等离子体物理基础(英)：Chapter 5 Distribution function of electrons in electric field

1、Chapter 5. Distribution function of electrons in electric field,5.1 Method for solving kinetic equation,Let us consider in the case where a homogeneous electric field is the source of disequilibrium. The electron velocity distribution function may depend only on the velocity and the angle between th

2、e directions of velocity and field,Therefore it is natural to represent this dependence as an expansion in orthogonal Legendre polynomials,5-1,With small anisotropy we can restrict ourselves to the first two terms in solving many problems. Then we have,5-2,where we have assumed that the axis is para

3、llel to field,The function defines the average electron energy and the average value of any other energy-dependent quantity . By averaging with the aid of (5-2), we obtain,5-3,In a similar manner we can ascertain that the average electron velocity is defined by the function . For the velocity compon

4、ent parallel to the electric field we get,5-4,The function is called the isotropic component, and the function,the directed component of the distribution function,For a homogeneous plasma in a homogeneous electric field the kinetic equation can be written as,5-5,Substituting (5-2) into (5-5), we tra

5、nsform the derivative,Then,5-6,The first equation is found by multiplying (5-6) by and integrating term by term over all the values of from to,As a result of integration the terms proportional to vanish, and the equation takes the form,or,5-7,where the following notation is introduced,5-8,The second

6、 equation is obtained by multiplying (5-6) by and integrating again over all the values of,In this case the terms independent of and proportional to vanish. Then,5-9,5-10,where,The components of the distribution function and can be found by solving the equation (5-7) and (5-9,A similar method for ex

7、panding the electron distribution can be applied to a more general case, where a magnetic field is present as well as the electric one,The distribution function may depend on all the velocity components,5-11,where a vector function with components , and is introduced,Substituting (5-11) into the kin

8、etic equation, we can give the resulting equations,5-13,The collision integrals and are obtained from the equalities,5-14,where is the angle between the axis and the velocity vector, and integration is performed over the solid angles covering all the velocity directions,5.2 Collision integrals for e

9、lectrons,5.2.1 Collision integral for elastic collision,We have obtained the term of electron-heavy particle collisions,The ratio of the elementary velocity space volumes for elastic collisions is determined by the change in velocity,5-15,Taking this into account, we obtain the integral of electron-

10、heavy particle elastic collisions,5-16,Using the representation of , we obtain the expression for and,5-17,5-18,From the well-known formula of spherical trigonometry, we find,5-19,Substituting this relation into (5-17) and (5-18) and integrating with respect to , we get,5-20,Fig.5-1,In elastic colli

11、sions with heavy particles the electron velocity change is very small. Assuming and , , , we obtain the following expression for,5-21,where is the collision frequency,The change of electron kinetic energy in elastic collisions of an electron with an atom is equal to,5-22,The integrand of (5-20) can

12、be represented as,5-23,Substituting (5-23) into (5-20), we obtain,5-24,When the average electron energy is comparable with that of the heavy particles, the distribution function of the heavy particles should be included,Calculations yield the following expression for,5-25,where is the temperature of

13、 the atoms,When it is much below the electron energy , the second term is much less than the first,5.2.2 Collision integral for inelastic collision,When the electron energy exceeds the excitation energy only slightly, one can assume that an inelastic collision results in a total loss of electron ene

14、rgy,The collision term takes the form,5-26,where is the frequency of inelastic collisions of the given type,Substituting (5-26) into the expression of collision integral, we find,5-27,Summing (5-27) over all the inelastic collisions with a large energy loss results in,5-28,where,Is the summary frequ

15、ency of inelastic collisions,5.3 Distribution function of electrons in electric field,Under stationary condition the equations for the components of the distribution functions in a constant electric field take the form,5-29,For the case where only elastic electron-atom collisions are substantial, we

16、 obtain,5-30,5-31,Substituting (5-30) into (5-31), we obtain the equation for the isotropic distribution function component,5-32,The above equation can be written as,5-33,5-34,where,is the flux determining energy acquisition in the field and,is the flux determining the energy losses on collision,5-3

17、5,Integrating the equation, we arrive at the constancy of the summary flux across the spherical surface of velocity space,5-36,Since the quantities and are finite as , the integration constant is zero. This means that the summary flux is zero as well,Substituting (5-34) and (5-35) into (5-36), we ob

18、tain,5-37,The solution of the equation has the form,5-38,At a constant collision frequency the distribution function is Maxwellian distribution,5-39,5-40,where,In strong fields, the first term in the denominator in (5-38) can be neglected. Then it takes the form,5-41,When is proportional to the velo

19、city, the distribution function in strong field has the form,5-42,From the normalization condition, we get,5-43,The distribution described by (5-42) is called the Druyvesteyn distribution,5.4 Effect of magnetic field on electron distribution function,In the presence of constant electric and magnetic

20、 fields the electron distribution function should be sought in the form,5-44,Assuming that the distribution function is independent of the time and coordinates, we can write equations for the functions as follows,5-45,5-46,where the collision integral and for the elastic collision have the following

21、 forms,5-47,5-48,Substituting (5-48) into (5-46), we get,5-49,Let us direct the axis OZ parallel to , and axis OX so that the vector lies in the plane XZ . Then the projections of (5-49) on the coordinate axis take the form,5-50,Solving these equations for , , and , we obtain,5-52,5-51,From (5-51),

22、we can find the projections of the directed electron velocity,For , integration can be done by parts. Then above equations are independent of the form of,5-53,As can be seen, the magnetic field does not affect the velocity component parallel to it. The velocity in the direction of the component perp

23、endicular to decreases, and a velocity component perpendicular to and appears,Substituting (5-51) into (5-45), we obtain the equation of the isotropic component,where,5-55,5-54,is the effective electric field. The equation (5-54) for will be formally the same as in the absence of a magnetic field,Fo

24、r the solution of the equation is,The electron temperature has the form,5-56,5.5 Electron distribution function in alternating electric field,Consider now the electron velocity distribution in an alternating electric field,5-57,We use the equations for and,5-58,where the collision integrals for the

25、elastic collision are,5-59,The collision integrals are of the order of,5-60,Since the relaxation (establishment) of the function and is caused by collision, the relaxation times are of the following order,for the function,and for the function,In the low-frequency case, the following inequalities hol

26、d,The inertial terms in both equations (5-58) are small compared with the collision terms, and they can be neglected,The functions and are the same as in a constant field by the instantaneous field value,For high frequencies, when the inequalities are fulfilled,the field alternation period is much less than the relaxation time of . This function can not catch up with the field alternation. Here is nearly constant in time,The equation for,is,5-61,5-62