Velocity changing collisions distribution

Our studies of the effect of velocity-changing collisions in an rf-laser double resonance experiment contribute to a new vista into the role of collisictis in laser spectroscopy of sub-level structures the limitation of the observation time of the active atoms due to narrow-bandwidth optical excitation and simultaneous velocity diffusion can be of importance for a variety of spectroscopic techniques that use a velocity-selective excitation and detection of either sublevel populations or sublevel coherence. On the other hand, the collisional velocity diffusion of sublevel coherence within an optical Doppler distribution can also give rise to new and surprising phenomena as will discussed in the next section. 

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

The reader may find the above result more convincing if he or she tries to picture how the initially equal densities of people in two rooms of very different sizes connected by identical doors through an anteroom will change in time if, when the doors are opened, people move from room to room with equal probability and equal velocity without collisions. It is also reassuring to observe that when the results are averaged over a realistic distribution of r the oscillations disappear and the concentrations behave monotonically, as they should. 

The pressure of a gas is the force per unit area exerted due to molecular collisions with the walls of the container. Since force is momentum change per unit time, pressure is determined by computing the momentum transfer per unit time per unit area. Consider a wall at x = L if at each collision there is perfect reflection the x component of velocity changes from w to — m and the net momentum transfer is Imu. While perfect reflection is not a reasonable assumption we know that the gas has an isotropic velocity distribution and from (1.8), F(w, p, w) = F(—u, v, w). Thus, on the average, the fraction of molecules leaving the walls with an x component of velocity —M is the same as that striking the wall with x component of velocity u. The net effect is a mean momentum transfer of 2mu. 

Our examination of collision events led to the conclusion that the rate of two-body collisions has a second order dependence on the concentration(s) of the species (Equation 6.7). We realize that a two-body collision event corresponds to an elementary second order (bimolecular) process, and hence, the rate law for a bimolecular process has the same second order dependence on concentration. A complete gas kinetic analysis of collisions requires that we recognize that molecules in a gas have a distribution of velocities. Since this distribution is prescribed by the temperature, we can ask what happens when the temperature is changed. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

In deriving this relation it has been assumed that the distribution function does not change significantly in a distance b, so that the distribution function describing the number of v2 particles is evaluated at the same point in space as that for the vx particles. Since the number of vx particles is f(r,v1,t)drdv1, the number of collisions between particles of velocity vx and v2 in At is... 

The expressions must now be integrated over all possible values of v2, b, and e to give the net change in the number of particles of velocity vx due to collisions in time At dividing by drdvxAJ, this gives the time rate of change in the distribution function due to collisions ... 

In the derivation of the Boltzmann equation, it was noted that the distribution function must not change significantly in times of the order of a collision time, nor in distances of the order of the maximum range of the interparticle force. For the usual interatomic force laws (but not the Coulomb force, which is of importance in ionized gases), this distance is less than about 10 T cm the corresponding collision times, which are of the order of the force range divided by a characteristic particle velocity (of the order of 10 cm/sec for hydrogen at 300° C), is about 10 12 seconds. 

Fig. 21. The product D-atom velocity-flux contour map, d

Now return to the system of N particles, the distribution function for which, pN, changes frequently because of a large number of successive instantaneous collisions. Each collision causes the N-particle distribution function to change, because of the change of position and velocity of the two hard spheres which collide. The effect of all these collisions is additive and each instantaneously alters the distribution, pN. The Liouville equation for hard spheres is... 

Hie singlet distribution ff changes when a collision occurs between the particle ot and any one of the remaining 3 = (N — 1) particles (labelled 2 during the collision process). The probability of particles ot and /3 being able to collide depends upon their mutual position in space and their velocity, which is described by the doublet density f"0. Now, in the Boltzmann equation analysis, this leads to a factorisation of the doublet density in terms of the singlet density of both particles. Because any correlation in the position and motion of these particles is lost by this procedure, an alternative approach must be tried to estimate the doublet density. 

A physical collision between fee reactive solutes A and B can lead to a change in the density of fee doublet distribution f B describing the mutual position and velocity of the reactants at a time t [285]. The physical collision is described by fee operator... 

To solve eqn. (294) for the doublet density, the hierarchy of the equation must be broken in a manner analogous to the super-position approximation of Kirkwood or that of Felderhof and Deutch [25], which was presented in Chap. 9, Sect. 5. Furthermore, it is not unreasonable to assume that the system is quite near to thermal equilibrium. Were the system at thermal equilibrium, then collisions would not change the velocity distribution of the particles and the equilibrium distribution would be of the usual Maxwellian form, 0 (v,), etc. These are the solutions of the psuedo-Liouville equation... 

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