(C.14)
iddtck=εkck+ϵjk|Vˆ(t)|jcj,

(p.239) where we have used the orthogonality of the eigenstates, k|j=δjk.

We now look for perturbative solutions of the previous equation. We first note that, in the absence of the perturbation, the amplitudes are described simply by ck(0)(t)=eiεkt/ck(0)(0), which suggests that it will be convenient to perform the change of variables,

(C.15)
ck(t)=eiεkt/bk(t),

and it is expected that, for ϵ1, the amplitudes bk will evolve slowly in time.[2] In this case, we can expand

(C.16)
bk=bk(0)+ϵbk(1)+,

where now the time derivatives of bk are of order 1. Replacing this expansion in (C.14) gives

(C.17)
ddtbk(0)=0,
(C.18)
ddtbk(1)=ijk|Vˆ(t)|jei(εkεj)t/bj(0).

As expected, the zeroth-order amplitudes state that the initial condition is preserved, bk(0)=δik, where |i is the state in which the system was initially prepared. At first order, therefore, we get

(C.19)
bk(1)(t)=i0tk|Vˆ(s)|iei(εkεi)s/ds,

from which the probability of finding the system in the state |k, |bk|2, is readily obtained.

C.2.2 Fermi golden rule

Two cases are of particular interest. The first consists of a constant perturbation Vˆ. In this case,

(C.20)
bk(t)=iϵk|Vˆ|i0tei(εkεi)s/ds=ik|V|iei(εkεi)t/1i(εkεi)/,

where we have replaced ϵVˆ=V. The probability of finding the system in a state |k at a time t after the perturbation was switched on is then

(C.21)
pk(t)=|bk(t)|2=k|V|i22sinεkεi2tεkεi22.

When the final states are discrete, oscillations take place. However, the picture is completely different when they are continuous or quasicontinuous, as is the case for quantum gases studied in Chapters 7, 8, and 9. In order to continue working with normalised functions, we consider finite volumes where the quantum states are quasicontinuous. The (p.240) probabilities pk are factored as a term that depends explicitly on the final state times the function F((εkεi)/,t), which depends only on the energy of the final state. For quasicontinuous states we can label them as |k=|ε,μ, where μ groups all the other quantum numbers for a given energy (for example, the direction of the momentum for free particles). Now, the probability of transit to a state in a window of ε and μ is

(C.22)
p(t)=μg(μ,ε)ε,μ|V|i22F((εεi)/,t)dε,

where g is the density of states and we have used the general rule, k=μg(μ,ε)dε. For large times, the function F is peaked for ε close to εi, and indeed it becomes a Dirac delta function, limtF((εεi)/,t)=2πtδ(εεi) (see Exercise C.3), which allows us to write,

(C.23)
p(t)=μg(μ,ε)ε,μ|V|i22πtδ(εεi)dε.

The probability increases linearly with time, allowing us to define the probability transition rate from the initial state |i to a quasicontinuous state |k as

(C.24)
Wik=2πk|V|i2δ(εkεi).

This important result, which states that, under constant perturbations, transitions can only take place between states with equal energy, is known as the Fermi golden rule.

The second case of interest is that of an oscillatory perturbation. As we are considering up to linear order in perturbation theory, it suffices to deal with monochromatic perturbations, V(t)=Vωsin(ωt) or V(t)=Vωcos(ωt). Both cases give, for long times, the transition rate,[3]

(C.25)
Wik=πk|Vω|i2δ(εkεiω)+δ(εkεiω).

This states that, under oscillatory perturbations, transitions can occur only between states that differ in energy by one quantum ω. The first term gives εk=εi+ω, meaning that the system absorbs one quantum from the field, while for the second term, εk=εiω, one quantum is emitted to the field. Note that both processes present equal transition rates.

Exercises

  1. (C.1) Collision rule for different masses. Show that, in a binary collision of particles with different masses m1 and m2, the collision rule (C.2) is also satisfied. To show this, it is convenient to write the (p.241) energy and momentum conservation equations in terms of the relative and centre-of-mass velocities.

  2. (C.2) Oscillatory perturbation. Derive eqn (C.25) for the perturbation V(t)=Vωsin(ωt).

  3. (C.3) Dirac delta function. Consider the function f(x,t)=1πtsin(xt)x2. Show that, as a function of x, it is normalised to unity and that when t it becomes concentrated around the origin, therefore becoming a Dirac delta distribution (see Fig. C.2).

C Scattering processes

Fig. C.2 The function f(x,t)=1πtsin(xt)x2 converges to a Dirac delta function for large t.

Notes:

(1) See Exercise C.1 to show that this relation also holds for particles with different masses.

(2) Indeed, this can be formally proved using multiple time scale methods (Cohen-Tannoudji et al., 1992).

(3) See Exercise C.2.