Particle Accelerator Physics pp 669-697 | Cite as

# Dynamics of Coupled Motion*

## Abstract

In linear beam dynamics transverse motion of particles can be treated separately in the horizontal and vertical plane. This can be achieved by proper selection, design and alignment of beam transport magnets. Fabrication and alignment tolerances, however, will introduce, for example, rotated quadrupole components where only upright quadrupole fields were intended. In other cases like colliding beams for high energy physics large solenoid detectors are installed at the collision points to analyse secondary particles. Such solenoids cause coupling which must be compensated.

## Keywords

Coupling Coefficient Beam Cross Section Linear Coupling Couple Beam Beam Dynamic

In linear beam dynamics transverse motion of particles can be treated separately in the horizontal and vertical plane. This can be achieved by proper selection, design and alignment of beam transport magnets. Fabrication and alignment tolerances, however, will introduce, for example, rotated quadrupole components where only upright quadrupole fields were intended. In other cases like colliding beams for high energy physics large solenoid detectors are installed at the collision points to analyse secondary particles. Such solenoids cause coupling which must be compensated. The perturbation caused creates a coupling of both the horizontal and vertical oscillation and independent treatment is no longer accurate. Such linear coupling can be compensated in principle by additional rotated or skew quadrupoles, but the beam dynamics for coupling effects must be known to perform a proper compensation.

Since coupling is caused by linear as well as nonlinear fields, we observe this effect in virtually any accelerator. In order to be able to manipulate coupling in a controlled and predictable way, we need to understand its dynamics in more detail. In this chapter, we will derive first the equations of motion for the two most general sources of coupling, the solenoid field and the field of a rotated quadrupole, solve the equations of motion and formulate modifications to beam dynamics parameters and functions of linear uncoupled motion.

## 20.1 Equations of Motion in Coupled Systems

*S*for the solenoid field not to be confused with the sine-like solution. In the following subsections we will derive separately the transformation through both rotated quadrupoles and solenoid magnets.

### 20.1.1 Coupled Beam Dynamics in Skew Quadrupoles

*z*varies between zero and the full length of the quadrupole, \(0 < z <\ell _{q}\). The coefficients \(a,b,c,\ldots D\) must be determined to be consistent with the initial parameters of the trajectories \((x_{\mathrm{0}},x_{\mathrm{0}}^{{\prime}},y_{\mathrm{0}},y_{\mathrm{0}}^{{\prime}})\). For

*z*= 0 we get

*f*. The matrix (20.9) then reduces to the simple form

*ℓ*

_{sq}= 0 but retained the linear terms in \(\ell_{\mathrm{sq}}\), which is a more appropriate thin-lens approximation for weak skew quadrupoles of finite length. Along the diagonal, the transformation matrix looks like a drift space of length \(\ell_{\mathrm{sq}}\) while the off-diagonal elements describe the coupling due to the thin skew quadrupole.

### 20.1.2 Particle Motion in a Solenoidal Field

*B*

_{s}, assumed to be colinear with the

*z*-direction, can be derived from ( 6.103)

*z*by derivatives with respect to the time, use the particle velocity

*v*, and replace \(\frac{\mathrm{d}} {\mathrm{d}z} \rightarrow \frac{1} {v} \frac{\mathrm{d}} {\mathrm{d}t}\). In a uniform solenoid field the equations of motion are then

*E*is the total particle energy. Multiplying (20.16) by \(\dot{x}\) and \(\dot{y}\), respectively, and adding both equations we get \(\mathrm{d}(\dot{x}^{2} + \dot{y}^{2})/\mathrm{d}t = 0\) or

*v*

_{t}or total transverse momentum of the particle

*cp*

_{t}stays constant during the motion in a uniform solenoid field. For \(\dot{x}_{\mathrm{0}} = 0\) and \(\dot{y}_{\mathrm{0}} = v_{\mathrm{t}}\), for example, the transverse velocities can be expressed by

*p*

_{t}=

*γ m v*

_{t}. The longitudinal motion is unaffected for not too strong solenoid fields and \(\dot{v}_{\mathrm{z}} = 0\) as can be derived from the Lorentz equation since all transverse field components vanish and

*z*-axis a distance

*p*

_{z}is the

*z*-component of the particle momentum.

*ϕ*may be a function of the independent variable

*z*. A single differential equation can be formed from (20.25) in complex notation

*z*

_{0}. We are able to eliminate two terms in the differential equation (20.28). Since a positive solenoid field generates Lorentz forces that deflect the particles onto counter clockwise spiraling trajectories, we have included the negative sign in (20.30) to remain consistent with our sign convention. From (20.30) it follows that \(\phi ^{{\prime}} = -\tfrac{1} {2}S(z)\) and \(\phi ^{{\prime\prime}} = -\tfrac{1} {2}S^{{\prime}}(z)\), which after insertion into (20.28) results in the simple equation of motion

*R*=

*v*+ i

*w*, we finally get two uncoupled equations

Introducing a coordinate rotation allow us to reduce the coupled differential equations (20.25) to the form of uncoupled equations of motion exhibiting focusing in both planes. At the entrance to the solenoid field *ϕ* = 0 and therefore *v*_{0} = *x*_{0} and *w*_{0} = *y*_{0}. To determine the particle motion through the solenoid field of length *L*_{s} we simply follow the particle coordinates (*v*, *w*) through the solenoid as if it were a quadrupole of strength \(k_{\mathrm{s}} = \tfrac{1} {4}\,S^{2}(L_{\mathrm{ s}})\) followed by a rotation of the coordinate system by the angle −*ϕ*(*L*_{s}) thus reverting to Cartesian coordinates (*x*, *y*).

### 20.1.3 Transformation Matrix for a Solenoid Magnet

*z*inside the solenoid magnet. The strength parameter in this case is \(\frac{1} {4}S^{2}\) assumed to be constant along the length of the magnet and the transformation matrix is

*ϕ*is proportional to the strength parameter and the sign of the solenoid field defines the orientation of the coordinate rotation. Fortunately, we need not keep track of the sign since the components of the focusing matrix \(\mathcal{M}_{\mathrm{s}}\) are even functions of

*z*and do not depend on the direction of the solenoid field.

*z*within the solenoid magnet

*z*

_{0}= 0 to

*z*finally is the product of (20.34) and (20.36)

This transformation matrix is correct inside the solenoid magnet but caution must be taken applying this transformation matrix for the whole solenoid by setting *z* = *L*_{s}. The result would be inaccurate because of a discontinuity caused by the solenoid fringe field. Only the focusing matrix \(\mathcal{M}_{\mathrm{s}}\) for the whole solenoid becomes a simple extension of (20.34) to the end of the solenoid by setting \(\phi (L_{\mathrm{s}}) =\varPhi = \frac{1} {2}SL_{\mathrm{s}}\).

*z*=

*L*

_{s}+

*ε*, where \(\epsilon \rightarrow 0\) the phase is

*ϕ*(

*L*

_{s}) =

*Φ*but the solenoid strength is now zero,

*S*= 0. Therefore, the rotation matrix (20.36) assumes the form

Comparing matrices (20.37), (20.39), we find no continuous transition between both matrices since only one matrix includes the effect of the fringe field. In reality, the fringe field is not a thin-lens and therefore a continuous transition between both matrices could be derived. To stay consistent with the rest of this book, however, we assume for our discussions hard-edge magnet models.

From the matrix (20.34) some special properties of particle trajectories in a solenoid can be derived. For \(\varPhi = \frac{1} {2}\pi\) a parallel beam becomes focused to a point at the magnet axis. A trajectory entering a solenoid with the strength \(\varPhi = \frac{1} {2}\,SL =\pi /2\) at say *y*_{0} will follow one quarter of a spiraling trajectory with a radius *ρ* = *y*_{0}∕2 and exit the solenoid at *x* = *y* = 0. Similarly, a beam emerging from a point source on axis and at the start of the solenoid field will have been focused to a parallel beam at the end of the solenoid. Such a solenoid is used to focus, for example, a divergent positron beam emerging from the target source and is called a *λ*∕4-lens or quarter-wavelength solenoid for obvious reasons.

*M*

_{21}and

*M*

_{43}element of the transformation matrix as we did for quadrupoles and other focusing devices and is with \(\phi = \frac{1} {2}SL_{\text{s }}\)

*S*

^{2}

*L*

_{s}= const.

The focal length is always positive and a solenoid will therefore always be focusing independent of the sign of the field or the sign of the particle charge.

Transformation matrices have been derived for the two most important coupling magnets in beam transport systems, the skew quadrupole and the solenoid magnet, which allows us now to employ linear beam dynamics in full generality including linear coupling. Using (4 × 4)-transformation matrices any particle trajectory can be described whether coupling magnets are included or not. Specifically, we may use this formalism to incorporate compensating schemes when strongly coupling magnets must be included in a particular beam transport line.

## 20.2 Betatron Functions for Coupled Motion

For the linear uncoupled motion of particles in electromagnetic fields we have derived powerful mathematical methods to describe the dynamics of single particles as well as that of a beam composed of a large number of particles. Specifically, the concept of phase space to describe a beam at a particular location and the ability to transform this phase space from one point of the beam transport line to another allow us to design beam transport systems with predictable results. These theories derived for particle motion in one degree of freedom can be expanded to describe coupled motion in both the horizontal and vertical plane.

## 20.3 Conjugate Trajectories

*u*(

*z*) stands for

*x*(

*z*) or

*y*(

*z*), and their derivatives

*z*= 0 and the point

*z*can be derived from

*ψ*

_{0}=

*ψ*(0) and

*ψ*=

*ψ*(

*z*). With (20.45), (20.47) we get

*u*

_{i}=

*u*

_{i}(

*z*) and

*u*

_{i0}=

*u*

_{i}(0). Finally, we can express the elements of the transformation matrix from

*z*= 0 to

*z*by

*z*= 0 and

*ψ*(0) = 0 and define an ellipse by the parametric vector equation

*ϕ*varies over a period of 2

*π*, the vector follows the outline of an ellipse. To parametrize this ellipse we calculate the area enclosed by the phase ellipse. The area element is d\(A = u^{{\prime}}\) d

*u*

_{0}, from (20.52) we get

The formalism of conjugate trajectories has not produced any new insight into beam dynamics that we did not know before but it is an important tool for the discussion of coupled particle motion and provides a simple way to trace individual particles through complicated systems.

Ripken [1] developed a complete theory of coupled betatron oscillations and of particle motion in four-dimensional phase space. In our discussion of coupled betatron motion and phase space transformation we will closely follow his theory. The basic idea hinges on the fact that the differential equations of motion provide the required number of independent solutions, two for oscillations in one plane and four for coupled motion in two planes, to define a two- or four-dimensional ellipsoid which serves as the boundary in phase space for the beam enclosed by it. Since the transformations in beam dynamics are symplectic, we can rely on invariants of the motion which are the basis for the determination of beam characteristics at any point along the beam transport line if we only know such parameters at one particular point.

*z*= 0 in parametric form. Due to the symplecticity of the transformations we find the area of the phase ellipse to be a constant of motion and we may describe the phase ellipse at any point

*z*along the beam line is given by (20.52). The Wronskian is a constant of motion normalized to unity in which case the phase ellipse (20.52) has the area \(A =\pi \epsilon\), where

*ε*is the beam emittance for the beam enclosed by the ellipse. The solutions are of the form (20.45) and forming the Wronskian we find the normalization

*π*the vector

**v**covers all points on the surface of a four-dimensional ellipsoid while the shape of the ellipse varies along the beam line consistent with the variation of the vector functions

**v**

_{i}. In this ansatz we chose two modes of oscillations indicated by the index I and II. If the oscillations were uncoupled, we would identify mode-I with the horizontal oscillation and mode-II with the vertical motion and (20.57) would still hold with

*χ*= 0 having only horizontal nonvanishing components while

**v**

_{3, 4}contain nonzero components only in the vertical plane for

*χ*=

*π*∕2. For independent solutions

**v**

_{i}of coupled motion, we try

*π*. For one-dimensional oscillations we know from the definition of the phase ellipse that the product \(\sqrt{\epsilon _{u}}\,\sqrt{\beta _{u}}\) is equal to the beam size or beam envelope

*E*

_{u}and \(\sqrt{\epsilon _{u}}\,\sqrt{\gamma _{u}}\) equal to the angular beam envelope

*A*

_{u}, where

*u*=

*x*or

*y*. These definitions of beam envelopes can be generalized to coupled motion but we find from (20.66) that the envelopes have two contributions. Each point on the phase ellipse for an uncoupled beam appears now expanded into an ellipse with an area \(\pi \epsilon _{_{\mathrm{II}}}\) as shown in Fig. 20.1.

In a real beam transport line we are not able to observe experimentally the four-dimensional phase ellipse. By methods of emittance measurements, however, we may determine the area for the projection of the four-dimensional ellipsoid onto the (*x* − *x*^{′}), the \((y - y^{{\prime}})\) or the (*x* − *y*)-plane.

*u*-plane occurs for \(\phi _{u_{\mathrm{I,II}}} = -\vartheta _{u_{\mathrm{I,II}}}\) and a projection angle

*χ*given by \(\sin ^{2}\chi = \frac{\epsilon _{u_{\mathrm{II}}}\,\beta _{u_{\mathrm{II}}}} {E_{u}} \,,\) where the beam envelope for coupled motion is given by

*x*=

*E*

_{x}which is the slope of the envelope \(E^{{\prime}}\). Taking the derivative of (20.67) we get

*u*=

*E*

_{u}and \(u^{{\prime}} = E_{u}^{{\prime}}\) into (20.71) we get for the emittance of the projection ellipse

The envelope functions can be measured noting that \(E^{2} =\sigma _{11},A^{2} =\sigma _{22}\) and \(EE^{{\prime}} = -\sigma _{12}\) where the *σ*_{ij} are elements of the beam matrix. Because of the deformation of the four-dimensional phase ellipse through transformations, we cannot expect that the projection is a constant of motion and the projected emittance is therefore of limited use.

*x*,

*y*)-plane which shows the actual beam cross section under the influence of coupling. For this projection we use the first and third equation in (20.66) and find an elliptical beam cross section. The spatial envelopes

*E*

_{x}and

*E*

_{y}have been derived before in (20.67) and become here

*y*-coordinate for

*E*

_{x}, which we denote by

*E*

_{xy}, can be derived from the third equation in (20.66) noting that now \(\vartheta _{y_{\mathrm{I,II}}} = -\phi _{x_{\mathrm{I,II}}}\),

*χ*is given by (20.69) and

*ψ*of the ellipse is determined by

The beam cross section of a coupled beam is tilted as can be directly observed, for example, through a light monitor which images the beam cross section by the emission of synchrotron light. This rotation vanishes as we would expect for vanishing coupling when \(\beta _{x_{\mathrm{II}}} \rightarrow 0\) and \(\beta _{y_{\mathrm{I}}} \rightarrow 0\). The tilt angle is not a constant of motion and therefore different tilt angles can be observed at different points along a beam transport line.

We have discussed Ripken’s theory [1] of coupled betatron motion which allows the formulation of beam dynamics for arbitrary strength of coupling. The concept of conjugate trajectories and transformation matrices through skew quadrupoles and solenoid magnets are the basic tools required to determine coupled betatron functions and the tilt of the beam cross section.

## 20.4 Hamiltonian and Coupling

In practical beam transport systems particle motion is not completely contained in one or the other plane although special care is being taken to avoid coupling effects as much as possible. Coupling of the motion from one plane into the other plane can be generated through the insertion of actual rotated magnets or in a more subtle way by rotational misalignments of upright magnets. Since such misalignments are unavoidable, it is customary to place weak rotated quadrupoles in a transport system to provide the ability to counter what is known as linear coupling caused by unintentional magnet misalignments. Whatever the source of coupling, we consider such fields as small perturbations to the particle motion.

The Hamiltonian treatment of coupled motion follows that for motion in a single plane in the sense that we try to find cyclic variables while transforming away those parts of the motion which are well known. For a single particle normalized coordinates can be defined which eliminate the *z*-dependence of the unperturbed part of the equations of motion. Such transformations cannot be applied in the case of coupled motion since they involve the oscillation frequency or betatron phase function which is different for both planes.

### 20.4.1 Linearly Coupled Motion

*H*

_{0}and the perturbation Hamiltonian for linear coupling

*c*

_{u}and

*ϕ*are of the form

*u*∕d

*z*and \(\partial H/\partial u = -\) d\(u^{{\prime}}/\) d

*z*and get

*a*= const. and

*ϕ*= const. and therefore \(\partial u/\partial z = \partial H_{0}/\partial u^{{\prime}}\) and\(\ \partial u^{{\prime}}/\partial z = -\partial H_{0}/\partial u\). With this we derive from (20.81)–(20.84) the equations

*z*is still the independent variable. The dynamics of linearly coupled motion becomes more evident after isolating the periodic terms in (20.86). For the trigonometric functions we set

*l*

_{x}and

*l*

_{y}are integers defined by

*ν*

_{0u}are the tunes for the periodic lattice, \(\varphi = 2\pi z/L\) and

*L*is the length of the lattice period. The first four terms in (20.90) are periodic with the period \(\varphi \left (L\right ) = 2\pi +\varphi \left (0\right )\). Inserting (20.90) into (20.88) we get with \(\psi _{u}\left (L\right ) = 2\pi \nu _{0u} +\psi _{u}\left (0\right )\)

*p*(

*z*) = k(

*z*) are periodic. After expanding (20.92) into a Fourier series

*l*≤ +1

*z*by the angle variable \(\varphi = 2\pi z/L\) and obtain the new Hamiltonian \(\tilde{H}_{1} = \frac{2\pi } {L}H_{1}\) or

*q*=

*r*defining the resonance condition for coupled motion \(\left (\varDelta _{q} \approx 0\right )\) or

*q*=

*r*. Neglecting all fast oscillating terms we apply one more canonical transformation \(\left (\phi _{u},a_{u}\right ) \rightarrow \left (\tilde{\phi }_{u},\tilde{a}_{u}\right )\) to eliminate the independent variable \(\varphi\) from the Hamiltonian. In essence, we thereby use a coordinate system that follows with the unperturbed particle and exhibits only the deviations from the ideal motion. From the generating function

From these equations we can derive criteria for the stability or resonance condition of coupled systems. Depending on the value of *l* we distinguish a sum resonance if *l* = +1 or a difference resonance if *l* = −1.

#### Linear Difference Resonance

*a*

_{x}and

*a*

_{y}will change such that one amplitude increases at the expense of the other but the sum of both will not change and therefore neither amplitude will grow indefinitely. Since

*a*

_{x}and

*a*

_{y}are proportional to the beam emittance, we note that the sum of the horizontal and vertical emittance stays constant as well,

Our discussion of linear coupling resonances reveals the feature that a difference resonances will cause an exchange of oscillation amplitudes between the horizontal and vertical plane but will not lead to beam instability. This result is important for lattice design. If emittance coupling is desired, one would choose tunes which closely meet the resonance condition. Conversely, when coupling is to be avoided or minimized, tunes are chosen at a save distance from the coupling resonance.

*w*and

*v*to describe oscillations, we assume that the motion in both planes depends on the initial conditions \(w_{\mathrm{0}},v_{\mathrm{0}}\) in both planes due to the effect of coupling. For simplicity, however, we study the dynamics of a particle which starts with a finite amplitudes

*w*

_{0}≠ 0 in the horizontal plane only and set

*v*

_{0}= 0. The ansatz for the oscillations be

*ν*. Inserting (20.110) into (20.109) the coefficients of the exponential functions vanish separately and we get from the coefficients of e\(^{\text{i}\nu \varphi }\) the two equations

*a*and

*c*to get the defining equation for the oscillation frequency

*a*,

*b*,

*c*,

*d*, we note that due to the initial conditions

*a*+

*b*= 1 and

*c*+

*d*= 0. Similar to (20.111) we derive another pair of equations from the coefficients of e

^{−iν φ}

The emittance coupling increases with the strength of the coupling coefficient and is equal to unity at the coupling resonance or for large values of *κ*. At the coupling resonance we observe complete exchange of emittances at the frequency *ν*. If on the other hand, the tunes differ and \(\varDelta _{\text{r}}\neq 0,\) there will always be a finite oscillation amplitude left in the horizontal plane because we started with a finite amplitude in this plane. A completely symmetric result would be obtained only for a particle starting with a finite vertical amplitude as well.

*w*by (20.110). Here, we are only interested in the oscillation frequencies of the particle motion and note that the oscillatory factor in (20.82) is Re\(\left [\mathrm{e}^{\text{i}(\psi _{x}+\phi _{x})}\right ]\). Together with other oscillatory quantities \(\mathrm{e}^{-\mathrm{i}\tilde{\phi }_{x}}\) and

*w*we get both in the horizontal and vertical plane terms with oscillatory factors

*u*stands for either

*x*or

*y*. The phase \(\psi _{u} =\nu _{u}\varphi\) and from (20.100) and

*l*= −1 for the difference resonance \(\tilde{\phi }_{u} =\phi _{u} \pm \frac{1} {2}\varDelta _{\text{r}}\varphi\). These expressions used in (20.118) define two oscillation frequencies

We have again found the result that under coupling conditions the betatron oscillations assume two modes. In a real accelerator only these mode frequencies can be measured while close to the coupling resonance. For very weak coupling \(\left (\kappa \approx 0\right )\) the mode frequencies are approximately equal to the uncoupled frequencies *ν*_{x, y}, respectively. Even for large coupling this equality is preserved as long as the tunes are far away from the coupling resonance or \(\varDelta _{\text{r}} \gg \kappa\).

The mode frequencies can be measured while adjusting quadrupoles such that the beam is moved through the coupling resonance. During this adjustment the detuning parameter \(\varDelta _{\mathrm{r}}\) varies and changes sign as the coupling resonance is crossed. For example, if we vary the vertical tune across a coupling resonance from below, we note that the horizontal tune or *ν*_{I} does not change appreciably until the resonance is reached, because \(-\varDelta _{r} + \sqrt{\varDelta _{\mathrm{r} }^{2 } +\kappa ^{2}} \approx 0\). Above the coupling resonance, however, \(\varDelta _{r}\) has changed sign and *ν*_{I} increase with \(\varDelta _{r}\). The opposite occurs with the vertical tune. Going through the coupling resonance the horizontal tune has been transformed into the vertical tune and vice versa without ever getting equal.

*κ*.

The coupling coefficient may be nonzero for various reasons. In some cases coupling may be caused because special beam characteristics are desired. In most cases, however, coupling is not desired or planned for and a finite linear coupling of the beam emittances is the result of rotational misalignments of upright quadrupoles. Where this coupling is not desired and must be minimized, we may introduce a pair or two sets of rotated quadrupoles into the lattice to cancel the coupling due to misalignments. The coupling coefficient (20.95) is defined in the form of a complex quantity. Both orthogonal components must therefore be compensated by two orthogonally located skew quadrupoles and the proper adjustment of these quadrupoles can be determined by measuring the width of the linear coupling resonance.

#### Linear Sum Resonance

*l*= +1 and get from (20.98) the resonance condition for a sum resonance

*u*∕d\(\varphi\), we get with (20.103), (20.104)

By a careful choice of the tune difference to avoid a sum resonance and careful alignment of quadrupoles, it is possible in real circular accelerators to reduce the coupling coefficient to very small values. Perfect compensation of the linear coupling coefficient eliminates the linear emittance coupling altogether. However, nonlinear coupling effects become then dominant which we cannot compensate for.

### 20.4.2 Higher-Order Coupling Resonances

The factors *l*_{x} and *l*_{y} are integers and the sum \(\left \vert l_{x}\right \vert + \left \vert l_{y}\right \vert\) is called the order of the resonance. In most cases it is sufficient to choose a location in the resonance diagram which avoids such resonances since circular accelerators are generally designed for minimum coupling. In special cases, however, where strong sextupoles are used to correct chromaticities, coupling resonances can be excited in higher order. For example, the difference resonance \(2\nu _{x} - 2\nu _{y}\) has been observed at the 400 GeV proton synchrotron at the Fermi National Laboratory [5].

### 20.4.3 Multiple Resonances

We have only discussed isolated resonances. In general, however, nonlinear fields of different orders do exist, each contributing to the stop-band of resonances. A particularly strong source of nonlinearities occurs due to the beam-beam effect in colliding-beam facilities where strong and highly nonlinear fields generated by one beam cause significant perturbations to particles in the other beam. The resonance patterns from different resonances are superimposed creating new features of particle instability which were not present in any of the resonances while treated as isolated resonances. Of course, if one of these resonances is unstable for any oscillation amplitude the addition of other weaker resonances will not change this situation.

Combining the effects of several resonances should cause little change for small amplitude oscillations since the trajectory in phase space is close to a circle for resonances of any order provided there is stability at all. Most of the perturbations of resonance patterns will occur in the vicinity of the island structures. When island structures from different resonances start to overlap, chaotic motion can occur and may lead to stochastic instability. The onset of island overlap is often called the Chirikov criterion after Chirikov [6], who has studied extensively particle motion in such situations.

It is beyond the scope of this text to evaluate the mathematical criteria of multi resonance beam dynamics. For further insight and references the interested reader may consult articles in [7, 8, 9, 10]. A general overview and extensive references can also be found in [11].

## Problems

**20.1 (S).** Consider a lattice made of 61 FODO cells with 90^{∘} per cell in both planes. The half cell length be *L* = 5 m and the full quadrupole length *ℓ* = 0. 2 m. Introduce a Gaussian distribution of rotational quadrupole misalignments. Calculate and plot the coupling coefficient for the ring and the emittance ratio as a function of the rms misalignment. If the emittance coupling is to be held below 1 % how must the lattice be retuned and how well must the quadrupoles be aligned? Insert two rotated quadrupoles into the lattice such that they can be used to compensate the coupling due to misalignments. Calculate the required quadrupole strength.

**20.2 (S).** Use the measurement in Fig. 20.2 and determine the coupling coefficient *κ*.

**20.3.** Can we rotate a horizontally flat 10 GeV beam by 90^{∘} with a solenoid? If yes, what is the strength of the solenoid and where along the *z*-axis do we have a flat vertical beam?

**20.4.** In circular accelerators rotated quadrupoles may be inserted to compensate for coupling due to misalignments. Assume a statistical distribution of rotational quadrupole errors which need to be compensated by special rotated quadrupoles. How many such quadrupoles are required and what criteria would you use for optimum placement in the ring?

**20.5.** Consider a point source of particles (e.g. a positron conversion target) on the axis of a solenoidal field. Determine the solenoid parameters for which the particles would exit the solenoid as a parallel beam. Such a solenoid is also called a *λ*∕4-lens, why? Let the positron momentum be 10 MeV/c. What is the maximum solid angle accepted from the target that can be focused to a beam of radius *r* = 1 cm? What is the exit angle of a particle which emerges from the target at a radius of 1 mm? Express the transformation of this *λ*∕4-lens in matrix formulation.

**20.6.** Choose a FODO lattice for a circular accelerator and insert at a symmetry point a thin rotated quadrupole. Calculate the tilt of the beam cross section at this point as a function of the strength of the rotated quadrupole. Place the same skew quadrupole in the middle of a FODO half cell and determine if the rotation of the beam aspect ratio at the symmetry point requires a stronger or a weaker field. Explain why.

**20.7.** Assume two cells of a symmetric FODO lattice and determine the betatron functions for a phase advance of 90^{∘} per cell. Now introduce a rotational misalignment of the first quadrupole by an angle *α* which generates coupling of the horizontal and vertical betatron oscillations: a.) Calculate and plot the perturbed betatron functions *β*_{I} and *β*_{II} and compare with the unperturbed solution. b.) If the beam emittances are *ε*_{I} = *ε*_{II} mm-mrad, what is the beam aspect ratio and beam rotation at the end of cell one and two with and without the rotation of the first quadrupole?

**20.8.** Use the Fokker-Planck equation and derive an expression for the equilibrium beam emittance of a coupled beam

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