**5.1 Statement of the problem. Dispersion relation**

In this section we pay special attention to systems, the geometry of which is similar to the geometry of plasma microwave sources and possible development of the new types of DSI in such systems. The simplest theoretical model of plasma microwave generators assumes relativistic e-beam propagating along axis of a plasma filled waveguide of radius *R*. The beam and plasma are assumed to be completely charge and current neutralized. In the waveguide cross-section the plasma and beam are annular, with mean radii *rp* and *rb*. Their thicknesses Δ*<sup>p</sup>* and Δ*<sup>b</sup>* are much smaller, than the mean radii. Strong external longitudinal magnetic field is assumed to freeze transversal motion of beam and plasma electrons.

For theoretical study of the problem we use an approach [10], which gives result for arbitrary level of beam-plasma coupling. This condition is obligatory for obtaining comprehensive results. The DR, which follows from the approach, has a form, which clearly shows interaction of the beam and plasma waves. The approach proceeds from equation for polarization potential *ψ*

$$\frac{\partial}{\partial t} \left( \Delta\_{\perp} + \hat{L} \right) \varphi = -4\pi \left( J\_{bx} + J\_{px} \right), \hat{L} = \frac{\partial^2}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \tag{42}$$

Here *Jbz*ð Þ¼ **r**⊥, *z*, *t pb*ð Þ **r**<sup>⊥</sup> *j bz*ð Þ *z*, *t* and *Jpz*ð Þ¼ **r**⊥, *z*, *t pp*ð Þ **r**<sup>⊥</sup> *j pz*ð Þ *z*, *t* are perturbations of the longitudinal current densities in the beam and plasma. Functions *pb*,*<sup>p</sup>*ð Þ **r**<sup>⊥</sup> describe transverse density profiles of the perturbations of the longitudinal currents in the beam and the plasma. For homogeneous beam/plasma *pb*,*<sup>p</sup>* � 1 for infinitesimal thin *pb*,*<sup>p</sup>* � *δ r* � *rb*,*<sup>p</sup>* (*δ* is Dirac function), Δ<sup>⊥</sup> is the Laplace operator over transverse coordinates, *z* is the longitudinal coordinate, *t* is the time, *c* is the speed of light. The longitudinal electric field expresses as *Ez* <sup>¼</sup> *<sup>L</sup>*^*ψ*. The equations for *j bz* and *j pz* are

$$\left(\frac{\partial}{\partial t} + V\_b \frac{\partial}{\partial x}\right)^2 j\_{bx} = \frac{\alpha\_b^2 \chi^{-3}}{4\pi} \frac{\partial}{\partial t} E\_x; \left(\frac{\partial}{\partial t} + \nu\right) j\_{px} = \frac{\alpha\_p^2}{4\pi} E\_x,\tag{43}$$

where *ω<sup>p</sup>*,*<sup>b</sup>* are the Langmuir frequencies for plasma and beam respectively, *ν* is the effective collision frequency in plasma, *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>V</sup>*<sup>2</sup> *b=c*<sup>2</sup> �1*=*<sup>2</sup> , *Vb* is the velocity of beam electrons.

The DR, which follows from the statement, is still very cumbersome (of integral type). To reduce the DR to a simple algebraic form one should make following expedient for theoretical model assumption: the plasma and the beam are not just thin but infinitesimal thin. In this case the DR becomes

$$D\_p(\boldsymbol{\alpha}, \boldsymbol{k}) D\_b(\boldsymbol{\alpha}, \boldsymbol{k}) = G \, \kappa^4 \delta \varepsilon\_p \delta \varepsilon\_b,\tag{44}$$

$$\begin{aligned} \text{where } D\_{p,b}(\boldsymbol{\omega}, \boldsymbol{k}) = \boldsymbol{k}\_{\perp p,b}^2 - \kappa^2 \delta \varepsilon\_{p,b}, \,\delta \varepsilon\_p = \frac{a\_p^2}{\omega(\boldsymbol{\omega} + \boldsymbol{i}\boldsymbol{\omega})}, \,\delta \varepsilon\_b = \frac{a\_b^2}{r^3(\boldsymbol{\omega} - \boldsymbol{k} V\_b^2)}, \kappa^2 = \boldsymbol{k}^2 - a\boldsymbol{\alpha}^2/c^2, \,\varepsilon\_b = \frac{\kappa^2}{r^3} \end{aligned}$$

*k* is the wave vector along axis, *ω* is the frequency, *k*⊥*<sup>p</sup>* and *k*⊥*<sup>b</sup>* play role of the zero order transversal wave numbers for plasma and beam [11, 14].

*New Types of Dissipative Streaming Instabilities DOI: http://dx.doi.org/10.5772/intechopen.98901*

$$k\_{\perp p,b}^2 = \left\{ r\_{p,b} \Delta\_{p,b} I\_l(\kappa r\_{p,b}) \left[ \frac{K\_l(\kappa r\_{p,b})}{I\_l(\kappa r\_{p,b})} - \frac{I\_l(\kappa R)}{K\_l(\kappa R)} \right] \right\}^{-1} \tag{45}$$

(*Il* and *Kl* are modified Bessel and Mac-Donald functions, *l* ¼ 0, 1, 2 … is the azimuthal wave numbers). *G* is the coupling parameter. It depends on the overlap of the plasma and the beam fields and shows efficiency of their interaction

$$\mathbf{G} = \begin{cases} \frac{I\_l(\kappa r\_b)K\_l(\kappa r\_p)I\_l(\kappa R) - K\_l(\kappa R)I\_l(\kappa r\_p)}{I\_l(\kappa r\_p)K\_l(\kappa r\_b)I\_l(\kappa R) - K\_l(\kappa R)I\_l(\kappa r\_b)} & r\_b \le r\_p\\ \frac{I\_l(\kappa r\_p)K\_l(\kappa r\_b)I\_l(\kappa R) - K\_l(\kappa R)I\_l(\kappa r\_b)}{I\_l(\kappa r\_b)K\_l(\kappa r\_p)I\_l(\kappa R) - K\_l(\kappa R)I\_l(\kappa r\_p)} & r\_p \le r\_b \end{cases} \tag{46}$$

An important property of *G* is: *G* =1 for *rp* ¼ *rb* and *G* < 1 in other cases. In long wavelength limit (for definiteness *l* ¼ 0 and *rb* ≤ *rp*) we have *G*≈ln *R=rp* � �*<sup>=</sup>* ln ð Þ *<sup>R</sup>=rb* , but in opposite limit *G*≈exp �2*κ rp* � *rb* � � � � � � (for arbitrary *<sup>l</sup>*).

### **5.2 Growth rates**

The DR (44) determines proper oscillations of transversally no uniform beamplasma waveguide. The changes of the physical character of beam-plasma interaction must reveal themselves on its solutions. *Dp*,*<sup>b</sup>*ð Þ¼ *ω*, *k* 0 are the DR for waveguide with thin annular plasma and e-beam respectively. The spectra of fast (+) and slow (�) waves are

$$a\_{\pm} = kV\_b(1 + \mathfrak{x}\_{\pm}); \mathfrak{x}\_{\pm} = \left(\sqrt{a}/\gamma\right)\left(\pm\sqrt{\beta^4 \gamma^2 a + 1} - \beta^2 \gamma\sqrt{a}\right);\tag{47}$$

where *<sup>β</sup>* <sup>¼</sup> *Vb=c*. The parameter *<sup>α</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *b=k*<sup>2</sup> ⊥*bV*<sup>2</sup> *<sup>b</sup>γ*<sup>3</sup> is familiar (see above). It determines the beam current value: *<sup>α</sup>* <sup>¼</sup> *Ib<sup>=</sup> <sup>γ</sup>*<sup>2</sup> ð Þ *<sup>I</sup>*<sup>0</sup> (*Ib* is the beam current, *<sup>I</sup>*<sup>0</sup> is the limiting current in vacuum waveguide). In the limit of under-limiting beams *x*� ! � ffiffiffi *<sup>α</sup>* <sup>p</sup> *<sup>=</sup>γ*. In opposite limit of over-limiting beam *<sup>x</sup>*<sup>þ</sup> <sup>¼</sup> <sup>1</sup>*=*2*β*<sup>2</sup> *<sup>γ</sup>*<sup>2</sup> and *<sup>x</sup>*� ¼ �2*β*<sup>2</sup> *α*. If one looks for solutions of (44) in form *ω* ¼ *kVb*ð Þ 1 þ *x* , *x*< < 1 it becomes

$$\left(\varkappa + q + i\frac{\nu}{ku}\frac{1 - 2\beta^2 \gamma^2 \varkappa}{2\gamma^2}\right)(\varkappa - \varkappa\_+)(\varkappa - \varkappa\_-) = G\frac{a}{2\gamma^4}\left(1 - 2\beta^2 \gamma^2 \varkappa\right)^2,\tag{48}$$

where *<sup>q</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> ⊥*pu*<sup>2</sup>*γ*<sup>2</sup>*=ω*<sup>2</sup> *<sup>p</sup>* � 1 � �*=*2*γ*2. The Eq. (48) presents sound way to study

instabilities in given system. First of all, it is easily seen that in conditions of growing negative energy wave *x*≈*x*� and collective Cherenkov resonance *q*≈ � *x*� the role of dissipation increases. For under-limiting e-beams *α* ≤1*=γ*<sup>2</sup> and in case of strong coupling *G* � 1 the DR (44) leads to the well-known conventional beam instabilities of no-dissipative and dissipative type. The growth rates of these instabilities have well-known dependencies on beam density � *n* 1*=*3 *<sup>b</sup>* and on dissipation (� <sup>1</sup>*<sup>=</sup>* ffiffi *ν* p ). Both for these instabilities proper oscillations of the beam are neglected. Only for explanation of the physical meaning of the DSI the conception of NEW should be invoked. However, if *G* < <1 (weak coupling) the growing of the NEW plays dominant role. In this case the growth rate of no-dissipative instability reaches its maximum under Collective Cherenkov resonance *<sup>q</sup>* <sup>¼</sup> ffiffiffi *α* p *=γ* and is equal

$$\left(\mathrm{Im}\rho\right)\_{\mathrm{und}}^{\left(\nu=0\right)} = \left(kV\_b/2\gamma\right)\left(\mathrm{G}\sqrt{a}/\gamma\right)^{\mathrm{i}\cdot\mathrm{i}}.\tag{49}$$

This expression coincides to (27). Dissipation coming into interplay transforms this instability to DSI of new type with growth rate (coincides to (29))

$$(\text{Im}\,\rho)^{(\nu\to\infty)}\_{\text{und}} = \mathcal{G}\sqrt{a}(kV\_b)^2/2\eta\nu\tag{50}$$

As it should be, this is the instability discovered under consideration of the classical problem of the initial perturbation development in weakly coupled beamplasma systems.

Of particular interest are limit of high, over-limiting currents of e-beam *γ*�<sup>2</sup> < <*α* < < 1. In this case the DR (44) takes the form

$$\left(\varkappa + q + i\frac{\nu}{ku}\frac{1 - 2\gamma^2\varkappa}{2\gamma^2}\right)(\varkappa + 2a) = -G\frac{a}{\gamma^2}(1 - 2\gamma^2\varkappa) \tag{51}$$

For *ν* ¼ 0 the analysis of (51) leads to following. Under single particle resonance we have either instability of negative mass type (under *G* � 1) with the growth rate Im*<sup>ω</sup>* <sup>¼</sup> *ku* ffiffiffi *α* p *=γ*, or stability (under *G* < < 1). But under collective Cherenkov effect *q* ¼ 2*α* the growth rate of developing instabilities is

$$(\text{Im}\,a)^{(\iota=0)}\_{\text{ovl}} = \begin{cases} \sqrt{3}kV\_b a & \text{for} \quad G \sim 1\\ 2kV\_b a \sqrt{G} & \text{for} \quad G << 1 \end{cases} \tag{52}$$

The instability (52) under *G* � 1 has mixed mechanism: it is caused simultaneously (i) by a-periodical modulation of the beam density in media with negative dielectric constant and (ii) by excitation of the NEW. But the lower expression is the growth rate of instability caused only by excitation of the NEW of overlimiting e-beam. The presence of dissipation intensifies the growing of the slow beam wave. Instability turns to be of dissipative type with growth rate that again is inverse proportional to dissipation.

$$(\text{Im}\,\rho)^{(\nu)}\_{\text{ovl}} = \mathcal{Z}(ku)^2 \text{G}\alpha/\nu \sim \alpha\_b^2/\nu \tag{53}$$

However, the dependence on the beam density is completely different. This is the same DSI, which develops in uniform cross-section beam-plasma waveguide under over-limiting currents. Instabilities of the same type may be substantiated for finite thicknesses of the beam and plasma layers in waveguide. In this case one must use perturbation theory based on smallness of coupling coefficient.

As follows from this section, in the geometry of microwave plasma sources, the development of both new DSI is possible. Basic parameters of the both new DSI, and the conditions of their development should be taken into account upon design of the high power, high frequency plasma microwave devices.

## **6. Conclusion**

Thus, based on very general initial assumptions, we have found out that the number of DSI in the beam-plasma interaction theory is not limited by the only previously known type. Two new, previously unknown types of DSI are presented. The new DSI reveal themselves in the analysis of solution of the problem of initial perturbation development. This problem is classical in the theory of instabilities.

## *New Types of Dissipative Streaming Instabilities DOI: http://dx.doi.org/10.5772/intechopen.98901*

The first new DSI is the dissipative instability under weak beam-plasma coupling. In absence of dissipation the instability in these systems is caused by the interaction of the beam NEW with the plasma. With an increase in the level of dissipation this instability gradually transforms to the new type of DSI. Its maximal growth rate depends on the beam Langmuir frequency *ω<sup>b</sup>* and the frequency of collisions in plasma *ν* as *ωb=ν*. This, more critical (as compared to conventional), inverse proportional dependence on *ν* is a result of superposition of two factors those lead to growth of the beam NEW: weak coupling and dissipation.

The second new type of DSI is dissipative instability of over-limiting e-beam in uniform cross section waveguide. With increase in the beam current, its space charge and inner degrees of freedom reveal themselves more efficiently. If the beam current becomes higher than the limiting current in vacuum waveguide then the instability mechanism changes. In uniform cross section beam-plasma waveguide the instability becomes due to a-periodical modulation of the beam density in medium with negative dielectric constant. In this case the increase in the level of dissipation leads to one more new type of DSI with the maximal growth rate � *<sup>ω</sup>*<sup>2</sup> *<sup>b</sup>=ν*.

The same types of DSI develop in systems having geometry, similar to microwave sources: cylindrical waveguide with thin annular beam and thin annular plasma. If the coupling between the beam and the plasma hollow cylinders is weak and the beam current is under-limiting the first type of DSI develops, but under over-limiting currents – the second. However, if the coupling of the beam and the plasma cylinders is strong, conventional type of DSI develops with well-known growth rate � *<sup>ω</sup>b<sup>=</sup>* ffiffi *ν* p .

Both new DSI are confirmed by conventional analysis of the respective DR.

Some words about the approach used. It has many advantages. First of all, it is based on very general initial assumptions and gives results regardless on geometry and specific parameters. The same approach is used for solving the same problem for conventional beam-plasma instabilities of all types (Cherenkov type, cyclotron type etc) [13], for the Buneman instability [22] etc. Obtained expressions for the spatial–temporal distribution of growing fields clearly show that with increase in the level of dissipation in background plasma, all these SI transform into DSI of conventional type. In addition, the analysis of obtained expressions gives much more detailed information on SI than other methods give. Part of the information on SI is not available in any other ways. The coincidence of other information to the results of conventional analysis confirms the validity of the approach (initial assumptions, mathematics etc).

Also, the presented approach shows that the DR describing the SI of given type can serve not only for solving of the initial/boundary problems and obtaining the dispersion curves. This point of view is very simplified. The approach shows that much more additional information is available from the DR. It, in fact, provides results on the initial perturbation development.

Summarizing, one can state that the presented approach can serve as an independent and very effective method for studying of any SI. There is no need to solve the problem again. One should only substitute the parameters of given instability in general expression for the field's space–time distribution. The usage of this approach instead of traditional initial/boundary problems gives complete picture of the instability development. At first glance, it might seem that this method of analyzing instabilities is more complicated. However, this complexity is only apparent. In addition, this complexity, if any, is overlapped by the completeness of the information received.

*Plasma Science and Technology*
