If the plasma is drifting, a Lorentz transformation can be performed to obtain the m-k relations in the reference frame in which the plasma is moving. In the analysis that follows, the comparisons will be. linearized and waves of the form e will be assumed. It turns out that at very low temperatures the electrons can have only one oscillation frequency, regardless of wavelength.
The inclusion of a magnetic field in the z direction in the analysis introduces transverse modes which are of most interest. Note also that these waves have a field component in the direction of propagation, which results from the tensor properties of the medium. Consider the ~k ratio of the wave whose field gyrates in the same sense as the ions for positive frequencies.
As P moves from a small value to a large value, the wave begins with a branch in the first quadrant. The case in which the electrons rotate in the same sense as the field for positive frequencies can be visualized in the same way as the previous case. However, a modification must be made. stationary plasma cutoff for positive frequency now occurs in disease = ill ce and state = ~. The evolution of the ill-k curves as the drift rate varies for this polarization can be seen in Figure 1.7 if one makes the conversion ill ~ ~ ,. k ~ -k Thus, one simply needs to rotate the curves in Fig. l.7 by n degrees to see the behavior for this polarization.
It can be seen that there is always a negative root near. corresponding to the fast cyclotron wave of the drifting electrons in a plasma.
SIDW WAVES IN BOUNDED PLASMAS, HIGH FREQUENCY CASE
We see that there are six equations for the field quantities and in general they are all non-zero, so there are no pure modes E and H except in special cases. Before exploring the general case, we will consider two simple examples; infinite and zero magnetic field. If the plasma were to be surrounded by a dielectric that is not free space, except for the zero-order angular mode, E and H waves appear.
The dispersion for the H states is the same as for free space, and a It is seen that there can be no slow wave for a plasma that completely fills a waveguide. If free space extended to infinity, which requires the continuity of the tangential fields at the boundary, one would obtain for the scattering.
The first bracketed term in equation 2.16 is the dispersion for the H modes; this will be ignored. The second term in brackets gives the distribution for the E modes, and using the recursion properties of the Bessel functions, becomes To get an idea of the range of validity of the quasi-static approximation, a comparison will be made for two of the cases.
For the first case, consider a plasma in an infinite magnetic field that completely fills a cylindrical waveguide. Now consider the solutions for a plasma column in free space without magnetic field for the angle-independent mode. Although the dispersion is complex, there is only one condition that must be met, namely that the imaginary part must be zero, since the real part of the dispersion is already zero.
From Appendix I it can be seen that the two transverse components of the electric field are Therefore, the waves are elliptically polarized, and the degree of ellipticity is a function of radius. The salient features of the dispersion can be demonstrated by considering the circularly symmetric state (n = 0) • In this case the dispersion is reduced to.
INTERACTION OF DRIFTING CHARGED PARTICLES AND A STATIONARY
In practice, to test the amplifying or evanescent character of the wave in question, k is plotted as a function. Because of the infinite magnetic field, the electrons have no effect on the wave. Considering the temperature of the plasma electrons introduces two more waves that are related to the longitudinal waves discussed in Chapter I.
If terms in the order y a r e ignored 2 compared to unity, two of the roots can be written. Again ignoring terms of the order y 2 , the transverse propagation constant can be determined approximately. However, the area of the curve that represents the right side of the dispersion relation is horizontal.
The slow ion beam cyclotron wave interacts with the plasma wave in which the electrons spin in the same sense as the field. The dispersion can be solved in the same way as before, substituting the plasma frequency of the ion beam instead of the plasma frequency of the electron beam. Incidentally, for this case there is still the interaction of the slow electron wave and the plasma wave near m = m.
In the system of a floating electron or ion beam through a plasma, there is instability only in the immediate vicinity of the ion and electron frequencies, respectively. For the + and - waves of the floating plasma in a plasma system, the frequencies are almost zero. For example, suppose that the electron beam densities of the two systems and the ion masses of the floating and stationary plasmas were the same.
The problem of propagation in a floating plasma can be solved by applying a Lorentz transformation to the solution of the system in which the charged particles are at rest. The general characteristics of the dispersion predicted by the quasi-static approach are verified, and the exact and quasi-static solutions are compared. The z direction is chosen as the direction of the B field and writing. the equations in AI-3 take the form. where the top sign refers to Q.l, 2 is eliminated from the above equations.
T is independent of the coordinate z and can also be used as the transverse operator in cylindrical coordinates. operator L can be written as. The speed can be expressed in terms of the current density from AII-1 and AII-2.
APPENDIX III
