Collisions Coulomb

For electron-ion or ion-ion collisions, the plane waves exp(i/c. R) are simply replaced by Coulomb waves to... 

RBS is based on collisions between atomic nuclei and derives its name from Lord Ernest Rutherford who first presented the concept of atoms having nuclei. When a sample is bombarded with a beam of high-energy particles, the vast majority of particles are implanted into the material and do not escape. This is because the diameter of an atomic nucleus is on the order of 10 A while the spacing between nuclei is on the order of 1 A. A small fraction of the incident particles do undergo a direct collision with a nucleus of one of the atoms in the upper few pm of the sample. This collision actually is due to the Coulombic force present between two nuclei in close proximity to each other, but can be modeled as an elastic collision using classical physics. 

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. 

Once the Fock operators have been constructed from a set of MSOs, this matrix equation is linear in its unknowns. Its coefficients are dependent on time in a way determined by the forces driving the electrons. These forces are the nuclear Coulomb potentials in molecular collisions or dynamics, but they could also be weak external fields. 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

All of the examples of singlet energy transfer we have considered take place via the long-range resonance mechanism. When the oscillator strength of the acceptor is very small (for example, n-> n transitions) so that the Fdrster critical distance R0 approaches or is less than the collision diameter of the donor-acceptor pair, then all evidence indicates that the transfer takes place at a diffusion-controlled rate. Consequently, the transfer mechanism should involve exchange as well as Coulomb interaction. Good examples of this type of transfer have been provided by Dubois and co-workers.(47-49)... 

In the foregoing, U is the interaction potential, M is the reduced mass of the colliding system, ftk and ftk are respectively the momentum of the projectile before and after the collision, ig and in are respectively the wavefunctions of the atom (or molecule) in the ground and nth excited states, and the volume element dt includes the atomic electron and the projectile. Since U for charged-particle impact may be represented by a sum of coulombic terms in most cases, Eq. (4.11) can be written as (Bethe, 1930 Inokuti, 1971)... 

We discuss briefly the factors that determine the intensity of the scattered ions. During collision, a low energy ion does not penetrate the target atom as deeply as in RBS. As a consequence, the ion feels the attenuated repulsion by the positive nucleus of the target atom, because the electrons screen it. In fact, in a head-on collision with Cu, a He+ ion would need to have about 100 keV energy to penetrate within the inner electron shell (the K or Is shell). An approximately correct potential for the interaction is the following modified Coulomb potential [lj ... 

Because long acquisition times are required for maximum resolution, it is essential that the ions can survive in the trap for extended periods of time. A main reason for ion loss is collisions with residual gas in the cell. Therefore, it is essential to keep the pressure as low as possible, preferably in the region of 10-9 torr or below. It is also important not to allow too many ions to enter the cell. When more than 106 to 107 ions are present in the cell, the coulomb repulsion can shift or broaden peaks in the mass spectra. By using an ion gate, the number of ions entering the cell can be limited in a controlled way. 

This problem is very difficult to solve in general however, we have to keep conditions (303) in mind, which we used in order to obtain the Fokker-Planck collision term (304). With this approximation, it is expected that the ions will exhibit random Brownian motion instead of free particle motion between two successive coulombic interactions. We shall thus refer to this model as the Brownian-static model (B.s.). 

Kunkel (K9) has derived expressions for the collision rate of charged particles allowing for both Coulomb attraction forces and ionic image forces. His final expressions, however, are not readily applied and he concludes only that the rate of growth of a particle over a period of several seconds is small if... 

In the derivations of the above equations, image forces (which are likely to be small compared to the Coulomb forces) were neglected and every collision was assumed to result in adhesion. Although some have questioned the validity of the latter assumption, there is no evidence to indicate that this is not, at least nominally, the case when dealing with particles smaller than 10 microns colliding at low velocities. While this subject of adhesion has never been investigated systematically, the bulk of all the data on aerosol flocculation and deposition would imply that for these conditions adhesion... 

For 1, de Broglie s wavelength is small enough compared to the classical collision radius b so that a wave packet can be constructed which, approximately, follows the classical Coulomb trajectory [3]. The opposite limit, where the Sommerfeld parameter Zie hv<, denotes the case of weak Coulomb interaction where the Born approximation may be expected to be valid. 

The relativistic regime differs fundamentally from what has been discussed so far, in that the cross sections for Coulomb scattering are not the same in quantum as in classical mechanics. Therefore, with the exception of the Fermi density effect -which is classical as far as the collision physics is concerned - classical arguments are less powerful in this regime. 

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