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1 Introduction to ion-optic This appendix is intended to provide a short introduction to the formalism describing the transport of charged particles in magnetic fields. Due to the analogies between ion-optic and geometrical optic, the formalism used when studying the behaviour of charged particles traversing a magnetic system is similar to that used in geometrical optic. A more detailed description of the Ion Optic can be found in reference []. This appendix is divided in three parts: in the first, we will explain very roughly the physical relation between magnetic and optic elements. In the second part we introduce the formalism and notation used to describe the trajectories of ions in magnetic fields. Finally, in the third part, some important concepts related to this topic will be defined. 2 Analogy between optical and magnetic systems In order to justify the formalism used in ion-optic, we will establish the physical relations between magnetic elements and optic elements. 2. Prism versus Dipole In geometric optic, a prism is the optic element which separates the different frequencies of a ray of white light. After the light crosses the prism, each ray with well defined frequency follows a trajectory with an angle depending on this frequency (figure left). In the ion-optic framework, the equivalent magnetic element is a dipole. In this case, a charged particle traversing the magnet follows a trajectory which is deflected depending on its momentum or charge (figure right). The curvature radius of the trajectory followed by the particle can be calculated with the following equation: ρ = p q () where is the magnetic field, p the momentum of the particle and q its charge. Figure illustrates schematically the behaviour of a ray of light and a charged particle traversing a prism and a dipole magnet, respectively.

2 ν ν 2 PRYSM ν 3 DIPOLE P 3 P2 P Figure : Left: White light (with a continuum spectrum of frequencies) enters into a prism. After crossing it, each frequency is split up in a different trajectory. Right: a bunch of charged particles with different momenta enters into a dipole. At the exit of the dipole, each particle with different momenta follows a different trajectory. 2.2 Lenses versus Quadrupoles In geometrical optic, the lenses can be classified in two kinds: lenses for focusing and lenses for defocusing. They are schematically depicted in figure 2 SOURCE SOURCE IMAGE FOCALIZING LENS DEFOCALIZING LENS Figure 2: Left: focalisation of a ray of light performed by a lens. defocalmisation of a ray of light. Right: The magnetic element which "focuses" or "defocuses" the trajectory of the charged particle is the quadrupole. Figure 3 shows the perpendicular section of the quadrupole respect to the trajectory followed by the beam of particles. The charged particle follows a trajectory depending on the direction of the field lines. This trajectory can be calculated by considering the Lorentz force: ~F = q ~v ~ (2) Due to the structure of the quadrupoles, a particle focused in x-direction will be defocused in y-direction and vice-versa. Therefore, to achieve a complete focalisation in both directions, it is necessary at least two quadrupoles 2

3 Y Y N S S N X X S N N S Figure 3: Cross section of two quadrupoles, one rotated 90 o other. respect to the rotated 90 o, one respect to the other. The intensity of the magnetic fields depends on the transversal distance from the position of the traversing particle to the symmetry axis of the quadrupole: it will vary from a minimum value equal to zero in the symmetry axis to a maximum value in the borders of the magnet. Figure 4 illustrates the action of two quadrupoles focusing a charged particle. 3

4 F D X Z Y D F Figure 4: Focalisation of the trajectory of a charged particle performed with two quadrupoles, one rotated 90 o respect to the other. x and y are the transversal coordinates of the particle respect to the symmetry axis of the magnets. The first quadrupole focuses in x direction and defocuses in y, whereas the second quadrupole defocuses in x and focuses in y. The focusing power is proportional to the transversal distance to the symmetry axis (z axis). 3 Ion-optical framework A beam of particles can be represented by a set of points in the phase-space. In this space, each particle has three coordinates giving their position and three coordinates specifying the momentum. From the region of phase-space populated by the beam, one can select a point to be the reference particle. The trajectory followed by this particle along the magnetic system is the central-trajectory and its momentum is the reference momentum p c. The coordinates of the other particles are defined respect to those of the reference particle: We will denote by s the distance along the central-trajectory. At any point of this trajectory we will define a vector ~z parallel to the reference momentum. Coordinates x and y are perpendicular to this vector and they specify the transversal distance to the central trajectory. The momentum of each particle is defined by its three components p x, p y 4

5 and p t. In ion-optic formalism, another three differentvariables related to the momentum are used to describe the motion of charged particles in magnetic fields: they are the tangent of the transversal angles and the relative deviation from the reference momentum. The tangent is the ratio of transversal to longitudinal momentum: tan() = p x p t (3) tan(ffi) = p y p t Since p x and p y are much smaller than p t,we can approximate the angles by the value of their tangent: ' tan() (4) ffi ' tan(ffi) These coordinates are represented in figure 5. Particle S Y X φ Z θ Central trajectory Reference particle Figure 5: Reference frame and coordinates defining the state of a charged particle traversing a magnetic system. The fractional momentum deviation (ffip) isdefinedby the following equation: ffip = p p c p c (5) where p c is the reference momentum. Since p / ρ, onecandefine the fractional magnetic rigidity deviation (ffiρ) to be used as an equivalent variable. 5

6 3. Matrix formalism In the following, we will describe the mathematical formalism used when studying the behaviour of a charged particle in a magnetic system. The trajectory followed by a charged particle along a magnetic element can be obtained solving its equations of motion for the different variables describing the @s = F x (x; ; y; ffi; ffiρ) (6) = F (x; ; y; ffi; ffiρ) = F y (x; ; y; ffi; ffiρ) = F ffi (x; ; y; ffi; ffiρ) where m is the mass of the particle and F x, F,... are functions which can depend on the variables needed to define the state of the particle. The solution of these equations is also expressed by complicated functions: x = f x (x 0 ; 0 ;y 0 ;ffi 0 ;s 0 ;ffiρ 0 ) (7) = f (x 0 ; 0 ;y 0 ;ffi 0 ;s 0 ;ffiρ 0 ) y = f y (x 0 ; 0 ;y 0 ;ffi 0 ;s 0 ;ffiρ 0 ) ffi = f ffi (x 0 ; 0 ;y 0 ;ffi 0 ;s 0 ;ffiρ 0 ) In order to simplify these solutions, we can expand them in a power series, including only lower order terms. The following equation shows this expansion up to first order terms: 0 x y 0 ffi 0 s 0 ffiρ 0 0 x (8) where the prime refers to the initial values of the variables. This simplification of the problem allows the description of the particle by avector of six coordinates ~ X =(x; ; y; ffi; s; ffiρ), whereas the transport carried out by the magnetic element is condensed in a matrix: 6

7 (x j x) (x j ) (x j y) (x j ffi) (x j s) (x j ffiρ) ( j x) ( j ) ( j ffiρ) (ffiρ j x) (ffiρ j ) (ffiρ j ffiρ) C A (0) Here we have used the rown notation [2], where (x j x) 0, (x j ) 0, an so on. Therefore, each magnetic element can be described by a matrix, whereas the state of the traversing particles is given by a vector. The product of this vector by the matrices describing the magnetic elements gives the evolution of the six coordinates specifying the state of the particle [2, 3]: x y ffi s ffiρ C A = (x j x) (x j ) (x j y) (x j ffi) (x j s) (x j p)) ( j x) ( j ) ( j p) : : : : (ffiρ j x) (ffiρ j ) (ffiρ j p) C A () x 0 0 y 0 ffi 0 s 0 ffiρ 0 C A The equation described above includes only first order terms from the expansion of the solutions. This is a rough simplification which must be improved in order to get a more precise description of the trajectories followed by the particles. Therefore, higher order terms are often included in the matrix formalism. The calculation of such matrices is a hard work which must be done by codes like GICO[4]. Nowadays, the matrix formalism is a very powerful tool used to design Monte-Carlo codes which simulates the pass of charged particles through magnetic systems. 7

8 4 Important concepts used in ion-optic 4. Image plane We define the focal plane as the place where the transversal x and y positions of the particle do not depend on its transversal angles and ffi, i.e.: (x j ) = (x j ffi) =0 (2) (y j ) = (y j ffi) =0 (3) In addition, if the transversal size of the beam reaches its minimumachievable value at this place, then the plane is called image plane. In most of the magnetic systems used in high and low energy physic such as spectrometers, accelerators or beam lines, the conditions to have an image plane can be achieved by using quadrupoles and sextupoles. 4.2 Magnification Magnification is the variation in the magnitude of a coordinate of the particle, from one image plane to the next one. For instance, (x j x) ij is the magnification of the x-coordinate from the image plane i to the image plane j. oth dipoles and quadrupoles contribute to define the magnification of the system. 4.3 Dispersion As mentioned in section A.., a bending magnet deflects the trajectory of a charged particle depending on its charge and momentum. Two identical particles, following the same trajectory and with different momenta will have a different curvature radius inside a bending magnet, so that both trajectories will be separated after traversing the magnet. The transversal distance between the central-trajectory and the trajectory of a particle with ffiρ=% behind a bending magnet is called dispersion. In the rown notation, the dispersion is defined as (x j ffiρ) and its value is expressed in cm=%. 4.4 Achromatism A magnetic system is achromatic when its dispersion is equal to zero, i.e. the deflection of particles does not depend on their momentum. In order to express the conditions fulfilled by such a system in the matrix formalism, one has to consider the evolution of both the transversal x-coordinate and the 8

9 relative momentum deviation of the traversing particle. The value of these two variables can be calculated with the following equation: x2 (x j x)02 (x j ffiρ) = 02 x0 (4) ffiρ 2 0 ffiρ 0 where index 0 refers to the state of the particle at the entrance of the system and the index 2tothestate at the exit. An achromatic system is always divided in two stages, both separated with a central image plane. The evolution of x and ffiρ from the entrance of the system to this central image plane are given by the next equation: x (x j x)0 (x j ffiρ) = 0 x0 (5) ffiρ 0 ffiρ 0 and the same can be done from the central image plane to the end of the system: x2 (6) ffiρ 2 = (x j x)2 (x j ffiρ) 2 0 x ffiρ In this two equations, the index refers to the central image plane. Therefore, the matrix 4 will be the product of the matrices 5 and 6, so that: (x j x)02 (x j ffiρ) 02 0 = (x j x)2 (x j ffiρ) 2 0 (x j x)0 (x j ffiρ) 0 0 (7) From this equation we can calculate the dispersion of the whole system: (x j ffiρ) 02 =(x j x) 2 (x j ffiρ) 0 +(x j ffiρ) 2 (8) Since this dispersion must be equal to zero (x j ffiρ) 02 = 0, we can deduce the conditions that must satisfy each stage, in order to have an achromatic system: (x j x) 2 (x j ffiρ) 0 = (x j ffiρ) 2 (9) 9

10 References [] David C. Carey. The Optics of the Charged Particle eams. Harwood Academic Publishers; Switzerland, 987. [2] K.L. rown, SLAC-Report No 9, Stanford (970). [3] S. Penner, Rev. Sci. Instr. 32 (96) 50. [4] H. Wollnik, Manual for GICO (University of Giefien, 990). 0