Chapter 3 Collisions in plasmas3.1 Application of conservation laws to particle collisionsThe momentum conservation law leads to the equality of the systems pre-and post-collision momenta or(3-1) orThe energy conservation law makes it possible to determine the change in the kinetic energy of the system(3-2)where is the net change in the internal particle energy as a result of collision.In elastic collisions, Among the inelastic collisions, one distinguishes collision of the first kind with is clearly equal to zero.from collisions of the second kind withIn the coordinates of the center of mass system, the coordinates of these particles is(3-5) and the center-of-mass velocity is(3-6)It remains constant in collision in conformity with the momentumconservation law. Therefore we can change to a reference system in which the center of inertia is at rest In such a system, we have(3-7)or(3-8) The relative particle velocity can be expressed asSubstituting Eq (3-8) into (3-9), we have(3-9)(3-10) (3-11) The particle velocities in the laboratory system are(3-12) The total kinetic energy is(3-13)The first term on the right-hand side is usually called the center-of-mass energy, and the second, the relative-motion energy; the quantity is calledthe reduced mass. (3-14)Using Eq (3-13), we can write the energy conservation law for collisionSince in collision the center-of-mass velocity and kinetic energy remain unchanged (), we get(3-15)By (3-15), only the part of the relative-motion energy can convert to internal energy.In elastic collision (), the relative velocity vector does not change its).value (The change in the momentum of a - type particle is (3-16)The projections of on the coordinate directions are where is called the scattering angle. Since the angledetermines the position of the interaction plane, the values are averaged over the anglein statistical consideration.Becausewe have, and (3-17) Substituting it into (3-16), we obtain(3-18)It is seen that the change in momentum is proportional to the relative velocity of the colliding particles. It also depends on the scattering angle in the factor. The change in the kinetic energy of a particle in the laboratory system as a result of collision can be written as(3-19)Substituting (3-18) into (3-19), we obtain (3-20)If the velocity distribution of the the third term in the parentheses is reduced to zero, thus - type is isotropic, then, upon averaging, or(3-21)where the coefficient(3-22)is called the energy transfer coefficient.It can be seen that energy exchange between the colliding particles is proportional to the difference of their initial energies, the angle factor, . and the coefficientFor the important case of collision of electrons with heavy particles,we have (3-24)Therefore the problem on collision in the center-of-mass system amounts to the problem on electron motion in the field of a fixed atom.The expressions for the energy and momentum transport in elastic collisions take the form is the electron scattering angle in collision with the fixed atom.(3-25)(3-26)hereIn inelastic collision, the energy conservation takes the form It means that the change in internal energy of the heavy particle is equal to the change in kinetic energy of the electron.(3-27)3.2 Differential scattering cross section3.2.1 Definition of differential cross sectionThe number of particles scattered per unit time into the solid angle is equal toThe proportionality factor , which has the dimension of area, characterizes the probability of particle scattering in a definite direction. It is called the differential scattering cross section.(3-28)3.2.2 Classical equation for differential cross sectionThe number of particles scattered per unit time into the solid angle is equal to the particle flux going through the element area perpendicular to the direction of the incident flux, so that(3-29)Comparing with (3-28), we obtain the equation of classical mechanics for the differential cross section(3-30)AFig.3-2 How to determine the relationship between and ? The equations of the colliding-particles motion arewhere is the particle interaction force. The difference of these equations become (3-31) (3-22) or (3-23) where is the relative-motion velocity and is the reduced mass. Introducing the potential , we can express the force as where is the distance between the colliding particles. The relationship between the scattering angle and the impact parameter can be obtained from the conservation laws. The energy conservation law has the formwhere is the pre-collision velocity, and are the velocity components in the polar system of coordinates. (3-35)The angular momentum conservation law can be represented as(3-36) Combining (3-35) and (3-36), we obtain(3-37) From (3-37), we obtain the equation for the trajectory(3-38) In the deducing process, we have used the expression Integrating, we find(3-39) Awhere where is the distance at which the denominator of the integral equation reduces to zero. With a given interaction potential U(r), the relationship between the scattering angle and the impact parameter can be obtainedFig.3-33.2.3 Quantum equation for differential cross sectionThe asymptotic representation of the wave function can be written as (3-40) The quantity is called the scattering amplitude. It is obtained by the Born approximation. (3-41) where the vector is the change in wave vector in collision; The differential cross section is (3-42) , thenif3.3 Integral characteristics of collisionsIntegrating the differential cross section over the solid angle, We obtain the total scattering cross section.(3-48)In the classical approach the total cross section determines the area of the surface perpendicular to the relative velocity in which particles collide with scattering at an arbitrary angle.The collision frequency takes the form (3-49)In accordance with the classical concept of the cross section it determines the number of times that an type particle impinges on targets formed by -type particles.The average inter-collisional time is (3-50)The mean free path can be written as(3-51) The total cross section and the related quantities are seldom applicable. The integral in the expression of the total cross section for collisions of electrons and ions with atoms or with each other diverges. It also neglects the difference in the effect of close and far collisions.We introduce the weighting factor into the integral cross section of elastic collisions. Then the expression for the cross section takes the form (3-52) The transport cross sectionThe effective collision frequency, the inter-collisional time, and the mean free path are (3-53)For inelastic processes the total collision cross section is It yields the number of inelastic acts of a given type (j).The total inelastic collision cross section3.4 Elastic collisions between charged particlesThe Coulomb potential is(3-54)We can determine the relationship between the scattering angle and the impact parameter , so that(3-55)Finishing the integral, we obtain (3-56)where the parameter is called the strong-interaction radius. The scattering angle corresponding to is equal to .The differential cross section is (3-57)This is the Rutherford equation.The transport cross section can be obtained by(3-58) Taking into account the relationship between the scattering angle and the impact parameter (3-59) for , we have (3-60)Substituting(3-60)into (3-58), we obtain(3-61) It diverges logarithmetically as . To eliminate the divergence one should limit the coulomb interaction radius. In a plasma, the field of each interacting particle is shielded by other charged particles. The size of the screening area is determined by the Debye radius. Therefore we can assume for collisions involving electrons, and for ion-ion collisions. The transport cross section can be written as where is called Coulomb logarithm in the form (3-63)(3-62)For different types of collision, the transport cross sections are determined as follows (3-64) for collisions involving electrons, for ion-ion collisions. whereand3.5 Elastic collisions of electrons with atoms The potential of electron-atom interaction can be written by (3-68)where the first term is determined by interaction of the electron with the nucleus, and the second, with atomic electrons. is the distribution of electron concentration in atom. The scattering amplitude from the Born approximation can be obtained by (3-69) where the vector is the change in wave vector in collision; .thenFinishing integration, we get (3-70) is the so-called atomic form factorwhere (3-71)The differential cross section is (3-72)3.6 Inelastic collisions of electrons with atomsFor some gases summary cross sections of inelastic processes near the threshold can be approximated in the following way with a error not exceeding 1020% (3-80) is the excitation threshold, is the coefficient, is the electron energy. where and (3-81) At electron energies greatly exceeding that of excitation we can use the Born approximation, which yields the following expression :where is the matrix element of the transition dipole momentum is a coefficient of the order of unity.and Summing the cross sections over all the possible transitions, we obtain the expression for the summary cross section of inelastic processes (3-82) is the average value of the square of the dipole momentum.whereThe most important processes of excitation for atoms and molecules in a plasma are by electron impact, and the following reactions are known to occur 1) Excitation2) Dissociation3.7 Elastic collisions of ions with atomsA free ion induces a dipole moment The potential of its interaction with an ion is.(3-83) Using the classical equation (3-84)and expanding(3-84) in the case , we have (3-85)Substituting (3-83) into (3-85), we get (3-86)The differential cross section is (3-87)From the dependence , we can determine the transport (3-88)cross sectionIt is inversely proportional to the square root of the kinetic energy of relative motion and depends exclusively on the polarizability of the atom .3.8 Charge exchange collisions of ions with atomsThe charge exchange effect, arising from electron exchange between the ion and the atom in the background gas, can evidently occur without any change in the internal energies. , the barrier height tendsFig.3-4As to zero. The probability that the electron is in any one of the atomic residues is equal to 1/2.This yields a crude estimate of the charge exchange cross section (3-89)3.9 Ionization processes3.9.1 Direct ionizationThe most important processes of ionization for atoms and molecules in plasma are by electron impact. When the electron energy exceed the ionization threshold, the following reactions are known to occur Experimental data on the ionization cross section have been obtained for many gases. Near the threshold the dependence can usually be considered as (3-90)It is of the following order to cm. (3-91)Within a rather wide energy range the ionization cross section of the atom can be approximated by the formulawhere is the number of electrons in the outer shell. The error of this approximation for atoms and ions does not exceed 20% in the range of energies less than 1 k eV.At high energy the ionization cross section can be found, using the Born approximation, (3-92)where and are numerical factors (for hydrogen, for instance, and )3.9.2 Penning ionizationPenning ionization is an ionization process caused by the collision between metastable particles and neutral particles which have a lower ionization energy than the excitation energy of the metastable particles. It is expressed as where is the excited atom or molecule. Penning ionization is significant in the longer mean free-path pressure regime, i.e.10 mTorr.3.9.3 Ionization by collisions among excited particles When an excited molecule collides with another excited molecule, the following ionization process can be expected if the sum of their excitation energies is larger the ionization energy of one of the moleculesThis can be an important ionization process in after-glow plasmas where the electron temperature is low. 3.9.4 Stepwise ionizationThe stepwise ionization may take place at and , which occurs in two stages, the first being excitation of the atom, and the second, ionization of the excited atom. We can write the scheme of this process3.10 Electron-ion collisional recombinationThe recombination process can be written in the form of a reaction . It is easy to see that it fails to occur in the absence of third bodies because the energy conservation law can not be satisfied simultaneously.Accordingly, the following processes of electron and ion recombination may take place1. 2. 3. 4.