Physics 301 3-Dec-2003 29-1The Maxwell Velocity DistributionThe beginning of chapter 14 covers some things we’ve already discus sed. Way back inlecture 6, we calcul ated the pressure for an ideal gas of non-interacting point particles byintegrating over the velocity distribution. In lecture 5, we discovered the M axwell velocitydistribution for non-interacting point particles. In homework 2, you worked out some ofthe velocity moments for the Maxwell distribution and you also worked out the distribu tionof velocities of gas molecules emerging from a small hole in an oven. Rather than derivethe earlier results again, we’ll just summar ize them here for convenient reference.First, of all, let f(t, r, v) d3r d3v be the number of molecules with position vectorin the sma ll volume element d3r centered at position r and velo ci ty vector in the smallvelocity volume d3v centered at velocity v at time t. The fun ction f is the distributionfunction (usually, we’ve assumed it to be a function of the energy, but we’re working upto bigger a nd better things !). When we h ave a Maxwell distribution,fMaxwell(t, r, v) = nm2πτ3/2e−m(v2x+ v2y+ v2z)/2τ,where m i s the molecular mass, and n is the concentration. The Maxwell distribution doesnot depend on t nor r. Integrating d3v over all velocities produces n, and integrating d3rmultiplies by the volume of the container, and we get N, the total number of particles. Ifwe want the probability distribution for just the velocity components, we ju st leave off then (and then we also don’t consider it to be a function of r).Sometim es, we want to know the probability distribution for the magni tude of thevelocity. We chang e var iables from vx, vy, vzto v, θ, φ , and dvxdvydvz= v2dv sin θ dθ dφ.We integra te over the angles, pick up a factor of 4π, a nd the probability distribution for visP (v) dv = 4πm2πτ3/2e−mv2/2τv2dv .With this distribu tion, one finds the most probable speed (peak of the pro bability 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 2003, Princeton University Physics De partment, Edward J. GrothPhysics 301 3-Dec-2003 29-2Cross SectionsThe next topic discussed in K& K is the m ean free path. We’re going to use this asan excuse to d iscuss 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 and what it means.Consider a situation in which two particlescan un dergo some sort of interaction when they“collide.” The interaction m ay be probab ilis-tic; examples might be a nuclear or chemicalreaction, an elastic scattering, 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 o f the incid entparticles (number per unit volume) is n. Theflux density of incident particles, or the particle current, is the number crossing a unit areain a un it tim e an d this is ju st nv. Note tha t 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 doubl e 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 intera ction may depend on the relative velocity, so it’s not strictly true that the r ate isproportional to v, but in a simple process, like the collis ion of hard spheres, it’s certainlytrue. We take as a starting point that the interaction rate is propor tional to the velocity,and effects having to do with the energy of the collision are included in the proportionality“constant.”The interaction rate is 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. Fo r 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 the dimensio nsof an area. R has the dimensi ons of a number (of interactions) per unit time, while 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,” perpen dicular to the incoming beam. Ifan incident particle hits this area pres ented 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 2003, Princeton University Physics De partment, Edward J. GrothPhysics 301 3-Dec-2003 29-3the direction (θ, φ)? In this case, the number is proportional to dΩ, an d the proportionalityconstant is often written as a d ifferential so the rate into dΩ isR(→ dΩ ) =dσdΩnv dΩ ,where dσ/dΩ is called the differential cross section. If one integrates over all solid anglesto find the rate for all interactions (no matter what the scattering direction), one hasσ =ZdσdΩdΩ ,and σ i s 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 bedescri bed by differential and total cross sections.If one knows the forces (the interaction) between the incident and target particles,one may calculate the cross sections. Cross sections may also be m easured. One uses atarget which has many target p articles. One counts the scattered particles or the outgoingreactants using particle detectors. The total number of events is just Nnvσ(v)t where Nis the numb er of target particles exposed to the incident beam and t is the d uration of theexperiment. This assumes that the chances of a single interaction are so small that thechances of interactions with two target particles are n egligible. Of course, sometimes thisisn’t true and a multiple