Collision velocity-changing

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. 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

Molecules in a gas are in constant motion at speeds on the order of the speed of a rifle bullet at equilibrium there is no net flow of gas and the motion is random. This motion produces collisions of the molecules with the walls of the vessel containing the gas, with a change in momentum of the gas molecule resulting from each collision. This change in momentum produces a force per unit area, or pressure on the wall. Consider those molecules with the component of velocity in the x direction between the value of vx and vx + dvx. The x direction is defined as the direction normal to the wall. The fraction of molecules with the x component of velocity in this range, denoted dN(vx)/N, is given by the density function, f(vx), where... 

Collision of particle a with causes f 23 to change because of the velocity change of v3 and v2s etc. But, in addition, collision of either a or 0 with another particle fi (labelled with co-ordinates r3 and v3) can also change f . This arises if a and (u or j3 and jti are so positioned and moving that they collide it is determined by f. Averaging the Liouville equation over the co-ordinates of (N — 3) particles (not a, 3 or p) then averaging over the possible position and velocity of theja particles shows [541, 542]... 

Fig. 5.7. The Cartesian x,y frame in the collision plane. Velocities of the atom of type A before and after the collision labeled 1 are v and X is the scattering angle. The x-axis is aligned with the direction of the velocity change, i>2 — i>i-...

Using this approach the +) and —) states are not coupled by the field of the ion, but are only split in energy. At high collision velocities the initial state 0) is simply projected onto the 0 + 1) state, a coherent superposition of +) and -) states, by the dipole matrix element. However, at lower velocities the change in energy of the +) and -) states during the collision allows the +) and -) states themselves to be populated rather than only a coherent superposition. The latter feature allows nondipole transitions at lower collision velocities, as observed experimentally. 

In studies of bulk gas samples, as in conventional microwave absorption experiments, one must take account of the fact that one is studying an assembly of molecules moving in different directions at different velocities, and suffering frequent collisions which change both the velocity and direction. It may be shown that for a gas at thermal equilibrium, the Doppler full line width Av at half-height is given by... 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

While the data base for pressure broadening is well established, far less is known about the rate constants for velocity changing collisions. Zagidullin et Copeland and Bauer used computer models to examine... 

The relative velocity changes from the value g2i before the collision to the value g2i after the collision. Since the collision is assumed to be elastic so that the internal energies of the molecules remain unchanged, the total kinetic energy Ec remains unchanged in a collision. 

At large Re the particles sediment only on a section of the leading part of the bubble surface because the normal component of the liquid velocity changes its sign at Oj < 7t / 2. It means that the hydrodynamic flow can prevent the sedimentation near the equator. As a result we can use the second term of Eq. (10.27) to estimate the collision efficiency... 

Another advantage of polarization spectroscopy is the suppression of the broad signal back-ground observed in saturated absorption (Fig. 16a), when collisions redistribute the velocities of the pumped atoms over the Doppler profile, because these velocity-changing collisions drastically reduce the laser-induced anisotropy (the curves in Figure 16 are obtained in the same experimental cell of neon). A good review about Polarization Spectroscopy and related phenomena can be found in ref. I 25 I. 

The line shapes of two-photon transitions are very simple as they are simply lorentzian curves, whereas the line shape in the saturation technique is quite complicated (its calculation involves the averaging of a non-linear effect which depends on the velocity component v ). In case of collisions, the two-photon line shape remains a lorentziai one and it is easy to measure the broadening and the shifts, whereas the velocity-changing collisions complicate still further the already complicated line shapes of saturation spectroscopy. 

Some techniques of laser spectroscopy, such as the method of separated fields optical Ramsey fringes. Sect. 9.4), coherent transient spectroscopy (Sect. 7.6), or polarization spectroscopy (Sect. 2.4) allow one to distinguish between phasechanging, velocity-changing, or orientation-changing collisions. 

A more detailed consideration of collisional broadening of Lamb dips or peaks must also take into account velocity-changing collisions. In Sect. 2.2 it was pointed out that only molecules with the velocity components = 0 di y/A can contribute to the simultaneous absorption of the two counterpropagating waves. The velocity vectors v of these molecules are confined to a small cone within the angles p <3ie around the plane v =0 (Fig. 8.3a), where... 

Fig. 8.3 Only molecules with velocity vectors within the angular range p < e around the plane z = 0 contribute to the line profile of the Lamb dip. Velocity-changing collisions that increase p to values p > e push the molecules out of resonance with the laser field...

The combined effect of both kinds of collisions gives a line profile with a kernel that can be described by a Lorentzian profile slightly broadened by soft collisions. The wings, however, form a broad background caused by velocity-changing collisions. The whole profile cannot be described by a single Lorentzian function. In Fig. 8.4 such a line profile is shown for the Lamb peak in the laser output Pl(co) at... 

Often the collisional broadening of the Lamb dip and of the Doppler profile can be measured simultaneously. A comparison of both broadenings allows the separate determination of the different contributions to line broadening. For phase-changing collisions there is no difference between the broadenings of the two different line profiles. However, velocity-changing collisions do affect the Lamb-dip profile (see above), but barely affect the Doppler profile because they mainly cause a redistribution of the velocities but do not change the temperature. 

Since the homogeneous width y of the Lamb-dip profile increases with pressure p, the maximum allowed deflection angle e in (8.1) also increases with p. A comparison of pressure-induced effects on the kernel and on the background profile of the Lamb dips and on the Doppler profile therefore yields more detailed information on the collision processes. Velocity-selective optical pumping allows the measurement of the shape of velocity-changing coUisional line kernels over the full thermal range of velocity changes [979]. 

An interesting phenomenon based on collisions between excited atoms and ground-state atoms is the macroscopic diffusion of optically pumped atoms realized in the optical piston [1014]. It is caused by the difference in cross sections for velocity-changing collisions involving excited atoms A or ground-state atoms A, respectively, and results in a spatial separation between optically pumped and unpumped atoms, which can be therefore used for isotope separation [1015]. 

---