Ion Optics

P(E) = Γ

(E−Er)²+ (Γ/2)² (2.20)

The resonance energy is given byEr.

It is now possible to plot the resonance strength and the width at half maximum ∆ of the Gamow window at different temperatures side by side with the nuclear core level scheme of the product nuclei.

2.4 Ion Optics

The general task in beam optics is to transport charged particles from point A to point B along a desired path. The collection of bending and focusing magnets installed along this ideal path are called the magnet lattice and the complete optical system including the bending and focusing parameters is called a beam transport system (see also [28]).

2.4.1 Particle Beam Guidance

To guide a charged particle along a predefined path, magnetic fields are used which deflect particles as determined by the equilibrium of the centrifugal force and Lorentz force

mγv²~k+q

~ v×B~

= 0 (2.21)

where~k= (kx, k_y,0) is the local curvature vector of the trajectory, which is pointing in the direction of the centrifugal force. Assuming that the magnetic field vectorB~ is oriented normal to the velocity vector~v, the treatment of linear beam dynamics is restricted to purely transverse fields. This has no fundamental reason other than to simplify the formulation of particle beam dynamics. The transverse components of the particle velocities for relativistic beams are small compared to the particle velocity.

A curvilinear coordinate system(x, y, z)following the ideal path is used. The direction of the particle is denoted with the coordinates: The bending radius for the particle trajectory in a magnetic field is from equation2.21withp=γmv

~kx,y=±qc βE

B~y,x (2.22)

and the angular frequency of revolution of a particle on a complete orbit normal to the fieldB is ωL=

qc EB

(2.23)

which is also called the cyclotron or Larmor frequency. The sign in equation2.22has been chosen to meet the definition of curvature in analytical geometry, where the curvature is negative, if the tangent to the trajectory rotates counterclockwise. Often, the beam rigidity, defined as

|Bρ|= p0

q (2.24)

is used to normalize the magnet strength. Using more practical units the expressions for the beam rigidity and bending radius become

Bρ[Tm] = 10

2.998βE[GeV] (2.25)

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12 CHAPTER 2. THEORY

and

1

ρ[m⁻¹] = B

Bρ= 0.2998Z A

|B[T]|

βE[GeV/u] (2.26)

where the sign for the bending radius was dropped. For relativistic particles this expression is further simplified sinceβ ≈1. The deflection angle in a magnetic field is

θ= Z dz

ρ (2.27)

or for a uniform field like in a dipole magnet of arc length l_m the deflection angle isθ=l_m/ρ.

2.4.2 Particle Beam Focusing

Similar to the properties of light rays, particle beams also have a tendency to spread out due to an inherent beam divergence. To keep the particle beam together and to generate specifically desired beam properties at selected points along the beam transport line, focusing devices are required. Any magnetic field, that deflects a particle by an angle proportional to its distance r from the axis of the focusing device, will act in the same way as a glass lens does in the approximation of paraxial, geometric optics for visible light. Iff is the focal length, the deflection angleαis defined by

α=−r

f (2.28)

A similar focusing property can be provided for charged particle beams by the use of azimuthal magnetic fieldsBφ with the property

α=−l

ρ =−qc

βEBφl=−qc

βEgrl (2.29)

wherelis the path length of the particle trajectory in the magnetic fieldBφandgis the field gradient defined by B_φ=gr or byg= dB_φ/dr.

In beam dynamics, it is customary to define an energy independent focusing strength. Similar to the definition of the bending curvature a focusing strengthK is define by

K= q pg= qc

βEg (2.30)

2.4.3 Equation of Motion

Magnetic fields are used to guide charged particles along a prescribed path or at least keep them close by. This path, or reference trajectory, is defined geometrically by straight sections and bending magnets only. Dipole magnets deflect the path and quadrupole and higher order magnets do not influence this path but provide the focusing forces necessary to keep all particles close to the reference path.

The most convenient coordinate system to describe particle motion is the curvilinear coordinate system as seen in Figure 2.3. The curvatures are functions of the coordinate z and are nonzero only where there are bending magnets. The equations of motion are derived in the horizontal plane only. The generalization to both horizontal and vertical plane is straightforward. Using the notation of Figure 2.4 the deflection angle of the ideal path is dφ0 = dz/ρ0 or utilizing the curvature to preserve the directionality of the deflection

dφ0= +k0dz (2.31)

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2.4. ION OPTICS 13

y Fig. 2.3 Locally defined right-handed

coordinate system used with bending design.

Here s is the distance along the design orbit, x is the distance from this orbit along the radius of curvature, and y is the distance from the design orbit out of the bend plane.

along the trajectory are a superposition of angle and offset perturbations. From these observations, we may strongly suspect that the perturbed trajectory is a harmonic oscillation, but we must first develop the proper analysis tools to verify this suspicion.

As stated in Chapter 1, as we will typically take the distance along the design orbit to be the independent variable (in this case indicated by

s), we implicitly

wish to analyze the charged particle dynamics near the design orbit. In the present case, this orbit is specified by a certain radius of curvature

R

(and thus a certain momentum

p₀ = qB₀R), and center of curvature,(x₀

,

y₀)

. With this choice of analysis geometry, we can locally define a new right-handed coordinate system

(x,y,s), as shown in Fig. 2.3. In this coordinate system, x

is the distance of the orbit under consideration from the design orbit, in the direction measured along the radius and normal to

s. The distancey

(formerly indicated by the coordinate

z

in Section 2.1) is measured from the design orbit

to the particle orbit under consideration, in the direction out of the bend plane.

^{1 1}This convention, in which the symboly is defined to be the distance out of the bend plane is typical of the American literature. European beam physicists more often use the symbolz instead, but we do not follow this conven-tion even though it connects more naturally to our previous discussion. This is because our adopted convention makes subsequent derivations somewhat easier to understand, and also because it allows the connection between linear accelerator and circular accel-erator coordinate systems to become more obvious.

The choice of a right-handed system in this case is a function of the direction of the bend, and in simple circular accelerators, one is free to construct the curvilinear coordinates once and for all. On the other hand, when we encounter bends in the opposing direction, as in chicane systems (see Chapter 3), we will choose to consistently define the coordinate

x, so that it is positive along

the direction away from the origin of the bend. As we will also choose to leave the vertical direction unchanged in this transformation, a left-handed coordinate system will result when the bend direction is changed.

The coordinate system shown in Fig. 2.3 is quite similar to a cylindrical coordinate system, with

x

related to the radial variable

ρ

by the definition

x≡ρ−R,s

replacing the azimuthal angle

φ (

ds

=R

d

φ)

, and

y, as previously

noted, replacing

z. Thus we can write the equations of motion for orbits in this

system by using the Lagrange–Euler formulation (see Problem 2.1), as

dp

_ρ

dt

= γm₀v²_φ

ρ −qv_φB₀

, (2.9)

where

v_φ =ρφ˙

is the azimuthal velocity.

Equation (2.9) can be cast as a familiar differential equation by using

x

as a small variable (x

R, which is also equivalent, as will be seen below, to

the paraxial ray approximation) to linearize the relation. This is accomplished through use of a lowest order Taylor series expansion of the motion about the design orbit equilibrium

(px =p_ρ =

0) at

ρ =R,

dp

x

dt

∼= −γ0m₀v²₀

R² x.

(2.10)

Figure 2.3: Locally defined right-handed coordinate system used with bending design. Heresis the distance along the design orbit,xis the distance from this orbit along the radius of curvature, andy is the distance from the design orbit out of the bend plane. [29]

5.3 Equation of Motion 107

Fig. 5.4 Particle trajectories

The ideal curvature 0 is evaluated along the reference trajectory u D 0 for a particle with the ideal momentum. In linear approximation with respect to the coordinates the path length element for an arbitrary trajectory is

dsD.1C0u/dzCO.2/; (5.22) where u D x or y is the distance of the particle trajectory from the reference trajectory in the deflecting plane.

The magnetic fields depend onzin such a way that the fields are zero in magnet free sections and assume a constant value within the magnets. This assumption results in a step function distribution of the magnetic fields and is referred to as the hard edge model, generally used in beam dynamics. The path is therefore composed of a series of segments with constant curvatures. To obtain the equations of motion with respect to the ideal path we subtract from the curvature for an individual particle the curvature0of the ideal path at the same location.

Sinceuis the deviation of a particle from the ideal path, we get for the equation of motion in the deflecting plane with respect to the ideal path from Fig.5.4 and (5.20), (5.21) withu⁰⁰D .d'=dzd'0=dz/,

u⁰⁰ D .1C0u/C0; (5.23)

where the derivations are taken with respect to z. In particle beam dynamics, we generally assume paraxial beams,u⁰² 1since the divergence of the trajectories u⁰ is typically of the order of 10³ rad or less and terms in u⁰² can therefore be neglected. Where this assumption leads to intolerable inaccuracies the equation of motion must be modified accordingly.

The equation of motion for charged particles in electromagnetic fields can be derived from (5.23) and the Lorentz force. In case of horizontal deflection, the curvature is D x and expressing the general field by its components, we have

Figure 2.4: Particle trajectories in deflecting systems. Reference pathz and individual particle tra-jectoryshave in general different bending radii [28]

wherek₀ is the curvature of the ideal path. The deflection angle for an arbitrary trajectory is then given by

dφ= +kds (2.32)

The ideal curvaturek₀is evaluated along the reference trajectory u= 0for a particle with the ideal momentum. In linear approximation with respect to the coordinates the path length element for an arbitrary trajectory is

ds= (1 +k0u)dz+O(2), (2.33)

whereu=xoryis the distance of the particle trajectory from the reference trajectory in the deflecting plane.

The magnetic fields depend onz in such a way that the fields are zero in magnet free sections and assume a constant value within the magnets. This assumption results in a step function distribution of the magnetic fields and is referred to as the hard edge model, generally used in beam dynamics. The path is therefore composed of a series of segments with constant curvatures. To obtain the equations of motion with respect to the ideal path the curvaturek₀ of the ideal path is subtracted from the

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14 CHAPTER 2. THEORY curvaturekfor an individual particle at the same location. Sinceuis the deviation of a particle from the ideal path, for the equation of motion in the deflecting plane with respect to the ideal path from Figure2.4 and equation2.31,2.32withu⁰⁰=−(dφ/dz−dφ0/dz),

u⁰⁰=−(1 +k0u)k+k0 (2.34)

where the derivations are taken with respect to z. Paraxial beams were assumed, u⁰² 1, since the divergence of the trajectoriesu⁰ is typically of the order of10⁻³rad or less and terms inu⁰² can therefore be neglected. Where this assumption leads to intolerable inaccuracies the equation of motion must be modified accordingly.

The equation of motion for charged particles in electromagnetic fields can be derived from equation 2.34 and the Lorentz force. In case of horizontal deflection, the curvature isk =k_x and expressing the general field by its components with equation 2.22

k_x= 1 1 +δ

k_0x+Kx+1 2mx²

(2.35)

where the field is expanded into components up to second order. The three lowest order multipoles, a bending magnet, a quadrupole and a sextupole were used. A real particle beam is never monochro-matic and therefore effects due to small momentum errors must be considered. This can be done by expanding the particle momentum in the vicinity of the ideal momentump0

1

p= 1

p₀(1 +δ)≈ 1

p₀(1−δ+. . .) (2.36) The horizontal plane (u=xand k=kx) is now applied to equation 2.34to get with equation 2.35 and2.36the equation of motion

x⁰⁰+ (K+k²_0x)x=k0x(δ−δ²) + (K+k²_0x)xδ−1

2mx²−k0Kx²+O(3) (2.37) The definitions of energy independent field strength parameters as defined in equation 2.30and2.22 was used. The equation of motion in the vertical plane can be derived in a similar way by setting u=y in equation2.34andk=ky. The equation of motion in the vertical plane thereby is

y⁰⁰−(K−k₀²_y)y=k_0y(δ−δ²)−(K−k²_0y)yδ+1

2my²+k₀Ky²+O(3) (2.38)

2.4.4 Solutions for the linear Equations of Motion

Equations 2.37 and 2.38 are the equations of motion for strong focusing beam transport systems, where the magnitude of the focusing strength is a free parameter. No general analytical solutions are available for arbitrary distributions of magnets. The best tool in the mathematical formulation of a solution to the equations of motion is the ability of magnet builders and alignment specialists to build magnets with almost ideal field properties and to place them precisely along a predefined ideal path.

In addition, the capability to produce almost monochromatic particle beams is of great importance for the determination of the properties of particle beams. As a consequence, all terms on the right-hand side of 2.37and 2.38can and will be treated as small perturbations and mathematical perturbation methods can be employed to describe the effects of these perturbations on particle motion.

The left-hand side of the equations of motion resembles that of a harmonic oscillator although with a time dependent frequency. By a proper transformation of the variables equations2.37and2.38can be expressed exactly in the form of the equation for a harmonic oscillator with constant frequency.

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2.5. DIRECTION COSINES AS A BEAM PROPERTY 15

Im Dokument Ion Optics Feasibility Study of the Astrophyscial 33,34m Cl(p, γ) Reactions at the CRYRING  (Seite 19-23)