等离子体物理基础(英)：Chapter 2 Single particle motions

1、Chapter 2 Single particle motions,2.1 Uniform magnetic field,In this case, the equation of motion for a particle is,2-1,Taking,to be the direction of,we have,2-2a,2-2b,2-2c,Differentiating (2-2a) and (2-2b), we obtain,2-3a,2-3b,2.1.1 Particle motion with E=0,where,Solving (2-3) and (2-2), we find,2-

2、4a,2-4b,2-4c,2-5,is the cyclotron frequency, and,is the speed perpendicular to,and,is an arbitrary phase,Integrating (2-4) yields the particle position,2-6c,2-6a,2-6b,Using (2-6a) and (2-6b), we have,2-7,where,2-8,is defined the Lamar radius,The direction of the gyration is always such that the magn

3、etic field generated by the charged particle is opposite to the externally imposed field. Plasmas are diamagnetic,Fig.2-1,2.1.2 Finite E,The equation of motion in uniform,and,2-11,2-9,We take,We obtain a uniform acceleration along,The equation for the transverse motion is,2-10,We let,Putting (2-11)

4、into (2-10), we have,2-12,Since,is the velocity of gyration, we obtain,2-13,Taking the cross product with B, we have,2-14,The electric drift velocity is,2-15,is perpendicular to both,and is independent of the mass and charge of the particles,It is important to note that,and,Fig.2-2,2.1.3 Gravitation

5、al field,The foregoing result can be applied to other forces by replacing,in the equation (2-15) by a general force,is then,The drift caused by,2-16,In particular, if,is the force of gravity,there is a drift,2-17,The drift,is perpendicular to both the force and,but it changes sign with the particle

6、charge,There is a net current density in the plasma given by,2-18,Fig.2-3,2.2 Non-uniform magnetic field,2.2.1 Gradient drift,The equation of motion for a particle in the non-uniform magnetic field is,2-19,The magnetic field in the neighborhood of the guiding center is expanded as,2-20,We let,2-21,w

7、ith the approximation,2-22,where,is the velocity of gyration,is the drift velocity,Substituting (2-20) and (2-22) in (2-19), we obtain,2-23,Averaging over a gyro-period, the rapidly rotating terms average to zero, so that,2-24,where denotes an average over a gyro-period,Cross multiplying (2-24) by,w

8、e obtain,and considering,2-25,For the motion of gyration, we have,2-26a,2-26b,where the magnetic field,is along z,Assuming the magnetic field,and substituting (2-26) in (2-25), we obtain,and,2-27,It can be generalized to,2-28,The drift velocity is perpendicular to both,Using the expression of magnet

9、ic moment,we can write (2-28) as,and grad-B and depends on,the sign of the charge,2-29,where,is the equivalent force on the particle,Fig.2-4,2.2.2 Curvature drift,Here we assume the lines of force to be curved with a constant radius of curvature R. The centrifugal force felt by the particles as they

10、 move along the field lines is,2-30,According to (2-16), this gives rise to a drift,2-31,component and,2-32,Thus,2-33,Using (2-28), we have,2-34,The total drift in a curved vacuum field is,2-35,Fig.2-5,2.3 Non-uniform electric field,2-36,This field distribution has a wavelength,The equation of motio

11、n is,2-37,whose transverse components are,2-38b,2-38a,By defining,we have,2-39,If the electric field is weak, we may use the undisturbed orbit to evaluate,That is,2-40,Expanding the cosine, we have,2-41,Using the Taylor expansions,for the small Lamar radius case,and,we can write,2-42,Substituting (2

12、-42) in (2-40), we have,2-43,Cosine and Sine terms,The solution of (2-43) is,2-44,2-45,Averaging over time, we obtain,The drift velocity in a non-uniform electric field is,2-47,Equation (2-45) gives,2-46,where,is the electric field at the guiding center,For an arbitrary variation of,2-48,we need onl

13、y replace,by,and write (2-47) as,The second term is called the finite-Lamar-radius effect,Since,is much larger for ions than for electrons,is no longer,independent of species,2.4 Time varying electric field,2-49,The equation of motion is,2-50,Assuming,to be in the z direction, we can write the trans

14、verse,components as,2-51b,2-51a,Defining,we have,2-52,The solution of (2-52) is,2-53,We can find,2-54a,2-54b,Assuming that,varies slowly, so that,we can obtain,the approximate expression,2-55b,2-55a,The first terms in (2-55) express the motion of gyration and the second terms,are the drift,is the dr

15、ift,We define the polarization drift as,2-56,is in opposite directions for ions and electrons,Since,there is a polarization current,is the mass density,2-57,where,2.5 Adiabatic constancy of the magnetic moment,2.5.1 Time varying,field,We consider the magnetic field to vary in time. The electric fiel

16、d associated with,is given by,2-58,The equation of motion for a charged particle is,2-59,Taking the dot product of the equation of motion with the transverse velocity,2-60,we have,where is the element of path along a particle trajectory,The change of kinetic energy in one gyration is obtained by int

17、egrating over one period,2-61,If the field changes slowly, we can replace the time integral by a line integral over the unperturbed orbit,2-62,2-63,2-64,The change rate of kinetic energy with time is just the average change during one period of gyration,where,Using,we have,2-65,or,Comparing (2-64) w

18、ith (2-65), we have the desired result,2-66,The magnetic moment is invariant in slowly varying magnetic field,2.5.2 Space varying field,Now we consider a magnetic field which is pointed primarily in z direction and whose magnitude varies in the z direction. The field is axisymmetric as shown in Fig.

19、2-6,Fig.2-6,The Lorentz force has a component along z given by,2-67,We have,for ions and,for electrons. Let,so we have,2-68,2-69,This yields,upon integrating with respect to,2-70,From the averaging procedure it is seen to be valid only for,2-71,Substituting (2-70) in (2-68), we have,2-73,2-72,or,Bec

20、ause the magnetic field does no work, the total kinetic energy of the particle is conserved,2-74,where,If the particle moves a distance,then,2-75,Differentiating (2-74) yields,hence (2-75) becomes,2-76,Integrating (2-76), we obtain,2-77,The magnetic moment is an adiabatic invariant that is approxima

21、tely conserved if the magnetic field changes slowly,2.6 Magnetic mirrors,As a particle moves from a weak-field region to a strong-field region, it sees an,constant,which particle will escape? Assume that a particle with,2-78,Conservation of energy requires,2-79,Combining (2-78) with (2-79), we have,2-81,2-80,or,Eq.(2-81) defines the boundary of a region in velocity space in the shape of a cone, called a loss cone,Assuming that the velocity distribution of particles is isotropic and the collisions of particles can be neglected, we can evaluate the loss proba