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Understanding the Focusing of Charged Particle Beams in A Understanding the focusing of charged particle beams in a solenoid magnetic field ͒ Vinit Kumara Beam Physics and Free Electron Laser Laboratory, Raja Ramanna Centre for Advanced Technology, Indore, 452013 India ͑Received 10 January 2009; accepted 14 April 2009͒ The focusing of a charged particle beam in a solenoid is typically explained by invoking the concept of a Larmour frame and using Busch’s theorem. Often, there is some confusion about how a uniform magnetic field of a long solenoid focuses the electron beam because it is generally understood that a uniform magnetic field can only guide charged particles. We perform a simple analysis of the dynamics of a charged particle beam in a solenoid and emphasize an intuitive understanding of some of the interesting features. © 2009 American Association of Physics Teachers. ͓DOI: 10.1119/1.3129242͔ I. INTRODUCTION exits this region to enter a field-free region. We also discuss the effect of Coulomb repulsion on the beam dynamics. We A solenoid is often used to focus charged particle beams in present some conclusions in Sec. IV. the low energy section of accelerators1 and other devices such as electron microscopes.2 The focusing of charged par- 3,4 ticle beams in a solenoid is explained in textbooks using II. BEAM DYNAMICS IN SOLENOID Busch’s theorem,5 which is a statement of conservation of canonical angular momentum in the axisymmetric magnetic Before we discuss the dynamics of a charged particle field of the solenoid. In a solenoid, the longitudinal magnetic beam in a solenoid magnetic field, we state the assumptions. field on the axis is peaked at the center of the solenoid, First, we ignore the space-charge force, that is, we neglect decreases toward the ends, and approaches zero far away the Coulomb repulsion between charged particles. Second, from the solenoid. In contrast, the radial magnetic field is we assume a cold beam, that is, the particles’ initial trans- peaked near the ends of the solenoid. In a simple model, the verse velocity is zero. In general, both assumptions are vio- longitudinal magnetic field can be assumed to be zero out- lated in reality. Also, particles will have nonzero transverse side the solenoid and uniform inside it. When a charged par- velocities, which will have a random distribution inside the ticle enters from the field-free region to the region of uni- beam. We will discuss the effect of removing these assump- form magnetic field in a solenoid, it starts rotating with the tions later in this paper. Larmour frequency, which equals half the cyclotron fre- For simplicity, we assume that the beam is cylindrical with quency in the uniform magnetic field. The dynamics of the a uniform distribution of particles. All the particles are as- charged particle is easily analyzed in the Larmour frame, sumed to have an initial velocity vz along the z-axis. The which rotates with the Larmour frequency around the axis of components of the axisymmetric magnetic field of a solenoid the solenoid. In the Larmour frame, the centrifugal force is are given by4 half of the Lorentz force, and there is a net focusing force toward the axis of the solenoid. r2 ͑ ͒ ͑ ͒ Љ͑ ͒ ͑ ͒ There is often some confusion about why a charged par- Bz r,z = B z − B z + ¯ , 1 ticle does not rotate with the cyclotron frequency inside the 4 solenoid, as is the case in a uniform magnetic field. It is also confusing how the uniform field in a solenoid provides fo- r r3 cusing because a charged particle in a uniform magnetic field B ͑r,z͒ =− BЈ͑z͒ + Bٞ͑z͒ + , ͑2͒ has a helical trajectory with a constant radius known as the r 2 16 ¯ Larmour radius. The answer to these questions often is bur- ied by the formal analysis in textbooks. where z is the distance along the solenoid axis, r is the radial In this paper, we take a fresh look at the problem and distance from the solenoid axis, and the prime denotes a perform a simple analysis. We use simple geometric concepts derivative with respect to z. The coordinate system used is to understand some of the interesting aspects of the dynamics shown in Fig. 1. Under the paraxial approximation, we will of charged particle beams in a solenoid magnetic field, in- keep only terms up to the first order term in r. We assume cluding the focusing of charged particle beams. The empha- ͑ ͒ Ͻ Ͻ ͑ ͒ that B z =B0 for 0 z L and B z =0 otherwise. We are sis in this paper is on developing a physical understanding of thus assuming that B͑z͒ abruptly drops to zero at the ends. different aspects of dynamics using simple mathematics. We will show later that our analysis can be generalized to an In Sec. II we analyze the beam dynamics of a charged arbitrary variation in B͑z͒. From Eq. ͑2͒ the expression for particle beam in an axisymmetric magnetic field. In Sec. III the magnetic field in this case in the paraxial approximation we discuss some special cases. We first apply this analysis to can be written as a thin lens and derive the formula for its focal length. We then apply this analysis to the case where an electron beam ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ generated in a region of uniform magnetic field subsequently Bz = B0 u z − u z − L , 3 737 Am. J. Phys. 77 ͑8͒, August 2009 http://aapt.org/ajp © 2009 American Association of Physics Teachers 737 x .A A´ .E z . A′ E′ . D.′ 2θ D. O´ O B y r .B′ .′ θ C . C O Fig. 1. Pictorial explanation of the focusing of a charged particle beam in a Fig. 2. Illustration of the relation between the Larmour frequency and the solenoid. The solid curve shows the periphery of the electron beam when it cyclotron frequency. Point O is on the axis of the solenoid and point OЈ is enters the solenoid. The dashed curve shows the periphery of the electron the center of the trajectory. beam after it travels some distance in the solenoid. The dotted curves show the trajectories of individual electrons. The coordinate axis is also shown. ␥m r ͯ v␪ ͯ 0 ͑ ͒ Rc = = . 6 eB0 2 r ͓␦͑ ͒ ␦͑ ͔͒ ͑ ͒ ͑ ͒ Br =− B0 z − z − L , 4 Equation 6 tells us that for every off-axis particle that en- 2 ters the magnetic field, the radius of curvature inside region ͑ ͒ Ͼ ͑ ͒ ␦͑ ͒ II is half of the initial radial displacement of the particle from where u z =1 for z 0 and u z =0, otherwise. Here, z is the solenoid axis. The projections of particle trajectories on the Dirac delta function. the x-y plane are shown in Fig. 1 by dotted curves. It is clear Ͻ ͑ ͒ We define three regions corresponding to z 0 region I , that every particle will just touch the solenoid axis and will Ͻ Ͻ ͑ ͒ Ͼ ͑ ͒ 0 z L region II , and z L region III . The trajectory of turn back as is shown. In Fig. 1 we have shown the cross the charged particle will be a straight line in regions I and III section of the periphery of the beam when it enters region II because these regions are field-free. The trajectory will be by a solid line, and four particles located at points A, B, C, helical in region II because it is a region of uniform magnetic and D on its periphery. As the beam travels through region II, field. We have to match the trajectory at the boundary be- the projection of the trajectories of these particles on the x-y tween regions I and II and between regions II and III. At the plane is shown by dotted circles. After the beam travels a boundary between regions I and II, the radial magnetic field certain distance in region II, these particles move from loca- ͑ ͒ is a Delta function as seen from Eq. 4 , which gives an tions A, B, C, and D to locations AЈ,BЈ,CЈ, and DЈ, respec- impulse in the azimuthal direction. The force in the azi- tively, as shown in Fig. 1. For clarity, we also have shown a ͑ ͒ muthal direction is given by −evzBr z , where e is the mag- particle located at E, which is inside the periphery of the nitude of the electronic charge. Due to this impulse, the in- beam. The trajectory of this particle is obtained by joining crement ⌬v␪ in the azimuthal velocity of the electron as it this point with the solenoid axis and then drawing a circle of crosses this boundary is given by diameter OE through these points. The particle moves to a location EЈ when the other particles move to AЈ,BЈ,... It eB ⌬ 0 ͑ ͒ is seen that the radius of the electron beam shrinks from OA v␪ = r0 , 5 2␥m to OAЈ, and the periphery of the beam shrinks to AЈBЈCЈDЈ, as shown by the dashed curve in Fig. 1. In this manner, the where r0 is the radial coordinate of the particle when it enters beam undergoes periodic focusing in the region of uniform ␥ /ͱ 2 / 2 region II, m is the electron’s rest mass, and =1 1−vz c magnetic field in the solenoid. is the usual Lorentz factor. The cyclotron frequency is de- As expected, the particles rotate with the cyclotron fre- ␻ /␥ fined as c =eB0 m and the Larmour frequency is defined quency about an axis passing through the center of their in- ␻ / ␥ ⌬ ␻ as L =eB0 2 m.
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