By Shulim E. Tsimring

This e-book specializes in a primary characteristic of vacuum electronics: the powerful interplay of the physics of electron beams and vacuum microwave electronics, together with millimeter-wave electronics. the writer publications readers from the roots of classical vacuum electronics to the latest achievements within the box. distinct awareness is dedicated to the physics and conception of relativistic beams and microwave units, in addition to the speculation and purposes of particular units.

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**Example text**

2) with pitch h ¼ (2p=vg )vB . It is evident that equations of the trajectory for x0 ¼ y0 ¼ z0 ¼ 0 are x ¼ r? sin(vg t) y ¼ r? 2 Motion of an electron in a homogeneous magnetic ﬁeld. 5 MOTION OF ELECTRONS IN HOMOGENEOUS STATIC FIELDS 25 The circle is called a Larmor circle, the radius is a Larmor radius, the center of the Larmor orbit is a guiding center, and vg is the gyrofrequency or cyclotron frequency. It is essential that the relativistic gyrofrequency depends on the particle’s kinetic energy: vg ¼ e0 c 2 hB B¼ w g (1:39) Therefore, electrons in magnetic ﬁelds behave as nonisochronous oscillators.

A number of particles in the unity phase volume), F¼ dn dP dq where dn is the number of particles in the phase volume dP dq. The value F denotes density in the phase space. In general, if there are N interacting particles, the Lagrangian for each particle depends on the coordinates and velocities of all particles, and the phase space has 6N dimensions. Each element of dPi dqi is really a six-dimensional element in Euclidean space. 6 LIOUVILLE THEOREM 11 where the 6N velocity v has the components P_ and q_ .

22 MOTION OF ELECTRONS IN EXTERNAL ELECTRIC AND MAGNETIC STATIC FIELDS Integrating the ﬁrst equation in Eq. 21), we obtain px ¼ mvx ¼ p0 (1:22) As follows from the second equation in Eq. 21), dpy dpy dx p0 dpy ¼ ¼ e0 E ¼ dt dx dt m dx (1:23) According to Eq. 2), the mass may be written as rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 W02 2 2 2 2 2 2 2 m¼ p þ m0 c ¼ p0 þ m0 c þ py ¼ þ p2y (1:24) c c c c2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where W0 ¼ c m20 c2 þ p20 is the initial kinetic energy of the particle.