Physics 301 7-Dec-2005 29-1The Maxwell Velocity DistributionThe beginning of chapter 14 covers some things we’ve already discussed. Way back inlecture 6, we calculated the pressure for an ideal gas of non-interacting point particles byintegrating over the velocity distribution. In lecture 5, we discovered t he Maxwell velocitydistribution for non-interacting p oint particles. In homework 2, you worked out some ofthe velocity moments for t he Max well distribution and you also worked out the distributionof velocities of gas molecules emerging from a small hole in an oven. Rather than derivethe earlier results again, we’ll just summarize them here for convenient reference.First, of all, let f(t, r, v) d3r d3v be the number of molecules with position vectorin the small volume element d3r centered at position r and velocity vector in the smallvelocity volume d3v centered at velocity v at time t. The function f is the distributionfunction (usually, we’ve assumed it to be a function of the energy, but we’re working upto bigg er and better things!). When we have a Maxwell distribution,fMaxwell(t, r, v) = nm2πτ3/2e−m(v2x+ v2y+ v2z)/2τ,where m is the molecular mass, and n is the concentration. The Maxwell distribution doesnot depend on t nor r. Integrating d3v over all velocities produces n, a nd integrating d3rmultiplies by the volume of the container, and we get N, the total number of particles. Ifwe want the probability distri butio n for just the velocity components, we just leave off then (and then we also don’t consider it t o be a function of r).Sometimes, we want to know the probability distribution for the magnitude of thevelocity. We change variables from vx, vy, vzto v, θ, φ, and dvxdvydvz= v2dv sin θ dθ dφ.We integrate over the angles, pick up a factor of 4π, and the probability distri butio n for visP (v) dv = 4πm2πτ3/2e−mv2/2τv2dv .With this distribution, one finds the most probable speed (peak of the probability distri-bution) isvmp=r2τm= 1.414rτm.The mean speed (denoted by ¯c by K&K) is¯c = hvi =r8τπm= 1.596rτm.The root mean square speed (which appears in the energy) isvrms=phv2i =r3τm= 1.732rτm.Copyrightc 2005, P rinceton University Physics Department, Edward J . GrothPhysics 301 7-Dec-2005 29-2Cross SectionsThe next topic discussed in K&K is the mean free path. We’re going to use this asan ex cuse to discuss cross sections and develop an expression for reaction rates based oncross sections and the Maxwell velocity distribution. The first thing we need to do is tosee where a cross section comes from a nd what it means.Consider a situation in which two particlescan undergo some sort of interaction when they“collide.” The interaction may be probabilis-tic; examples might be a nuclear or chemicalreaction, an elastic scatt ering, or an inelasticscattering (which means one or both particlesmust be in an excited state after the interac-tion). Suppose one has a stationary target par-ticle, and one shoots other particles at it withvelocity v. The concentration of the incidentparticles (number per unit volume) i s n. Theflux density of incident particles, or the particle current, is the number crossing a unit areain a unit time and this is just nv. Note that this has the dimensions of number per unitarea per unit time. Now the number of interactions per second must be proportional tonv. If we double the density, we have twice as many particles per second with a chance tointeract. Similarly, if we double the velocity, particles arrive at twice the rate and we havetwice as many particles per second with a chance to interact. Actually, the “strength” ofthe interaction may depend on the relative velocity, so it’s not strictly t rue that the rate isproportional to v, but in a simple process, like the collision of hard spheres, it’s certainlytrue. We take as a starting point that the interaction rate is proportional to the velocity,and effects having to do wit h the energy of the collision are included in the proportionality“constant.”The i nteraction rate i s thenR = σ(v)nv ,where σ is NOT the entropy. Instead it’ s the proportionality constant and is called thecross section. We’ve explicitly shown that it might depend on v. There can be morecomplicated dependencies. For example, if the incident and target particles have spin, thecross section might depend on the spins of the particles as well as the relative velocity.The proportionality constant is called a cross section because it must have t he dimensionsof an area. R has the dimensions of a number ( of interactions) per unit time, whil e nvhas dimensions of number per unit area per unit time. The cross section can be thoughtof as an area that the target particle “holds up,” perpendicular to the incoming beam. Ifan incident parti cl e hits this area presented by the target, an interaction occurs.Now some jargon. It’s often the case that one considers a scattering, in which case thequestion asked is how many particles per second scatter into the solid angle dΩ centered onCopyrightc 2005, P rinceton University Physics Department, Edward J . GrothPhysics 301 7-Dec-2005 29-3the direction (θ, φ)? In this case, the number is proportional to dΩ, and the proportionalityconstant is often written as a differential so the rate into dΩ isR(→ dΩ) =dσdΩnv dΩ ,where dσ/dΩ is called t he differential cross section. If one integrates over all solid angl esto find the rate for all interactions (no mat ter what the scattering direction), one hasσ =ZdσdΩdΩ ,and σ is called the total cross section in this case. Of course, a reaction (not just ascattering) may result in particles headed into a solid angle dΩ, so reactions may also bedescribed by differential and tota l cross sections.If one knows the forces (the interaction) between the incident and target particles,one may calculate the cross sections. Cross sections may a lso be measured. One uses atarget which has many target particles. One counts the scattered particles or the outgoingreactants using particle detectors. The total number of events is just N nvσ(v)t where Nis the number of target particles exposed to the incident beam and t is the duration of theexperiment. This assumes that the chances of a single interaction are so small that thechances of interactions with two target particles are negligible. Of course, sometimes thisisn’t true and a multiple scattering correction