By Richard J. Briggs
This learn considers the instabilities that outcome whilst an electron beam is injected right into a plasma. a few varied versions of the method are thought of, and all instabilities are categorised in line with whether or not they are convective instabilities (amplifying waves) or nonconvective (absolute) instabilities. The research additionally analyzes the instabilities in unbounded beam-plasma platforms and in platforms of finite quantity transverse to the electron move and offers an in depth attention of the potential for a robust interplay with the ions in a hot-electron plasma. moreover, the writer offers mathematical standards for picking out absolute instabilities and amplifying waves. those standards are dependent basically on an research of the dispersion equation of the process and aren't constrained to beam-plasma systems.Two issues must be acknowledged approximately this ebook: the bankruptcy on absolute and convective instabilities makes a tremendous contribution to the sector. moment, it's going to be mentioned that the theoretical effects are diminished to a sort which lead them to available to an experimentalist. Plasma physicists and digital engineers can be drawn to this paintings.
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This examine considers the instabilities that outcome while an electron beam is injected right into a plasma. a few diversified types of the method are thought of, and all instabilities are categorized based on whether or not they are convective instabilities (amplifying waves) or nonconvective (absolute) instabilities.
Additional info for Electron-Stream Interaction with Plasmas
A brief indication of this gener al appr oa ch in the c a s e of distinguishing between ab solute and conve ctive instabilitie s wa s given by Landau and Lifshitz . 45 ) Even in the context o f his fo rmulation, however , Sturr o ck did not car efully con s ide r the effe cts of the b r anch points of w(k) and k(w) when defo rming contour s in the complex k and w plane s , as was pointe d out in Refer enc e s 51 and 5 2 . The s e s ingularitie s ar e of impor tance in any disper s ion equation which i s o f higher o r de r than fir s t i n w and k; fo r thi s r ea s on the r e i s little c o r r e spond ence of Stur r o ck' s r e sult with the r e sult s of the pr e s ent for mula tion .
24 Amplifying Wave s and Ab solute Ins tability a system which is "unstabl e " by thi s definition will always "blow Upll in amplitude , even though it may app ear to d e c r ease in time at a fixed point b e cau s e it could conve ct along the s ys tem a s it blows up . The validity of this statement i s not obvious from the analysis in Se ction 2 . 3 , s inc e we always inve stigated the asymp totic r e spon s e a t a fixed point i n space . The s tatement i s an im portant one , however , and is e s tablished in the p r e s ent s e ction.
W e then br ing Wr down fr om lar g e po s itive value s to the de s ir e d r eal fr equency ( slightly below Wpp in the pre s ent cas e ) , using the r e al w diagram t o follow the r o ot o f inter e s t ( Figur e 2 . 2 3) . In this p r o ce s s , we r etain the infinite s imal negative imaginary part of w in o r de r to avoid going dir e ctly through any "indeterminate " point s like wpp . For the r e a s on s dis cus s e d in the following para graph, the u s e o f thi s s mall imaginary part of w tells us how to j oin corre ctly the k roots on the oppo s ite side s of such s ingular point s .