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**

The most generally used magnets that introduce coupling in beam transport systems are rotated quadrupoles and solenoid magnets and we will restrict our discussion of coupled beam dynamics to such magnets defining the realm of linear coupling. Equations (6.95), (6.96) include all linear and nonlinear coupling terms up to third order while longitudinal fields are treated in Sect. 6.6. The equations of motion in

This chapter has been made Open Access under a CC BY 4.0 license. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-319-18317-6\_28

the presence of upright and rotated quadrupoles as well as solenoid fields are

$$\begin{array}{l}\mathbf{x}'' + k\mathbf{x} = -\underline{\mathbf{k}}\mathbf{y} + \mathbf{S}\mathbf{y}' + \frac{1}{2}\mathbf{S}'\mathbf{y},\\\mathbf{y}'' - k\mathbf{y} = -\underline{\mathbf{k}}\mathbf{x} - \mathbf{S}\mathbf{x}' - \frac{1}{2}\mathbf{S}'\mathbf{x}\end{array} \tag{20.1}$$

where the solenoid field is expressed by

$$S(z) = \frac{e}{p} B\_s(z) \,. \tag{20.2}$$

We use in this chapter the symbol *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*

The distribution of rotated or skew quadrupoles and solenoid magnets is arbitrary and therefore no analytic solution can be expected for the differential equations (20.1). Similar to other beam line elements, we discuss solutions for the equations of motion within individual magnets only and assume that strength parameters within hard-edge model magnets stay constant. We discuss first solutions of particle motion in skew quadrupoles alone and ignore solenoid fields. The equations of motion for skew quadrupoles are from (20.1)

$$\begin{aligned} \mathbf{x}'' + \underline{\underline{k}} \mathbf{y} &= \mathbf{0} \,, \\ \mathbf{y}'' + \underline{\underline{k}} \mathbf{x} &= \mathbf{0} \, . \end{aligned} \tag{20.3}$$

These equations look very similar to the equations for ordinary upright quadrupoles except that the restoring forces now depend on the particle amplitude in the other plane. We know the solution of the equation of motion for an upright focusing and defocusing quadrupole and will try to apply these solutions to (20.3). Combining the observation that each quadrupole is focusing in one plane and defocusing in the other with the apparent mixture of both planes for a skew quadrupole, we will try an ansatz for (20.3) which is made up of four principal solutions

$$\begin{cases} \varkappa = a \cos \varphi + \frac{b}{\sqrt{k}} \sin \varphi + c \cosh \varphi + \frac{d}{\sqrt{k}} \sinh \varphi \text{ },\\ \text{y} = A \cos \varphi + \frac{B}{\sqrt{k}} \sin \varphi + C \cosh \varphi + \frac{D}{\sqrt{k}} \sinh \varphi \text{ }, \end{cases} \tag{20.4}$$

where ' D p*k z* and the variable *z* varies between zero and the full length of the quadrupole, 0 < *z* < `*q*. The coefficients *a*; *b*; *c*;::: *D* must be determined to be consistent with the initial parameters of the trajectories .*x*0; *x*0 0; *y*0; *y*0 0/. For *z* D 0 we get

$$\begin{aligned} \mathbf{x}\_0 &= a + c \,, \qquad \mathbf{y}\_0 = A + C \,, \\ \mathbf{x}\_0' &= b + d \, , \qquad \mathbf{y}\_0' = B + D \, . \end{aligned} \tag{20.5}$$

Solutions (20.4) must be consistent with (20.3) from which we find

$$\begin{array}{ll} a = A \, , & c = -C \, , \\ b = B \, , & d = -D \, . \end{array} \tag{20.6}$$

From (20.5), (20.6) we get finally the coefficients consistent with the initial conditions and the differential equations (20.3)

$$\begin{array}{ll} a = A = \frac{1}{2} (\mathbf{x}\_0 + \mathbf{y}\_0) \end{array}, \quad b = B = \frac{1}{2} (\mathbf{x}\_0' + \mathbf{y}\_0') \,, \\ c = -C = \frac{1}{2} (\mathbf{x}\_0 - \mathbf{y}\_0) \,, \; d = -D = \frac{1}{2} (\mathbf{x}\_0' - \mathbf{y}\_0') \,. \end{array} \tag{20.7}$$

With these definitions the transformation through a skew quadrupole is

$$
\begin{pmatrix} \mathbf{x} \\ \mathbf{x}' \\ \mathbf{y} \\ \mathbf{y}' \end{pmatrix} = \mathcal{M}\_{\mathbf{sq}} \begin{pmatrix} \mathbf{x}\_0 \\ \mathbf{x}'\_0 \\ \mathbf{y}\_0 \\ \mathbf{y}'\_0 \end{pmatrix}, \tag{20.8}
$$

where Msq is the transformation matrix for a skew quadrupole,

$$\mathcal{M}\_{\text{sq}}(s|0) = \frac{1}{2} \begin{pmatrix} \mathcal{C}^{+} & \frac{1}{\sqrt{k}} \mathcal{S}^{+} & \mathcal{C}^{-} & \frac{1}{\sqrt{k}} \mathcal{S}^{-} \\ -\sqrt{\underline{k}} \mathcal{S}^{-} & \mathcal{C}^{+} & -\sqrt{\underline{k}} \mathcal{S}^{+} & \mathcal{C}^{-} \\ \mathcal{C}^{-} & \frac{1}{\sqrt{\underline{k}}} \mathcal{S}^{-} & \mathcal{C}^{+}(\varphi) & \frac{1}{\sqrt{k}} \mathcal{S}^{+} \\ -\sqrt{\underline{k}} \mathcal{S}^{+} & \mathcal{C}^{-} & -\sqrt{\underline{k}} \mathcal{S}^{-} & \mathcal{C}^{+} \end{pmatrix},\tag{20.9}$$

with C˙ D C˙.'/ D cos ' ˙ cosh ' and S˙ D S˙.'/ D sin ' ˙ sinh ' and ' D p*kz*.

This transformation matrix is quite elaborate and becomes useful only for numerical calculations on computers. We employ again thin lens approximation where the quadrupole length vanishes `sq ! 0 in such a way as to preserve the integrated magnet strength or the focal length *f* . The matrix (20.9) then reduces to the simple form

$$\mathcal{M}\_{\rm sq}(0 \mid \ell\_{\rm sq}) = \begin{pmatrix} 1 & \ell\_{\rm sq} & 0 & 0 \\ 0 & 1 & -1/f & 0 \\ 0 & 0 & 1 & \ell\_{\rm sq} \\ -1/f & 0 & 0 & 1 \end{pmatrix},\tag{20.10}$$

where the focal length is defined as *f* 1 D *k* `sq. Note, that we have not set `sq D 0 but retained the linear terms in `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 `sq while the off-diagonal elements describe the coupling due to the thin skew quadrupole.

#### *20.1.2 Particle Motion in a Solenoidal Field*

The equations of motion in a solenoid can be derived from (6.92a), neglecting all transverse beam deflection and electric fields

$$\begin{array}{l} \chi^{\prime\prime} - \frac{1}{2} \frac{\chi^{\prime}}{\underline{\chi}^{\prime 2}} \frac{d\underline{\chi}^{\prime 2}}{d\underline{\chi}} = \frac{e}{p} \underline{z}^{\prime} (\text{y}^{\prime}B\_{\text{s}} - B\_{\text{y}}) \,, \\\chi^{\prime\prime} - \frac{1}{2} \frac{\chi^{\prime}}{\underline{\chi}^{\prime 2}} \frac{d\underline{\chi}^{\prime 2}}{d\underline{\chi}} = \frac{e}{p} \underline{z}^{\prime} (B\_{\text{x}} - \underline{\chi}^{\prime}B\_{\text{s}}) \,, \end{array} \tag{20.11}$$

where the solenoid field component *B*s, assumed to be colinear with the *z*-direction, can be derived from (6.103)

$$\mathcal{B} = \begin{pmatrix} -\frac{1}{2}B\_s' \ge, -\frac{1}{2}B\_s' \circ, B\_s \end{pmatrix} \,. \tag{20.12}$$

Following the same derivation as in Sect. 6.5, the general equations of motion in a solenoid field including up to third-order terms are

$$\begin{split} \mathbf{x}^{\prime\prime} &= + \frac{e}{p} \mathbf{B}\_{\mathbf{s}} \mathbf{y}^{\prime} + \frac{1}{2} \frac{e}{p} \mathbf{B}\_{\mathbf{s}}^{\prime} \mathbf{y} \\ &+ \frac{1}{4} \frac{e}{p} (2 \mathbf{x}^{\prime2} \mathbf{y}^{\prime} \mathbf{B}\_{\mathbf{s}} + \mathbf{x}^{\prime2} \mathbf{y} \mathbf{B}\_{\mathbf{s}}^{\prime} + 2 \mathbf{y}^{\prime3} \mathbf{B}\_{\mathbf{s}} + \mathbf{y} \mathbf{y}^{\prime2} \mathbf{B}\_{\mathbf{s}}^{\prime}) + \mathcal{O}(4) \,, \\ \mathbf{y}^{\prime\prime} &= -\frac{e}{p} \mathbf{B}\_{\mathbf{s}} \mathbf{x}^{\prime} - \frac{1}{2} \frac{e}{p} \mathbf{B}\_{\mathbf{s}}^{\prime} \mathbf{x} \\ &- \frac{1}{4} \frac{e}{p} \left( 2 \mathbf{x}^{\prime} \mathbf{y}^{\prime2} \mathbf{B}\_{\mathbf{s}} + \mathbf{x} \mathbf{y}^{\prime2} \mathbf{B}\_{\mathbf{s}}^{\prime} + 2 \mathbf{x}^{\prime3} \mathbf{B}\_{\mathbf{s}} + \mathbf{x} \mathbf{x}^{\prime2} \mathbf{B}\_{\mathbf{s}}^{\prime} \right) + \mathcal{O}(4) \, . \end{split} \tag{20.14}$$

Considering only linear terms, the equations of motion in a solenoidal field simplify to

$$\begin{array}{l} \mathbf{x}^{\prime\prime} = + \frac{c}{p} B\_{\mathbf{s}} \mathbf{y}^{\prime} + \frac{1}{2} \frac{c}{p} B\_{\mathbf{s}}^{\prime} \mathbf{y} \,, \\ \mathbf{y}^{\prime\prime} = -\frac{c}{p} B\_{\mathbf{s}} \mathbf{x}^{\prime} - \frac{1}{2} \frac{c}{p} B\_{\mathbf{s}}^{\prime} \mathbf{x} \,, \end{array} \tag{20.15}$$

exhibiting clearly coupling terms. In a uniform field, where *B*0 s D 0, the particle trajectory assumes the form of a helix parallel to the axis of the solenoid field.

The equations of motion (20.15) have been derived under the assumption of paraxial rays so that we can set v vz. In a solenoid field this approximation is not generally acceptable since we may, for example, be interested in using a solenoid to focus particles emerging from a target at large angles. We therefore replace all derivatives with respect to *z* by derivatives with respect to the time, use the particle velocity v, and replace d d*z* ! 1 v d d*t* . In a uniform solenoid field the equations of motion are then

$$\begin{aligned} \ddot{\boldsymbol{x}} &= + \left( \frac{\boldsymbol{\varepsilon}}{\rho} \boldsymbol{B}\_{\mathrm{s}} \boldsymbol{v} \right) \dot{\boldsymbol{y}} = \boldsymbol{\omega}\_{\mathrm{L}} \dot{\boldsymbol{y}}, \\ \ddot{\boldsymbol{y}} &= - \left( \frac{\boldsymbol{\varepsilon}}{\rho} \boldsymbol{B}\_{\mathrm{s}} \boldsymbol{v} \right) \dot{\boldsymbol{x}} = - \boldsymbol{\omega}\_{\mathrm{L}} \dot{\boldsymbol{x}}, \end{aligned} \tag{20.16}$$

where the Larmor frequency !L is defined by

$$
\rho\_{\rm l.} = \frac{eB\_{\rm s}v}{p} = \frac{ec^2}{E}B\_{\rm s} \,, \tag{20.17}
$$

and *E* is the total particle energy. Multiplying (20.16) by *x*P and *y*P, respectively, and adding both equations we get d.*x*P2 C P*y*2/=d*t* D 0 or

$$
\dot{\mathbf{x}}^2 + \dot{\mathbf{y}}^2 = v\_t^2 = \text{const.}\tag{20.18}
$$

The transverse particle velocity vt or total transverse momentum of the particle *cp*t stays constant during the motion in a uniform solenoid field. For *x*P0 D 0 and *y*P0 D vt, for example, the transverse velocities can be expressed by

$$\begin{aligned} \dot{x} &= v\_{\text{l}} \sin \omega\_{\text{L}} t \\ \dot{y} &= v\_{\text{l}} \cos \omega\_{\text{L}} t \end{aligned} \tag{20.19}$$

and the solutions of the equations of motion are

$$\begin{aligned} \mathbf{x}(t) &= \mathbf{x}\_0 - \frac{v\_{\parallel}}{\frac{\alpha\_{\perp}}{\alpha\_{\perp}}} \cos \boldsymbol{\omega}\_{\perp} t \,, \\ \mathbf{y}(t) &= \mathbf{y}\_0 + \frac{v\_{\parallel}}{\alpha\_{\perp}} \sin \boldsymbol{\omega}\_{\perp} t \,. \end{aligned} \tag{20.20}$$

The amplitude of the oscillating term in (20.20) is equal to the radius of the helical path

$$
\rho\_{\rm h} = \frac{p\_{\rm l}}{eB\_{\rm s}}\,,\tag{20.21}
$$

where we have used the Larmor frequency (20.17) and set the transverse momentum *p*t D *m*vt: The longitudinal motion is unaffected for not too strong solenoid fields and vPz D 0 as can be derived from the Lorentz equation since all transverse field components vanish and

$$z(t) = z\_0 + v\_z t \,. \tag{20.22}$$

The time to complete one period of the helix is

$$T = \frac{2\pi}{\alpha\_{\rm L}}\tag{20.23}$$

during which time the particle moves along the *z*-axis a distance

$$
\Delta z = 2\pi \frac{p\_z}{eB\_s} \,, \tag{20.24}
$$

where *pz* is the *z*-component of the particle momentum.

The solutions of the equations of motion for a solenoid magnet are more complex since we must now include terms that depend on the slope of the particle trajectories as well. Ignoring skew quadrupoles the differential equations of motion in a solenoid magnet becomes from (20.15)

$$\begin{array}{l} \mathbf{x}^{\prime\prime} - S(\mathbf{z})\mathbf{y}^{\prime} - \frac{1}{2}S^{\prime}(\mathbf{z})\,\mathbf{y} = \mathbf{0} \,\, , \\ \mathbf{y}^{\prime\prime} + S(\mathbf{z})\,\mathbf{x}^{\prime} + \frac{1}{2}S^{\prime}(\mathbf{z})\,\mathbf{x} = \mathbf{0} \,\, . \end{array} \tag{20.25}$$

Coupling between both planes is obvious and the variation of coordinates in one plane depends entirely on the coordinates in the other plane. We note a high degree of symmetry in the equations in the sense that both coordinates change similar as a function of the other coordinates. This suggests that a rotation of the coordinate system may help simplify the solution of the differential equations. We will therefore try such a coordinate rotation in complex notation by defining

$$R = \left(\mathbf{x} + \mathbf{i}\,\mathbf{y}\right) \mathbf{e}^{-\mathbf{i}\,\phi(\mathbf{z})},\tag{20.26}$$

where the rotation angle may be a function of the independent variable *z*. A single differential equation can be formed from (20.25) in complex notation

$$(\mathbf{x} + \mathbf{i}\mathbf{y})'' + \mathbf{i}\, S(\mathbf{z})\,(\mathbf{x} + \mathbf{i}\,\mathbf{y})' + \mathbf{i}\,\frac{1}{2}\,S'(\mathbf{z})\,(\mathbf{x} + \mathbf{i}\,\mathbf{y}) = \mathbf{0}.\tag{20.27}$$

The rotation (20.26) can now be applied directly to (20.27) and with

$$(\mathbf{x} + \mathbf{i}\mathbf{y})' = R'\mathbf{e}^{\mathbf{i}\phi} + \mathbf{i}\phi' R\,\mathbf{e}^{+\mathbf{i}\phi}$$

and

$$(\mathbf{x} + \mathbf{i}\mathbf{j})'' = R''\mathbf{e}^{\dot{\mathbf{e}}\phi} + 2\,\mathbf{i}\,\phi'R'\mathbf{e}^{\dot{\mathbf{e}}\phi} + \mathbf{i}\,\phi''R\,\mathbf{e}^{\dot{\mathbf{e}}\phi} - \phi'^2R\,\mathbf{e}^{\dot{\mathbf{e}}\phi}.\tag{20.28}$$

After insertion into (20.26) and sorting of terms

$$R' - [S(z)\phi' + \phi'^2]R + \mathrm{i}\, 2[\phi' + \frac{1}{2}S(z)]R' + \mathrm{i}\, [\phi'' + \frac{1}{2}S'(z)]R = 0\,\,. \tag{20.29}$$

At this point, the introduction of the coordinate rotation allows a great simplification (20.28) by assuming a continuous rotation along the beam axis with a rotation angle defined by

$$\phi(z) = -\frac{1}{2} \int\_{z\_0}^{z} S(\xi) \,\mathrm{d}\xi \tag{20.30}$$

where the solenoid field starts at *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 0 D 1 2 *S*.*z*/ and 00 D 1 2 *S*0 .*z*/, which after insertion into (20.28) results in the simple equation of motion

$$\left(R^{\prime\prime} + \frac{1}{4}S^2(z)\right)R = 0\,. \tag{20.31}$$

With *R* D v C i*w*, we finally get two uncoupled equations

$$v'' + \frac{1}{4}S^2(z)\,v = 0\,,\tag{20.32}$$

$$\left(\boldsymbol{w}'' + \frac{1}{4}\boldsymbol{S}^2(z)\,\boldsymbol{w} = 0\,.$$

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 D 0 and therefore v0 D *x*0 and *w*0 D *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*s D 1 4 *S*2.*L*s/ followed by a rotation of the coordinate system by the angle .*L*s/ thus reverting to Cartesian coordinates .*x*; *y*/.

#### *20.1.3 Transformation Matrix for a Solenoid Magnet*

Similar to the transformation through quadrupoles and other beam transport magnets, we may formulate a transformation matrix for a solenoid magnet. Instead of uncoupled 2 - 2 transformation matrices, however, we must use 4 - 4 matrices to include coupling effects. Each coordinate now depends on the initial values of all coordinates, *x*.*z*/ D .*x*0; *x*0 0; *y*0; *y*0 0/, etc. The transformation through a solenoid is performed in two steps in which the first is the solution of (20.32) in the form of the matrix Ms, and the second is a coordinate rotation introduced through the matrix Mr. The total transformation is therefore

$$
\begin{pmatrix} \mathbf{x} \\ \mathbf{x}' \\ \mathbf{y} \\ \mathbf{y}' \end{pmatrix} = \mathcal{M}\_{\mathbf{r}} \mathcal{M}\_{\mathbf{s}} \begin{pmatrix} \mathbf{x}\_{0} \\ \mathbf{x}'\_{0} \\ \mathbf{y}\_{0} \\ \mathbf{y}'\_{0} \end{pmatrix}. \tag{20.33}
$$

In analogy to the transformation through an upright quadrupole, we get from (20.32) the transformation matrix Ms from the beginning of the solenoid field at *z*0 to a point *z* inside the solenoid magnet. The strength parameter in this case is 1 4 *S*2 assumed to be constant along the length of the magnet and the transformation matrix is

$$\mathcal{M}\_{\mathbf{s}}(\boldsymbol{z}|\boldsymbol{z}) = \begin{pmatrix} \cos\phi & \frac{2}{\mathcal{S}}\sin\phi & 0 & 0\\ -\frac{\mathcal{S}}{2}\sin\phi & \cos\phi & 0 & 0\\ 0 & 0 & \cos\phi & \frac{2}{\mathcal{S}}\sin\phi\\ 0 & 0 & -\frac{\mathcal{S}}{2}\sin\phi & \cos\phi \end{pmatrix},\tag{20.34}$$

where D 1 2 *Sz*. The next step is to introduce the coordinate rotation Mr which we derive from the vector equation

$$\mathbf{(x+i\,y)} = (\mathbf{v} + \mathbf{i}\,\mathbf{w})\,\mathbf{e}^{-i\phi(z)},\tag{20.35}$$

where the vectors are defined like *x* D .*x*; *x*0 /, etc. Note that the value of the rotation angle 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 matrixMs are even functions of *z* and do not depend on the direction of the solenoid field.

By separating (20.35) into its real and imaginary part and applying Euler's identity e˛ D cos ˛ C i sin ˛, we get for the rotation matrix at the point *z* within the solenoid magnet

$$\mathcal{M}\_{\mathbf{r}} = \begin{pmatrix} \cos\phi & 0 & \sin\phi & 0\\ -\frac{S}{2}\sin\phi & \cos\phi & \frac{S}{2}\cos\phi & \sin\phi\\ -\sin\phi & 0 & \cos\phi & 0\\ \frac{S}{2}\cos\phi & -\sin\phi & -\frac{S}{2}\sin\phi \cos\phi \end{pmatrix} \tag{20.36}$$

The total transformation matrix for a solenoid magnet from *z*0 D 0 to *z* finally is the product of (20.34) and (20.36)

$$\mathcal{M}\_{\text{sol}}(0|z$$

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* D *L*s. The result would be inaccurate because of a discontinuity caused by the solenoid fringe field. Only the focusing matrix Ms for the whole solenoid becomes a simple extension of (20.34) to the end of the solenoid by setting .*L*s/ D ˚ D 1 2 *SL*s.

Due to the solenoid fringe field, which in hard-edge approximation adopted here is a thin slice, the rotation matrix exhibits a discontinuity. For *z* D *L*s C , where ! 0 the phase is .*L*s/ D ˚ but the solenoid strength is now zero, *S* D 0. Therefore, the rotation matrix (20.36) assumes the form

$$\mathcal{M}\_{\mathbf{f}} = \begin{pmatrix} \cos\Phi & 0 & \sin\Phi & 0\\ 0 & \cos\Phi & 0 & \sin\Phi\\ -\sin\Phi & 0 & \cos\Phi & 0\\ 0 & -\sin\Phi & 0 & \cos\Phi \end{pmatrix} \tag{20.38}$$

Notice that this matrix at the solenoid entrance is just the unit matrix. This does not mean that we ignored the entrance fringe field, it only indicates that this effect is already included in (20.37). After multiplication of (20.34) with (20.38), the transformation matrix for a complete solenoid magnet is finally

$$\mathcal{M}\_{\text{sol}}(0|L) = \begin{pmatrix} \cos^2\Phi & \frac{1}{\mathcal{S}}\sin 2\Phi & \frac{1}{2}\sin 2\Phi & \frac{2}{\mathcal{S}}\sin^2\Phi\\ -\frac{S}{4}\sin 2\Phi & \cos^2\Phi & -\frac{S}{2}\sin^2\Phi & \frac{1}{2}\sin 2\Phi\\ -\frac{1}{2}\sin 2\Phi & -\frac{2}{\mathcal{S}}\sin^2\Phi & \cos^2\Phi & \frac{1}{\mathcal{S}}\sin 2\Phi\\ \frac{S}{2}\sin^2\Phi & -\frac{1}{2}\sin 2\Phi & -\frac{S}{4}\sin 2\Phi & \cos^2\Phi \end{pmatrix}. \tag{20.39}$$

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 ˚ D 1 2 a parallel beam becomes focused to a point at the magnet axis. A trajectory entering a solenoid with the strength ˚ D 1 2 *SL* D -=2 at say *y*0 will follow one quarter of a spiraling trajectory with a radius D *y*0=2 and exit the solenoid at *x* D *y* D 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 =4-lens or quarter-wavelength solenoid for obvious reasons.

The focusing properties of the whole solenoid are most notable when the field strength is weak and the focal length is long compared to the length of the solenoid. In this case, the focal length can be taken immediately from the *M*21 and *M*43 element of the transformation matrix as we did for quadrupoles and other focusing devices and is with D 1 2 *SL*s

$$\frac{1}{f\_{\mathbf{x}}} = M\_{21} = -\frac{1}{2} \mathcal{S} \sin \phi \,\cos \phi \,\,, \tag{20.40}$$

$$\frac{1}{f\_{\mathbf{y}}} = M\_{43} = -\frac{1}{2} \, \text{S} \, \sin \phi \, \cos \phi \,. \tag{20.41}$$

In contrast to quadrupole magnets, the focal length of a solenoid magnet is the same in both planes and is in thin-lens approximation, D 1 2 *SL*s ! 0 while *S*2*L*s Dconst.

$$\frac{1}{f\_{\rm sol}} = \frac{1}{4} S^2 L\_s = \frac{1}{4} \left( \frac{e}{p} \right)^2 B\_s^2 L\_s \,. \tag{20.42}$$

The thin lens transformation matrix for a weak solenoid is thereby

$$\mathcal{M}\_{\text{sol}}(0|L) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ -\frac{1}{f\_{\text{sol}}} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -\frac{1}{f\_{\text{sol}}} & 1 \end{pmatrix} . \tag{20.43}$$

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**

Lattice functions have been defined to express solutions to the equations of motion for individual trajectories. Conversely, there must be a way to express these lattice functions by the principal solutions of the equation of motion. This would enable us to determine lattice functions for coupled particle motion by integrating the equations of motion for two orthogonal trajectories. To do this, we start from the differential equation of motion in normalized coordinates for which two linearly independent principal solutions are given by

$$\begin{aligned} \, \_2w\_1(\varphi) &= \cos \left( \nu \varphi \right) \,, \\ \, \_2w\_2(\varphi) &= \sin \left( \nu \varphi \right) \, . \end{aligned} \tag{20.44}$$

For simplicity, we set the initial amplitudes equal to unity and get in regular coordinates with *u*.*z*/ D *w* pˇ.*z*/ the conjugate trajectories are

$$\begin{array}{l} \mu\_1(z) = \sqrt{\beta(z)} \cos \psi(z), \\ \mu\_2(z) = \sqrt{\beta(z)} \sin \psi(z), \end{array} \tag{20.45}$$

where *u*.*z*/ stands for *x*.*z*/ or *y*.*z*/; and their derivatives

$$\begin{array}{l} \mu\_{1}'(z) = -\frac{\alpha(z)}{\sqrt{\beta(z)}} \cos \psi(z) - \frac{1}{\sqrt{\beta(z)}} \sin \psi(z) \,, \\\mu\_{2}'(z) = -\frac{\alpha(z)}{\sqrt{\beta(z)}} \sin \psi(z) + \frac{1}{\sqrt{\beta(z)}} \cos \psi(z) \, . \end{array} \tag{20.46}$$

Using (20.45), (20.46) all lattice functions can be expressed in terms of conjugate trajectories like

$$\begin{aligned} \beta(z) &= \mu\_1^2(z) + \mu\_2^2(z) \ , \\ \alpha(z) &= -\mu\_1(z)\,\mu\_1'(z) - \mu\_2(z)\,\mu\_2'(z) \ , \\ \gamma(z) &= \mu\_1'^2(z) + \mu\_2'^2(z) \ . \end{aligned} \tag{20.47}$$

The betatron phase advance D 0 between the point *z* D 0 and the point *z* can be derived from

$$\cos(\psi - \psi\_0) = \cos\psi \cos\psi\_0 + \sin\psi \sin\psi\_0 \ ,$$

where 0 D .0/ and D .*z*/. With (20.45), (20.47) we get

$$\cos\psi(z) = \frac{\mu\_1(z)}{\sqrt{\beta(z)}} = \frac{\mu\_1(z)}{\sqrt{\mu\_1^2(z) + \mu\_2^2(z)}}\tag{20.48}$$

and similarly,

$$\sin \psi(z) = \frac{\mu\_2(z)}{\sqrt{\beta(z)}} = \frac{\mu\_2(z)}{\sqrt{\mu\_1^2(z) + \mu\_2^2(z)}} \,. \tag{20.49}$$

The betatron phase advance then is given by

$$\cos\left(\psi - \psi\_0\right) = \frac{u\_1 u\_{10} + u\_2 u\_{20}}{\sqrt{u\_1^2 + u\_2^2}\sqrt{u\_{10}^2 + u\_{20}^2}}\tag{20.50}$$

where *ui* D *ui*.*z*/ and *ui*0 D *ui*.0/. Finally, we can express the elements of the transformation matrix from *z* D 0 to *z* by

$$\mathcal{M}(\boldsymbol{z}|0) = \begin{pmatrix} M\_{11} \ M\_{12} \\ M\_{21} \ M\_{22} \end{pmatrix} = \begin{pmatrix} \boldsymbol{\mu}\_{1} \boldsymbol{\mu}\_{20}' - \boldsymbol{\mu}\_{2} \boldsymbol{\mu}\_{10}' \, \boldsymbol{\mu}\_{10} \, \boldsymbol{\mu}\_{2} - \boldsymbol{\mu}\_{1} \boldsymbol{\mu}\_{20} \\ \boldsymbol{\mu}\_{1}' \boldsymbol{\mu}\_{10}' - \boldsymbol{\mu}\_{2}' \boldsymbol{\mu}\_{20}' \, \boldsymbol{\mu}\_{10} \, \boldsymbol{\mu}\_{2}' - \boldsymbol{\mu}\_{20} \boldsymbol{\mu}\_{1}' \end{pmatrix}. \tag{20.51}$$

The two linearly independent solutions (20.45) also can be used to define and characterize the phase space ellipse. At the start of a beam line we set *z* D 0 and .0/ D 0 and define an ellipse by the parametric vector equation

$$\mathbf{u}(0) = a \begin{bmatrix} \mathbf{u}\_1(0) \ \cos \phi - \mathbf{u}\_2(0) \ \sin \phi \end{bmatrix},\tag{20.52}$$

where

$$\mathbf{u}(0) = \begin{pmatrix} u\_0 \\ u\_0' \end{pmatrix} \qquad \text{and} \qquad \mathbf{u}\_i(0) = \begin{pmatrix} u\_{i0} \\ u\_{i0}' \end{pmatrix} . \tag{20.53}$$

As varies over a period of 2-; 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* D *u*0 d*u*0, from (20.52) we get

$$\mathrm{d}u\_0 = a \left[ u\_{10} \sin \phi - u\_{20} \cos \phi \right] \mathrm{d}\phi \tag{20.54}$$

and the area enclosed by the ellipse is

$$\begin{split} A &= 2 \, a^2 \int\_0^\pi (u'\_{10} \cos \phi - u'\_{20} \sin \phi) \left( u\_{10} \sin \phi - u\_{20} \cos \phi \right) \mathrm{d}\phi \\ &= a^2 \pi \left( u\_{10} u'\_{20} - u'\_{10} u\_{20} \right) = a^2 \pi \ , \end{split} \tag{20.55}$$

since the expression in the brackets is the Wronskian, which we choose to normalize to unity. The Wronskian is an invariant of the motion and therefore the area of the phase ellipse along the beam transport line is preserved. The vector equation (20.52) describes the phase ellipse enclosing a beam with the emittance *a*2 D .

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.

Before we discuss coupled motion in more detail it might be useful to recollect some salient features of linear beam dynamics. The concept of conjugate trajectories can be used to define a phase ellipse at *z* D 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* D -, where 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

$$
\beta \,\,\phi' = 1 \,\,\tag{20.56}
$$

as we would expect.

To describe coupled motion we try analogous to (20.52) the ansatz

$$\mathbf{v}(z) = \sqrt{\epsilon\_{\text{l}}} [\mathbf{v}\_{\text{l}}(z) \cos \vartheta\_{\text{l}} - \mathbf{v}\_{2}(z) \sin \vartheta\_{\text{l}}] \cos \chi \tag{20.57}$$

$$+ \sqrt{\epsilon\_{\text{ll}}} [\mathbf{v}\_{3}(z) \cos \vartheta\_{\text{ll}} - \mathbf{v}\_{4}(z) \sin \vartheta\_{\text{l}}] \sin \chi.$$

As the independent variables ; #I and #II vary from 0 to 2 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 D 0 having only horizontal nonvanishing components while **v**3;4 contain nonzero components only in the vertical plane for D -=2. For independent solutions **v***i* of coupled motion, we try

$$\begin{array}{ll} \text{x}\_{\text{l}}(\text{z}) = \sqrt{\beta\_{\text{xl}}(\text{z})} \cos \phi\_{\text{xl}}(\text{z}), & \text{y}\_{\text{l}}(\text{z}) = \sqrt{\beta\_{\text{yn}}(\text{z})} \cos \phi\_{\text{yl}}(\text{z}) \text{ },\\ \text{x}\_{\text{2}}(\text{z}) = \sqrt{\beta\_{\text{xl}}(\text{z})} \sin \phi\_{\text{xl}}(\text{z}), & \text{y}\_{\text{2}}(\text{z}) = \sqrt{\beta\_{\text{yn}}(\text{z})} \sin \phi\_{\text{yl}}(\text{z}) \text{ },\\ \text{x}\_{\text{3}}(\text{z}) = \sqrt{\beta\_{\text{x}\text{ll}}(\text{z})} \cos \phi\_{\text{xl}}(\text{z}), & \text{y}\_{\text{3}}(\text{z}) = \sqrt{\beta\_{\text{yn}}(\text{z})} \cos \phi\_{\text{yn}}(\text{z}) \text{ },\\ \text{x}\_{\text{4}}(\text{z}) = \sqrt{\beta\_{\text{x}\text{ll}}(\text{z})} \sin \phi\_{\text{xl}}(\text{z}), & \text{y}\_{\text{4}}(\text{z}) = \sqrt{\beta\_{\text{yn}}(\text{z})} \sin \phi\_{\text{yn}}(\text{z}), \end{array} \tag{20.58}$$

which is consistent with the earlier definitions of conjugate trajectories. Earlier in this section we defined conjugate trajectories to be independent solutions normalized to the same phase ellipse and developed relationships between these trajectories and betatron functions. These relationships can be expanded to coupled motion by defining betatron functions for both modes of oscillations similar to (20.47)

$$
\beta\_{\mathbf{x\_l}} = \mathbf{x\_l^2} + \mathbf{x\_2^2}, \qquad \qquad \beta\_{\mathbf{x\_{ll}}} = \mathbf{x\_3^2} + \mathbf{x\_4^2}, \tag{20.59}
$$

$$
\beta\_{\rm y1} = \text{y}\_1^2 + \text{y}\_2^2,\qquad\qquad\beta\_{\rm yn} = \text{y}\_3^2 + \text{y}\_4^2.\tag{20.60}
$$

The phase functions can be defined like (20.48) by

$$\cos\phi\_{\mathbf{x}\_{\parallel}} = \frac{\mathbf{x}\_{\parallel}}{\sqrt{\mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{2}}}, \qquad \cos\phi\_{\mathbf{x}\_{\parallel}} = \frac{\mathbf{x}\_{3}}{\sqrt{\mathbf{x}\_{3}^{2} + \mathbf{x}\_{4}^{2}}}, \tag{20.61}$$

$$\cos \phi\_{\rm yl} = \frac{\mathbf{y}\_1}{\sqrt{\mathbf{y}\_1^2 + \mathbf{y}\_2^2}}, \qquad \cos \phi\_{\rm yl} = \frac{\mathbf{y}\_3}{\sqrt{\mathbf{y}\_3^2 + \mathbf{y}\_4^2}}. \tag{20.62}$$

All other lattice functions can be defined in a similar way. By following the conjugate trajectories and utilizing the .4 - 4/-transformation matrices including coupling effects we are able to determine the betatron functions at any point along the coupled beam transport line. To correlate parameters of the four-dimensional phase ellipse with quantities that can be measured, we write the solutions in the form

$$\begin{array}{ll} \chi\_{\rm l}(\boldsymbol{z}) = \sqrt{\beta\_{\rm l\rm l}(\boldsymbol{z})} \cos \phi\_{\rm l\rm l}(\boldsymbol{z}), & \boldsymbol{\chi}\_{\rm l}(\boldsymbol{z}) = \sqrt{\beta\_{\rm l\rm l}(\boldsymbol{z})} \sin \phi\_{\rm l\rm l}(\boldsymbol{z}) \end{array}, \\\begin{array}{ll} \chi\_{\rm l}(\boldsymbol{z}) = \sqrt{\beta\_{\rm l\rm l}(\boldsymbol{z})} \cos \psi\_{\rm l\rm l}(\boldsymbol{z}), & \boldsymbol{\chi}\_{\rm l}'(\boldsymbol{z}) = \sqrt{\beta\_{\rm l\rm l}(\boldsymbol{z})} \sin \psi\_{\rm l\rm l}(\boldsymbol{z}) \end{array}, \end{array} \tag{20.63}$$

and similar for all other solutions. Comparing the second equations in (20.63) with the derivative of the first equations we find the definitions

$$\gamma\_{\rm xl} = \frac{\beta\_{\rm xq}^2 \phi\_{\rm xq}^{\prime 2} + \alpha\_{\rm xq}^2}{\beta\_{\rm xl}} \tag{20.64}$$

and

$$
\psi\_{\rm xl} = \phi\_{\rm xl} - \arctan \frac{\beta\_{\rm xl} \phi\_{\rm xl}'}{\alpha\_{\rm xl}} \,. \tag{20.65}
$$

The other parameters *x*II , etc. are defined similarly and the phase ellipse (20.57) can now be expressed by the four-dimensional vector

$$\mathbf{v}(z) = \sqrt{\epsilon\_I} \begin{pmatrix} \sqrt{\beta\_{xl}} \cos \left(\phi\_{xl} + \vartheta\_I\right) \\ \sqrt{\gamma\_{xl}} \cos \left(\psi\_{xl} + \vartheta\_I\right) \\ \sqrt{\beta\_{yl}} \cos \left(\phi\_{yl} + \vartheta\_I\right) \\ \sqrt{\gamma\_{yl}} \cos \left(\psi\_{yl} + \vartheta\_I\right) \end{pmatrix} \cos \chi \tag{20.66}$$

$$+ \sqrt{\epsilon\_H} \begin{pmatrix} \sqrt{\beta\_{xl}} \cos \left(\phi\_{xl} + \vartheta\_H\right) \\ \sqrt{\gamma\_{xl}} \cos \left(\psi\_{xl} + \vartheta\_H\right) \\ \sqrt{\beta\_{yl}} \cos \left(\phi\_{yl} + \vartheta\_H\right) \\ \sqrt{\gamma\_{yl}} \cos \left(\psi\_{yl} + \vartheta\_H\right) \end{pmatrix} \sin \chi$$

This vector covers all points on the surface of the four-dimensional ellipsoid as ; #I and #II vary independently from 0 to 2-. For one-dimensional oscillations we know from the definition of the phase ellipse that the product p*u* pˇ*u* is equal to the beam size or beam envelope *Eu* and p*u* p*u* equal to the angular beam envelope *Au*, where *u* D *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 -II as shown in Fig. 20.1.

In a real beam transport line we are not able to observe experimentally the fourdimensional 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*0 /, the .*y y*0 / or the .*x y*/-plane.

To do that we note in (20.66) that the maximum amplitude of a particle in the *u*plane occurs for *u*I;II D #*u*I;II and a projection angle given by sin2 D *u*II ˇ*u*II *Eu* ; where the beam envelope for coupled motion is given by

$$E\_{\rm u} = \sqrt{\epsilon\_{\rm u\eta} \beta\_{\rm u\eta} + \epsilon\_{\rm u\eta} \beta\_{\rm u\eta}} \,. \tag{20.67}$$

**Fig. 20.1** Phase space ellipse for coupled motion

Similarly, we get from the second component of (20.66) the angular envelope

$$A\_u = \sqrt{\epsilon\_{u\_{\rm l}} \chi\_{u\_{\rm l}} + \epsilon\_{u\_{\rm ll}} \chi\_{u\_{\rm ll}}} \tag{20.68}$$

for *u*I;II D #Q *u*I;II and a projection angle given by

$$
\sin^2 \chi = \frac{\epsilon\_{\text{ull}} \beta\_{\text{ull}}}{A\_{\text{u}}} \,. \tag{20.69}
$$

To completely determine the phase ellipse we calculate also the slope *x*0 for the particle at *x* D *Ex* which is the slope of the envelope *E*0 . Taking the derivative of (20.67) we get

$$E'\_{\mu} = -\frac{\epsilon\_{\text{u\\_}\text{\\_}} + \epsilon\_{\text{u\\_}\text{\\_}}\alpha\_{\text{u\\_}}}{\sqrt{\epsilon\_{\text{u\\_}}\beta\_{\text{u\\_}} + \epsilon\_{\text{u\\_}}\beta\_{\text{u\\_}}}} \cdot \tag{20.70}$$

Expressing the equation of the phase ellipse in terms of these envelope definitions we get

$$A\_{\
u}^{2}\mu^{2} - 2\, E\_{\mu}'E\_{\mu}\mu\mu' + E\_{\mu}^{2}\mu'^{2} = \epsilon\_{\mu}^{2} \tag{20.71}$$

and inserting *u* D *Eu* and *u*0 D *E*0 *u* into (20.71) we get for the emittance of the projection ellipse

$$
\epsilon\_u = E\_u \sqrt{A\_u^2 - E\_u'^2}.\tag{20.72}
$$

The envelope functions can be measured noting that *E*2 D 11; *A*2 D 22 and *EE*0 D 12 where the *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.

A more important and obvious projection is that onto the .*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 *Ex* and *Ey* have been derived before in (20.67) and become here

$$E\_{\rm x} = \sqrt{\epsilon\_{\rm xq} \beta\_{\rm xq} + \epsilon\_{\rm xq} \beta\_{\rm xq}} \,, \tag{20.73}$$

$$E\_{\rm y} = \sqrt{\epsilon\_{\rm y\_1} \beta\_{\rm y\_1} + \epsilon\_{\rm y\_{\rm y}} \beta\_{\rm y\_{\rm y}}} \,. \tag{20.74}$$

The *y*-coordinate for *Ex*, which we denote by *Exy*, can be derived from the third equation in (20.66) noting that now #*y*I;II D *x*I;II , is given by (20.69) and

$$E\_{\rm xy} = \frac{\epsilon\_{\rm 1} \sqrt{\beta\_{\rm x\_{\rm I}} \beta\_{\rm y\_{\rm I}}} \cos \Delta \phi\_{\rm i} + \epsilon\_{\rm 1} \sqrt{\beta\_{\rm x\_{\rm II}} \beta\_{\rm y\_{\rm II}}} \cos \Delta \phi\_{\rm i}}{\sqrt{\epsilon\_{\rm x\_{\rm I}} \beta\_{\rm x\_{\rm I}} + \epsilon\_{\rm x\_{\rm II}} \beta\_{\rm x\_{\rm II}}}},\tag{20.75}$$

where I;II D *x*I;II *y*I;II . The beam cross section is tilted due to coupling whenever *Ex*;*y* ¤ 0. The tilt angle of the ellipse is determined by

$$\tan 2\psi = \frac{2\,E\_\text{x}\,E\_\text{xy}}{E\_\text{x}^2 - E\_\text{y}^2} \tag{20.76}$$

or more explicitly

$$\tan 2\psi = 2 \frac{\epsilon\_{\rm l} \sqrt{\beta\_{\rm l\rm l} \beta\_{\rm y}} \cos \Delta \phi\_{\rm l} + \epsilon\_{\rm ll} \sqrt{\beta\_{\rm x\rm ll} \beta\_{\rm y\rm l}} \cos \Delta \phi\_{\rm l\rm l}}{\epsilon\_{\rm x\_{\rm l}} \Delta \beta\_{\rm l} + \epsilon\_{\rm x\_{\rm l\rm l}} \Delta \beta\_{\rm l\rm l}} \tag{20.77}$$

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 ˇ*x*II ! 0 and ˇ*y*I ! 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*

We will derive some properties of coupled motion for the case of linear coupling introduced, for example, by a rotated quadrupole. Equations of linearly coupled motion are with *k* D *p*.*z*/ of the form

$$\begin{aligned} \mathbf{x}'' + k\,\mathbf{x} &= -p(\mathbf{z})\,\mathbf{y} \,, \\ \mathbf{y}'' - \mathbf{y}\,\mathbf{x} &= -p(\mathbf{z})\,\mathbf{x} \,, \end{aligned} \tag{20.78}$$

which can be derived from the Hamiltonian for linearly coupled motion

$$H = \frac{1}{2} \left. \mathbf{x}'^2 + \frac{1}{2} \left. \mathbf{y}'^2 + \frac{1}{2} k \mathbf{x}^2 - \frac{1}{2} k \mathbf{y}^2 + p(\mathbf{z}) \mathbf{x} \mathbf{y} \right. \tag{20.79}$$

This Hamiltonian is composed of an uncoupled Hamiltonian *H*0 and the perturbation Hamiltonian for linear coupling

$$H\_1 = p(\mathbf{z}) \propto \mathbf{y} \,. \tag{20.80}$$

The solutions for the uncoupled equations with integration constants *cu* and are of the form

$$\begin{array}{l} u(z) = c\_u \sqrt{\beta\_u} \cos \left[ \psi\_u(z) + \phi \right] \,, \\ u'(z) = -\frac{c\_u}{\sqrt{\beta\_u}} \left\{ \alpha\_u(z) \cos \left[ \psi\_u(z) + \phi \right] + \sin \left[ \psi\_u(z) + \phi \right] \right\} \,. \end{array} \tag{20.81}$$

Applying the method of variation of integration constants, we try the ansatz

$$\begin{cases} u(z) = \sqrt{2a\_{\mathfrak{u}}(z)} \sqrt{\beta\_{\mathfrak{u}}} \cos \left[ \psi\_{\mathfrak{u}}(z) + \phi(z) \right], \\ u'(z) = -\sqrt{\frac{2a\_{\mathfrak{u}}(z)}{\beta\_{\mathfrak{u}}}} \left\{ \alpha\_{\mathfrak{u}}(z) \cos \left[ \psi\_{\mathfrak{u}}(z) + \phi(z) \right] + \sin \left[ \psi\_{\mathfrak{u}}(z) + \phi(z) \right] \right\}, \end{cases} \tag{20.82}$$

for the coupled motion. Now we use the integration constants.*a*;/ as new variables and to show that the new variables are canonical, we use the Hamiltonian equations @*H*=@*u*0 Dd*u*=d*z* and @*H*=@*u* D d*u*0 =d*z* and get

$$\frac{\partial H}{\partial u'} = \frac{\partial H\_0}{\partial u'} + \frac{\partial H\_1}{\partial u'} = \frac{\mathbf{d}u}{\mathbf{d}z} = \frac{\partial u}{\partial z} + \frac{\partial u}{\partial a}\frac{\partial a}{\partial z} + \frac{\partial u}{\partial \phi}\frac{\partial \phi}{\partial z}.\tag{20.83}$$

A similar expression exists for the second Hamiltonian equation of motion

$$\frac{\partial H}{\partial u} = \frac{\partial H\_0}{\partial u} + \frac{\partial H\_1}{\partial u} = -\frac{\mathbf{d}u'}{\mathbf{d}z} = -\frac{\partial u'}{\partial z} - \frac{\partial u'}{\partial a}\frac{\partial a}{\partial z} - \frac{\partial u'}{\partial \phi}\frac{\partial \phi}{\partial z} \,. \tag{20.84}$$

For uncoupled oscillators we know that *a* D const. and D const. and therefore @*u*=@*z* D @*H*0=@*u*0 and @*u*0 =@*z* D @*H*0=@*u*. With this we derive from (20.81)– (20.84) the equations

$$\begin{array}{ll}\frac{\partial H\_{1}}{\partial \phi} = \frac{\partial H\_{1}}{\partial u}\frac{\partial u}{\partial \phi} + \frac{\partial H\_{1}}{\partial u'}\frac{\partial u'}{\partial \phi} = -\frac{\mathbf{d}a}{\mathbf{d}z},\\ \frac{\partial H\_{1}}{\partial a} = +\frac{\partial H\_{1}}{\partial u}\frac{\partial u}{\partial a} + \frac{\partial H\_{1}}{\partial u'}\frac{\partial u'}{\partial a} = \frac{\mathbf{d}\phi}{\mathbf{d}z},\end{array} \tag{20.85}$$

demonstrating that the new variables .; *a*/ are canonical variables and (20.82) are canonical transformations. Applying (20.82) to the perturbation Hamiltonian (20.80) with appropriate indices to distinguish between horizontal and vertical plane, the perturbation Hamiltonian becomes

$$H\_1 = 2\, p(z)\sqrt{\beta\_x \beta\_\text{y}}\sqrt{a\_x a\_\text{y}}\cos\left(\psi\_x + \phi\_x\right)\cos\left(\psi\_\text{y} + \phi\_\text{y}\right)\,,\tag{20.86}$$

where *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

$$\cos\left(\psi\_u + \phi\_u\right) = \frac{1}{2}\left[\mathbf{e}^{\mathbf{i}(\psi\_u + \phi\_u)} + \mathbf{e}^{-\mathbf{i}(\psi\_u + \phi\_u)}\right] \tag{20.87}$$

and the Hamiltonian takes the form

$$H\_1 = \frac{1}{2} p(z) \sqrt{\beta\_x \beta\_y} \sqrt{a\_x a\_y} \sum\_{l\_x, l\_y} \mathbf{e}^{\mathbf{i} \left[ l\_x (\psi\_x + \phi\_x) + l\_y (\psi\_y + \phi\_y) \right]},\tag{20.88}$$

where the non-zero integers *lx* and *ly* are integers defined by

$$l\_x, l\_y \exists \ (-1, 1) \;. \tag{20.89}$$

Similar to the one-dimensional case we try to separate constant or slowly varying terms from the fast oscillating terms and expand the exponent in (20.88) like

$$\begin{split} l\_{\mathbf{x}}\psi\_{\mathbf{x}} + l\_{\mathbf{y}}\psi\_{\mathbf{y}} - l\_{\mathbf{x}}\upsilon\_{0\mathbf{x}}\varphi - l\_{\mathbf{y}}\upsilon\_{0\mathbf{y}}\varphi \\ + l\_{\mathbf{x}}\upsilon\_{0\mathbf{x}}\varphi + l\_{\mathbf{y}}\upsilon\_{0\mathbf{y}}\varphi + l\_{\mathbf{x}}\phi\_{\mathbf{x}} + l\_{\mathbf{y}}\phi\_{\mathbf{y}} \end{split} \tag{20.90}$$

where *u* D 0*u*'; 0*u* are the tunes for the periodic lattice, ' D 2*z*=*L* and *L* is the length of the lattice period. The first four terms in (20.90) are periodic with the period ' .*L*/ D 2- C ' .0/. Inserting (20.90) into (20.88) we get with *u* .*L*/ D 2- 0*u* C *u* .0/

$$\overline{H}\_{1} = \frac{1}{2} \sum\_{l\_{x}, l\_{\circ}} p(z) \sqrt{\beta\_{x} \beta\_{\circ}} \mathbf{e}^{\mathbf{i} \left[ l\_{x} \psi\_{x} + l\_{\circ} \psi\_{\circ} - l\_{x} v\_{0x} \varphi - l\_{\circ} v\_{0\gamma} \varphi \right]}$$

$$\times \sqrt{a\_{x} a\_{\circ}} \sum\_{l\_{x}, l\_{\circ}} \mathbf{e}^{\mathbf{i} \left[ l\_{\circ} v\_{0x} \varphi + l\_{\circ} v\_{0\gamma} \varphi + l\_{\circ} \psi\_{\gamma} + l\_{\circ} \phi\_{\gamma} \right]} \,. \tag{20.91}$$

In this form we recognize the periodic factor

$$A\left(\varphi\right) = p(z)\sqrt{\beta\_x \beta\_y} \mathbf{e}^{\mathbf{i}\left[l\_x \psi\_x + l\_y \psi\_y - l\_x \upsilon\_{0x} \varphi - l\_y \upsilon\_{0y} \varphi\right]} \tag{20.92}$$

since betatron functions and perturbations *p*.*z*/ D*k*.*z*/ are periodic. After expanding (20.92) into a Fourier series

$$\frac{L}{2\pi}A\left(\varphi\right) = \sum\_{q} \kappa\_{q,l\_{\chi}l\_{\chi}} \mathbf{e}^{\mathrm{iq}N\varphi} \tag{20.93}$$

coupling coefficients can be defined by

$$\begin{split} \kappa\_{q,l\_{\nu}l\_{\mathcal{V}}} &= \frac{1}{2\pi} \int\_{0}^{2\pi} \frac{L}{2\pi} A\left(\varphi\right) \mathbf{e}^{-iqN\varphi} \mathrm{d}\varphi \\ &= \frac{1}{2\pi} \int\_{0}^{L} \underline{k} \sqrt{\beta\_{x} \wp\_{\mathcal{V}}} \mathbf{e}^{\left[l\_{\boldsymbol{x}}\psi\_{\boldsymbol{x}} + l\_{\boldsymbol{y}}\psi\_{\boldsymbol{\gamma}} - \left(l\_{\boldsymbol{x}}v\_{0\boldsymbol{x}} + l\_{\boldsymbol{y}}v\_{0\boldsymbol{\gamma}} - qN\right)\frac{2\pi}{\mathcal{T}}z\right]} \mathrm{d}z \end{split} \tag{20.94}$$

Since *q*;1;1 D *q*;1;1 and *q*;1;1 D *q*;1;1, we have with 1 *l* C1

$$\kappa\_{q,l} = \frac{1}{2\pi} \int\_0^L \underline{k} \sqrt{\beta\_x \beta\_y} \,\mathrm{e}^{\mathrm{i}\left[\psi\_x + l\,\psi\_\gamma - \left(v\_{0x} + l\,v\_{0\gamma} - qN\right)\frac{2\pi}{L}z\right]} \,\mathrm{d}z \,. \tag{20.95}$$

The coupling coefficient is a complex quantity indicating that there are two orthogonal and independent contributions which require also two orthogonally independent corrections. Now that the coupling coefficients are defined in a convenient form for numerical evaluation we replace the independent variable *z* by the angle variable ' D 2*z*=*L* and obtain the new Hamiltonian *H*Q 1 D 2- *L H*1 or

$$\tilde{H} = \sum\_{q} \kappa\_{q,l} \sqrt{a\_x a\_y} \cos \left(\phi\_x + l\phi\_y + \Delta\_q \varphi\right) \,, \tag{20.96}$$

where

$$
\Delta\_q = \upsilon\_{0\mathbf{x}} + l\,\upsilon\_{0\mathbf{y}} - qN. \tag{20.97}
$$

Most terms in (20.96) are fast oscillating and therefore cancel before any damage can be done to particle stability. One term, however, is slowly varying for *q* D *r* defining the resonance condition for coupled motion *q* 0 or

$$
\nu N \approx \nu\_{0\mathbf{x}} + l \,\nu\_{0\mathbf{y}} \,. \tag{20.98}
$$

In this resonant case, the quantity r is the distance of the tunes from the coupling resonance as defined by (20.97) with *q* D *r*. Neglecting all fast oscillating terms we apply one more canonical transformation .*u*; *au*/ ! Q*u*; *a*Q *u* to eliminate the independent variable ' 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

$$G = \tilde{a}\_x \left(\phi\_x + \frac{1}{2}\Delta\_\mathbf{r}\phi\right) + \tilde{a}\_\mathbf{y} \left(\phi\_\mathbf{y} + l\frac{1}{2}\Delta\_\mathbf{r}\phi\right) \tag{20.99}$$

we get for the new variables

$$\begin{array}{ll}\tilde{\phi}\_{\mathbf{x}} = \frac{\partial G}{\partial \mathbf{a}\_{\mathbf{x}}} = \phi\_{\mathbf{x}} + \frac{1}{2}\Delta\_{\mathbf{r}}\varphi \; , \; \tilde{a}\_{\mathbf{x}} = \frac{\partial G}{\partial \phi\_{\mathbf{x}}} = a\_{\mathbf{x}} \; , \\\ \tilde{\phi}\_{\mathbf{y}} = \frac{\partial G}{\partial a\_{\mathbf{y}}} = \phi\_{\mathbf{y}} + l\frac{1}{2}\Delta\_{\mathbf{r}}\varphi \; , \; \tilde{a}\_{\mathbf{y}} = \frac{\partial G}{\partial \phi\_{\mathbf{y}}} = a\_{\mathbf{y}} \; , \end{array} \tag{20.100}$$

and the new Hamiltonian for the rotating coordinate system is

$$
\tilde{H}\_{\rm t} = \tilde{H} + \frac{\partial G}{\partial \varphi} = \tilde{H} + \frac{1}{2} \Delta\_{\rm t} a\_{\rm x} + l \frac{1}{2} \Delta\_{\rm t} a\_{\rm y} \,. \tag{20.101}
$$

For simplicity we drop the tilde on the amplitudes and use *ax*; *ay* : The resonant Hamiltonian becomes after this transformation

$$\tilde{H}\_{\rm r} = \frac{1}{2}\Delta\_{\rm r} \left(a\_{\rm x} + la\_{\rm y}\right) + \kappa\_{\rm r,l} \sqrt{a\_{\rm x}a\_{\rm y}} \cos\left(\tilde{\phi}\_{\rm x} + l\,\tilde{\phi}\_{\rm y}\right) \tag{20.102}$$

and the equations of motion are

$$\begin{array}{lcl}\frac{\partial a\_{\mathbf{t}}}{\partial \boldsymbol{\phi}} = -\frac{\partial H\_{\mathbf{t}}}{\partial \boldsymbol{\Phi}\_{\mathbf{x}}} = \kappa\_{\mathbf{t},l}\sqrt{a\_{\mathbf{x}}a\_{\mathbf{y}}}\sin\left(\tilde{\boldsymbol{\phi}}\_{\mathbf{x}} + l\,\tilde{\boldsymbol{\phi}}\_{\mathbf{y}}\right)\,, \\\frac{\partial a\_{\mathbf{y}}}{\partial \boldsymbol{\phi}} = -\frac{\partial H\_{\mathbf{t}}}{\partial \boldsymbol{\phi}\_{\mathbf{y}}} = l\,\kappa\_{\mathbf{r},l}\sqrt{a\_{\mathbf{x}}a\_{\mathbf{y}}}\sin\left(\tilde{\boldsymbol{\phi}}\_{\mathbf{x}} + l\,\tilde{\boldsymbol{\phi}}\_{\mathbf{y}}\right)\,, \end{array} \tag{20.103}$$

and

$$\begin{array}{ll}\frac{\partial\tilde{\phi}\_{\mathbf{r}}}{\partial\varphi} = \frac{\partial\tilde{H}\_{\mathbf{r}}}{\partial a\_{\mathbf{r}}} = \frac{1}{2}\Delta\_{\mathbf{r}} + \frac{1}{2}\kappa\_{\mathbf{r},l}\sqrt{\frac{a\_{\mathbf{r}}}{a\_{\mathbf{r}}}}\cos\left(\tilde{\phi}\_{\mathbf{r}} + l\,\tilde{\phi}\_{\mathbf{y}}\right),\\ \frac{\partial\tilde{\phi}\_{\mathbf{y}}}{\partial\varphi} = \frac{\partial\tilde{H}\_{\mathbf{r}}}{\partial a\_{\mathbf{y}}} = l\frac{1}{2}\Delta\_{\mathbf{r}} + \frac{1}{2}\kappa\_{\mathbf{r},l}\sqrt{\frac{a\_{\mathbf{r}}}{a\_{\mathbf{y}}}}\cos\left(\tilde{\phi}\_{\mathbf{x}} + l\,\tilde{\phi}\_{\mathbf{y}}\right)\end{array} \tag{20.104}$$

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* D C1 or a difference resonance if *l* D 1.

#### **Linear Difference Resonance**

In case of a difference resonance .*l* D 1/ we add both Eqs. (20.103) and get

$$\frac{\mathrm{d}}{\mathrm{d}\varphi} \left( a\_{\mathrm{x}} + a\_{\mathrm{y}} \right) = 0 \,. \tag{20.105}$$

The coupled motion is stable because the sum of both amplitudes does not change. Both amplitudes *ax* and *ay* 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 *ax* and *ay* are proportional to the beam emittance, we note that the sum of the horizontal and vertical emittance stays constant as well,

$$
\epsilon\_x + \epsilon\_y = \text{const.}\tag{20.106}
$$

The resonance condition (20.98) for a difference resonance becomes [2]

$$
\nu\_x - \nu\_y m\_r = N.\tag{20.107}
$$

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.

There exists a finite stop-band width also for the coupling resonance just as for any other resonance and we have all the mathematical tools to calculate that width. Since the beam is not lost at a difference coupling resonance, we are also able to measure experimentally the stop-band width by moving the tunes through the resonance. The procedure becomes obvious after linearizing the equations of motion (20.103), (20.104). Following a suggestion by Guignard [3], we define new variables similar in form to normalized coordinates

$$\begin{array}{l} w = \sqrt{a\_x} \mathbf{e}^{\dot{\phi}\_x}, \\ v = \sqrt{a\_y} \mathbf{e}^{\dot{\phi}\_y}. \end{array} \tag{20.108}$$

Taking derivatives of (20.108) with respect to ' and using (20.103), (20.104) we get after some manipulation the linear equations

$$\begin{array}{l} \frac{d\mathbf{w}}{d\boldsymbol{\varphi}} = \mathbf{i}\frac{1}{2} \left(\boldsymbol{\kappa}\boldsymbol{v} + \boldsymbol{\Delta}\_{\mathbf{r}}\boldsymbol{w}\right) \,, \\\frac{d\mathbf{v}}{d\boldsymbol{\varphi}} = \mathbf{i}\frac{1}{2} \left(\boldsymbol{\kappa}\boldsymbol{w} - \boldsymbol{\Delta}\_{\mathbf{r}}\boldsymbol{v}\right) \,, \end{array} \tag{20.109}$$

where we have set for simplicity r;1 D .

These equations can be solved analytically and will provide further insight into the dynamics of coupled oscillations. We will look for characteristics of coupled motion which do not depend on initial conditions but are general for all particles. Expecting the solutions *w* and v to describe oscillations, we assume that the motion in both planes depends on the initial conditions *w*0; v0 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 v0 D 0. The ansatz for the oscillations be

$$\begin{aligned} \mathcal{W}(\varphi) &= \mathcal{W}\_0 \left( a \operatorname{\mathbf{e}}^{\mathrm{i}\nu\varphi} + b \operatorname{\mathbf{e}}^{-\mathrm{i}\nu\varphi} \right), \\ \mathcal{V}(\varphi) &= \mathcal{W}\_0 \left( c \operatorname{\mathbf{e}}^{\mathrm{i}\nu\varphi} + d \operatorname{\mathbf{e}}^{-\mathrm{i}\nu\varphi} \right), \end{aligned} \tag{20.110}$$

where we define an as yet undefined frequency . Inserting (20.110) into (20.109) the coefficients of the exponential functions vanish separately and we get from the coefficients of ei ' the two equations

$$\begin{aligned} 2\nu a &= \kappa c + \Delta\_t a \,, \\ 2\nu c &= \kappa a + \Delta\_t c \,, \end{aligned} \tag{20.111}$$

from which we may eliminate the unknowns *a* and *c* to get the defining equation for the oscillation frequency

$$\nu = \frac{1}{2}\sqrt{\Delta\_{\mathbf{r}}^2 + \kappa^2}. \tag{20.112}$$

While determining the coefficients *a*; *b*; *c*; *d*; we note that due to the initial conditions *a* C *b* D 1 and *c* C *d* D 0. Similar to (20.111) we derive another pair of equations from the coefficients of ei '

$$\begin{aligned} 2\nu b &= \kappa d - \Delta\_{\text{f}} b \,, \\ 2\nu d &= \kappa b + \Delta\_{\text{f}} d \,, \end{aligned} \tag{20.113}$$

which completes the set of four equations required to determine with (20.112) the four unknown coefficients

$$\begin{array}{l} a = \frac{2v + \Delta\_t}{4v}, \ b = \frac{2v - \Delta\_t}{4v}, \\ c = \frac{\kappa}{4v}, \qquad d = -\frac{\kappa}{4v}, \end{array} \tag{20.114}$$

With this, the solutions (20.110) become

$$\begin{array}{l} \left(w(\boldsymbol{\varphi}) = \boldsymbol{w}\_{0}\cos\nu\boldsymbol{\varphi} + \mathrm{i}\,\boldsymbol{w}\_{0}\frac{\Delta\_{t}}{2\boldsymbol{v}}\sin\nu\boldsymbol{\varphi}\right),\\ \left(v(\boldsymbol{\varphi}) = \boldsymbol{r}\right) + \mathrm{i}\,\boldsymbol{w}\_{0}\frac{\boldsymbol{\kappa}}{2\boldsymbol{v}}\sin\nu\boldsymbol{\varphi}\,,\end{array} \tag{20.115}$$

and by multiplication with the complex conjugate and (20.108) we get expressions for the coupled beam emittances .*u* D 2*au*/

$$\begin{cases} a\_x = a\_{x0} \frac{1}{4\nu^2} \left(\Delta\_\mathbf{r}^2 + \kappa^2 \cos^2 \nu \varphi\right) \\ a\_\circ = a\_{x0} \frac{\kappa^2}{4\nu^2} \sin^2 \nu \varphi \ . \end{cases} \tag{20.116}$$

The ratio of maximum values for beam emittances in both planes under the influence of linear coupling is from (20.116)

$$\frac{\epsilon\_{\rm y}}{\epsilon\_{\rm x}} = \frac{\kappa^2}{\Delta\_{\rm r}^2 + \kappa^2} \,. \tag{20.117}$$

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 . At the coupling resonance we observe complete exchange of emittances at the frequency . If on the other hand, the tunes differ and r ¤ 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.

We may now collect all results and derive the particle motion as a function of time or '. For example, the horizontal particle position is determined from (20.82) where we set p*ax* D *w*eiQ*x* and further replace *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 ei. *x*C*x*/ . Together with other oscillatory quantities eiQ*x* and *w* we get both in the horizontal and vertical plane terms with oscillatory factors

$$\operatorname{Re}\left[\mathbf{e}^{i(\psi\_{\mu}+\phi\_{\mu}-\bar{\phi}\_{\mu}\pm\upsilon\varphi)}\right] \tag{20.118}$$

where the index *u* stands for either *x* or *y*. The phase *u* D *u*' and from (20.100) and *l* D 1 for the difference resonance Q*u* D *u* ˙ 1 2 r'. These expressions used in (20.118) define two oscillation frequencies

$$\nu\_{I,H} = \nu\_{x,y} \mp \frac{1}{2} \Delta\_{\mathbf{r}} \pm \nu \tag{20.119}$$

or with (20.112)

$$\nu\_{I,H} = \nu\_{x,\mathbf{y}} \mp \frac{1}{2} \Delta\_{\mathbf{f}} \pm \frac{1}{2} \sqrt{\Delta\_{\mathbf{f}}^2 + \kappa^2} \,. \tag{20.120}$$

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 . 0/ the mode frequencies are approximately equal to the uncoupled frequencies  *x*;*y*, respectively. Even for large coupling this equality is preserved as long as the tunes are far away from the coupling resonance or r .

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 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 *I* does not change appreciably until the resonance is reached, because *r* C p2 r C 2 0. Above the coupling resonance, however, *r* has changed sign and *I* increase with *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.

Actual tune measurements [4] are shown in Fig. 20.2 as a function of the excitation current of a vertically focusing quadrupole. The vertical tune change is proportional to the quadrupole current and so is the parameter *r*. While increasing the quadrupole current, the vertical tune is increased and the horizontal tune stays practically constant. We note that the tunes actually do not cross the linear coupling resonance during that procedure, rather the tune of one plane is gradually transformed into the tune of the other plane and vice versa. Both tunes never become equal and the closest distance is determined by the magnitude of the coupling coefficient .

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

**Fig. 20.2** Measurements of mode frequencies as a function of detuning for linearly coupled motion [4]

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**

To complete the discussion, we will now set *l* D C1 and get from (20.98) the resonance condition for a sum resonance

$$
\nu\_x + \nu\_y = m\_r N \,. \tag{20.121}
$$

Taking the difference of both Eqs. (20.103), we get

$$\frac{d}{d\rho} \left( a\_{\mathbf{x}} - a\_{\mathbf{y}} \right) = 0 \,, \tag{20.122}$$

which states only that the difference of the emittances remains constant. Coupled motion in the vicinity of a sum resonance is therefore unstable allowing both emittances to grow unlimited. To solve the equations of motion (20.103), (20.104), we try the ansatz

$$
\mu = \sqrt{a\_x} \mathbf{e}^{i\phi\_x} + \mathbf{i}\sqrt{a\_y} \mathbf{e}^{i\phi\_y}.\tag{20.123}
$$

From the derivative d*u*=d', we get with (20.103), (20.104)

$$\frac{\mathrm{d}\mu}{\mathrm{d}\varphi} = \mathrm{i}\_{\mathrm{Z}}^{\mathrm{I}} \left( \Delta\_{r} \, \mu - \kappa \, \mu^{\star} \right) \,, \tag{20.124}$$

and for the complex conjugate

$$\frac{\mathrm{d}u^{\*}}{\mathrm{d}\varphi} = -\mathrm{i}\frac{1}{2}\left(\Delta\_{r}\,\boldsymbol{u}^{\*} + \kappa\,\boldsymbol{u}\right)\,. \tag{20.125}$$

Solving these differential equations with the ansatz

$$
\mu = a \text{ e}^{\text{iv}\varphi} + b \text{ e}^{-\text{iv}\varphi}, \tag{20.126}
$$

and the complex conjugate

$$
\mu^\* = a \text{ e }^{-\text{i}\nu\psi} + b \text{ e }^{\text{i}\nu\psi}, \tag{20.127}
$$

we get after insertion into (20.124), (20.125) analogous to (20.111) the oscillation frequency

$$\nu = \frac{1}{2}\sqrt{\Delta\_{\mathbf{r}}^2 - \kappa^2}. \tag{20.128}$$

This result shows that motion in the vicinity of a linear sum resonance becomes unstable as soon as the detuning is less than the coupling coefficient. The condition for stability is therefore

$$
\Delta\_{\mathbf{f}} > \kappa \,. \tag{20.129}
$$

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*

So far all discussions on coupled motions and resonances have been based on linear coupling effects caused by rotated quadrupole fields. For higher-order coupling the mathematical treatment of the beam dynamics is similar although more elaborate. The general form of the *n*th-order resonance condition (20.98) is

$$l\_{\mathbf{x}}\nu\_{\mathbf{x}} + l\_{\mathbf{y}}\nu\_{\mathbf{y}} = m\_{\mathbf{r}}N \qquad \text{with} \qquad |l\_{\mathbf{x}}| + \left|l\_{\mathbf{y}}\right| \le n\,. \tag{20.130}$$

The factors *lx* and *ly* are integers and the sum j*lx*j C ˇ ˇ*ly* ˇ ˇ 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 *x* 2*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 [7–10]. 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 90ı per cell in both planes. The half cell length be *L* D 5 m and the full quadrupole length ` D 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 :

**20.3.** Can we rotate a horizontally flat 10 GeV beam by 90ı 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 =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* D 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 =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 90ı per cell. Now introduce a rotational misalignment of the first quadrupole by an angle ˛ which generates coupling of the horizontal and vertical betatron oscillations: a.) Calculate and plot the perturbed betatron functions ˇI and ˇII and compare with the unperturbed solution. b.) If the beam emittances are I D 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

### **References**

# **Part VIII Intense Beams**

# **Chapter 21 Statistical and Collective Effects\***

Transverse and longitudinal beam dynamics as discussed in earlier chapters is governed by purely single-particle effects where the results do not depend on the presence of other particles or any interactive environment. Space-charge effects were specifically excluded. This restriction is sometimes too extreme and collective effects must be taken into account where significant beam intensities are desired. In most applications high beam intensities are desired and it is therefore prudent to test for the appearance of space charge and other intensity effects.

Collective effects can be divided into two distinct groups according to the physics involved. The compression of a large number of charged particles into a small volume increases the probability for collisions of particles within the same beam. Because particles perform synchrotron and betatron oscillations, statistical collisions occur in longitudinal, as well as transverse phase space often causing a mixing of phase space coordinates. The other group of collective effects includes effects which are associated with electromagnetic fields generated by the collection of all particles in a beam.

The study and detailed understanding of the cause and nature of collective effects or collective instabilities with corrective measures is important for a successful design of the accelerator. Most accelerator design and developments are conducted to eliminate collective effects as much as possible through self-imposed limitation on the performance or installation of feedback systems and other stabilizing control mechanisms. Beyond that, we also must accept limitations in beam performance imposed by nature or lack of understanding and technological limits. Pursuit of accelerator physics is the attempt to explore and push such limits as far as nature and general understanding of the subject allows.

This chapter has been made Open Access under a CC BY 4.0 license. For details on rights and licenses please read the Correction https://doi.org/10.1007/978-3-319-18317-6\_28

#### **21.1 Statistical Effects**

Coupling of individual particles to the presence of other particles may occur through very short range forces in collisions with each other. In this section, we will discuss statistical effects related to the finite number of particles and from collision processes within a particle bunch.

#### *21.1.1 Schottky Noise*

Electrical current is established by moving charged particles. The finite electrical charge and finite number of particles gives rise to statistical variations of the electrical current. This phenomenon has been observed and analyzed by Schottky [1] and we will discuss this Schottky noise in the realm of particle dynamics in circular accelerators. The information included in the Schottky noise is of great diagnostic importance for the nondestructive determination of particle beam parameters, a technique which has been developed at the CERN Intersecting Storage Ring (ISR) [2] and has become a standard tool of beam diagnostics.

We consider a particle *k* with charge *q* orbiting in an accelerator with the angular revolution frequency !*k* and define a particle line density by 2-*R*.*t*/ D 1 where 2-*R* is the circumference of the ring. On the other hand, we may describe the orbiting particle by delta functions

$$q = q \int\_0^{2\pi} \sum\_{m = -\infty}^{+\infty} \delta(\omega\_k t + \theta\_k - 2\pi m) \,\mathrm{d}\theta \,\,\mathrm{d}\theta$$

where !*k* is the angular revolution frequency of the particle *k* and *k* its phase at time *t* D 0. The delta function can be expressed by a Fourier series and the line-charge density at time *t* becomes

$$q\lambda\_k(t) = \frac{q}{2\pi\overline{R}} \left[ 1 + 2\sum\_{n=0}^{\infty} \cos(n\phi\_k t + n\theta\_k) \right]. \tag{21.1}$$

From a pick up electrode close to the circulating particle, we would obtain a signal with a frequency line spectrum ! D *n*!*k* where *n* is an integer. In a real particle beam there are many particles with a finite spread of revolution frequencies !*k* and therefore the harmonic lines *n*!*k* spread out proportionally to *n*. For not too high harmonic numbers the frequency spreads do not yet overlap and we are able to measure the distribution of revolution frequencies. Tuning the spectrum analyzer to !, we observe a signal with an amplitude proportional to *N*.!=*n*/ ı! *n* where *N*.!=*n*/ is the particle distribution in frequency space and ı! the frequency resolution of the spectrum analyzer. The signal from the pick up electrode is proportional to the line-charge density which is at the frequency ! from (21.1)

$$q\lambda\_{\rm rms}(\omega) = \frac{\sqrt{2}}{2\pi\overline{R}}\sqrt{N(\omega/n)\frac{\delta\omega}{n}}\tag{21.2}$$

and has been derived first by Schottky for a variety of current sources [1]. The spread in the revolution frequency originates from a momentum spread in the beam and measuring the Schottky spectrum allows its nondestructive determination.

Individual particles orbiting in an accelerator perform transverse betatron oscillations which we describe, for example, in the vertical plane by

$$\mathbf{y}\_k(t) = a\_k \cos(\nu\_k \alpha\_k t + \psi\_k) \,, \tag{21.3}$$

where *ak* is the amplitude and *k* the phase of the betatron oscillation for the particle *k* at time *t* D 0. The difference signal from two pick up electrodes above and below the particle beam is, in linear approximation, proportional to the product of the betatron amplitude (21.3) and the line-charge density (21.1) and of the form

$$D\_k(t) = A\_k \sum\_{n=0}^{\infty} \cos[(n - \nu\_k)(\omega\_k t + \phi\_k)]\tag{21.4}$$

$$+ A\_k \sum\_{n=0}^{\infty} \cos[(n + \nu\_k)(\omega\_k t + \varphi\_k)]\,,$$

where we have ignored terms at frequencies *n*!*k*. The transverse Schottky signal is composed of two side bands for each harmonic at frequencies

$$
\rho = \left(\boldsymbol{n} \pm \boldsymbol{\nu}\_k\right) \boldsymbol{\alpha}\_k \,. \tag{21.5}
$$

which are also called the fast wave for ! D .*n* C *k*/!*k* and the slow wave for ! D .*n k*/!*k*.

The longitudinal Schottky noise depends on the rms contribution of all particles which are spread over a range of revolution frequencies due to a momentum spread and over betatron frequencies by virtue of the chromaticity. For !rms D c!0ırms and rms D *y*ırms where !0 is the revolution frequency of the bunch center, ırms D *p*rms=*p*0 the rms relative momentum error, c the momentum compaction and *y* the vertical chromaticity, the frequency distribution of the signal from the pick up is

$$
\omega = \left[ n \pm \left( \upsilon\_{\jmath 0} + \xi\_{\jmath} \delta\_{k} \right) \right] \left( \alpha\_{0} + \eta\_{\circledast} \alpha\_{0} \delta\_{k} \right) \tag{21.6}
$$

$$
= \left( n \pm \upsilon\_{\jmath 0} \right) \alpha\_{0} + \left[ \left( n \pm \upsilon\_{\jmath 0} \right) \eta\_{\circledast} \pm \xi\_{\jmath} \right] \alpha\_{0} \delta\_{k} + O(\delta^{2}) \, .
$$

The momentum spread ı*k* causes a frequency spread which is different for the slow and fast wave. For example, for positive chromaticity above transition, c < 0 and the frequency spreads add up for the slow wave and cancel partially for the fast wave. This has been verified experimentally for a coasting proton beam in the ISR [2].

A transverse Schottky scan may exhibit the existence of weak resonances which may dilute the particle density, specifically in a coasting proton or ion beam. To control coasting beam instabilities, it is desirable to make use of Landau damping by introducing a large momentum and tune spread. This tune spread, however, can be sufficiently large to spread over higher order resonances and blow up that part of the beam which oscillates at those resonance frequencies. A Schottky scan can clearly identify such a situation as reported in [2].

In this text we are able to touch only the very basics of Schottky noise and the interested reader is referred to references [3–6] for more detailed discussions on the theory and experimental techniques to obtain Schottky scans and how to interpret the signals.

#### *21.1.2 Stochastic Cooling*

The "noise" signal from a circulating particle beam includes information which can be used to drive a feedback system in such a way as to reduce the beam emittance, longitudinal as well as transverse. Due to the finite number of particles in a realistic particle beam, the instantaneous center of a beam at the location of a pick up electrode exhibits statistical variations. This statistical displacement of a slice of beam converts to a statistical slope a quarter betatron wavelength downstream. The signal from the small statistical displacement of the beam at the pick up electrode can be amplified and fed back to the beam through a kicker magnet located an odd number of quarter wavelength downstream, assuming that the statistical variations do not smear out between pick up electrode and kicker. Van der Meer [7] proposed this approach to reduce the transverse proton beam emittance in ISR for increased luminosity and the process is now known as stochastic cooling.

This process of correction is not a statistical process and we must ask ourselves if this is an attempt to circumvent Liouville's theorem. It is not. Due to the finite number of particles in the beam, the phase space is not uniformly covered by particles but rather exhibits many holes. The method of stochastic cooling detects the moment one of these holes appears on one or the other side of the beam in phase space. At the same moment, the whole emittance is slightly shifted with respect to the center of the phase space and this shift can be both detected and corrected. The whole process of stochastic cooling therefore only squeezes the "air" out of the particle distribution in phase space. The most prominent application of this method occurs in the cooling of an antiproton beam to reach a manageable beam emittance for injection into high energy proton antiproton colliders. To discuss this process in more detail, theoretically as well as technically, would exceed the scope of this text and the interested reader is referred to a series of articles published in [8].

#### *21.1.3 Touschek Effect*

The concentration of many particles into small bunches increases the probability for elastic collisions between particles. This probability is further enhanced considering that particles perform transverse betatron as well as longitudinal synchrotron oscillations. In each degree of freedom, we have acceptance limits and if a particle's oscillation amplitude exceeds such limits due, for example, to a collision with another particle one or both particles can get lost. In this section, we discuss the process of single collisions where the momentum transfer is large enough to lead to the loss of both particles involved in the collision and postpone the discussion of multiple collisions with small momentum transfer to the next section.

We may consider two collision processes which could lead to beam loss. First, we observe two particles performing synchrotron oscillations and colliding headon in such a way that they transfer their longitudinal momentum into transverse momentum. This collision process is insignificant in particle accelerators because the longitudinal motion includes not enough momentum to increase the betatron oscillation amplitude enough for particle loss. On the other hand, transverse oscillations of particles represent large momenta and a transfer into longitudinal momenta can lead to the loss of both particles. This effect was discovered on the first electron storage ring ever constructed [9, 10] and we therefore call this the Touschek effect.

In this text, we will not pursue a detailed derivation of the collision process and refer the interested reader to references [11–13]. Of particular interest is the expression for the beam lifetime as a result of particle losses due to a momentum transfer into the longitudinal phase space exceeding the rf-bucket acceptance of *p*=*p*0jrf. Whenever such a transfer occurs both particles involved in the collision are lost. The beam decay rate is proportional to the number of particles in the bunch and the beam current therefore decays exponentially. Last, but not least, a loss occurs only if there is sufficient momentum in the transverse motion to exceed the rf-momentum acceptance. We assume the momentum acceptance to be limited by the rf-voltage and combining these parameters in a collision theory results in a beam lifetime for a Gaussian particle distribution given by

$$\frac{1}{\tau} = -\frac{1}{N\_{\rm b}} \frac{\mathrm{d}N\_{\rm b}}{\mathrm{d}t} = \frac{r\_{\rm c}^2 c \, N\_{\rm b}}{8\pi \, \sigma\_{\rm x} \sigma\_{\rm y} \sigma\_{\ell}} \frac{\lambda^3}{\chi^2} D(\epsilon), \tag{21.7}$$

where *r*c is the classical particle radius, *x*; *y*; ` are the standard values of the Gaussian bunch width, height and length, respectively, and 1 D *p*=*p*0jrf the momentum acceptance parameter. The function *D*./ (Fig. 21.1) is defined by [13]

$$D(\epsilon) = \sqrt{\epsilon} \left[ -\frac{3}{2} \mathbf{c}^{-\epsilon} + \frac{\epsilon}{2} \int\_{\epsilon}^{\infty} \frac{\ln u}{u} \mathbf{c}^{-u} \, \mathrm{d}u \right. \tag{21.8}$$

$$+ \frac{1}{2} (3\epsilon - \epsilon \ln \epsilon + 2) \int\_{\epsilon}^{\infty} \frac{\mathbf{c}^{-u}}{u} \, \mathrm{d}u \right],$$

**Fig. 21.1** Touschek lifetime function *D* ./

where the argument is

$$
\epsilon = \left(\frac{\Delta p\_{\rm rf}}{\gamma \,\sigma\_p}\right)^2 \qquad \text{with} \qquad \sigma\_\text{p} = \frac{mc\gamma \,\sigma\_x}{\beta\_x}. \tag{21.9}
$$

Particle losses due to the Touschek effect is particularly effective at low energies and where the rf-acceptance is small. For high particle densities *N*b=.*x y* `/ the rf-acceptance should therefore be maximized. This seems to be the wrong thing to do because the bunch length is reduced at the same time and the particle density becomes even higher but a closer look at (21.7) shows us that the Touschek lifetime increases faster with rf-acceptance than it decreases with bunch length.

#### *21.1.4 Intra-Beam Scattering*

The Touschek effect describes collision processes which lead to immediate loss of both colliding particles. In reality, however, there are many other collisions with only small exchanges of momentum. While these collisions do not lead to immediate particle loss, there might be sufficiently many during a damping time in electron storage rings or during the storage time for proton and ion beams to cause a significant increase in the bunch volume, or in the case of a coasting beam an increase in beam cross section. During the discussion of the Touschek effect we neglected the transfer from the longitudinal momentum space into transverse momentum space because the transverse momentum acceptance is larger than the longitudinal acceptance and particles are generally not lost during such an exchange. This is not appropriate any more for the multiple Touschek effect or intra-beam scattering where we are interested in all collisions.

The multiple Touschek effect was observed in the first ever constructed storage ring, AdA (Anello di Accumulatione) in Frascati, Italy. The Touschek effect had been expected and analyzed before but did give too pessimistic beam lifetimes compared to those observed in AdA. A longer beam lifetime had been obtained because of multiple elastic scattering between particles increasing the bunch volume and thereby reducing the Touschek effect [10].

During the exchange of momentum as a consequence of collisions between particles within the same bunch or beam, each degree of freedom can increase its energy or temperature because the beam is able to absorb any amount of energy from the rf- system. We are particularly interested in the growth times of transverse and longitudinal emittances to asses the long-term integrity of the particle beam. The multiple Touschek effect or intra-beam scattering has been studied extensively [14, 15] and we will not repeat here the derivations but merely recount the results.

The growth time of the beam emittances for Gaussian particle distributions are for the longitudinal phase space or momentum and bunch distribution [14, 15]

$$\sigma\_p^{-1} = \frac{1}{2\sigma\_p^2} \frac{\mathrm{d}\sigma\_p^2}{\mathrm{d}t} = A \frac{\sigma\_h^2}{\sigma\_p^2} f(a, b, c) \,, \tag{21.10}$$

where the particle bunch density is expressed by

$$A = \frac{r\_\text{\textdegree}cN\_\text{b}}{64\pi^2\sigma\_z\sigma\_p\sigma\_x\sigma\_y\sigma\_{x'}\sigma\_{y'}\beta^3\,\gamma^4} \tag{21.11}$$

with the standard dimensions of a Gaussian distribution for the bunch length *z*, the relative momentum spread *p*, horizontal and vertical betatron amplitudes *x*; *y* and divergences *x*0 ; *y*0 and number of particles per bunch *N*b. The constants *r*c and ˇ D v=*c*, finally, are the classical particle radius and velocity in units of the velocity of light.

The function

$$f(a,b,c) = 8\pi \int\_0^1 \left\{ \ln \left[ \frac{c^2}{2} \left( \frac{1}{\sqrt{p}} + \frac{1}{\sqrt{q}} \right) \right] - 0.577... \right\} \frac{1 - 3x^2}{\sqrt{pq}} \,\mathrm{d}x \tag{21.12}$$

where

$$\begin{array}{ll} p = a^2 + \varkappa^2(1 - a^2) \,, & q = b^2 + \varkappa^2(1 - b^2) \,, \\ a = \frac{\sigma\_h}{\wp \sigma\_{\varkappa'}} \,, & b = \frac{\sigma\_h}{\wp \sigma\_{\jmath'}} \,, \\ \sigma\_h^2 = \frac{\sigma\_\wp^2 \sigma\_\wp^2}{\sigma\_\wp^2 + \eta^2 \sigma\_\wp^2} \,, & c^2 = \beta^2 \sigma\_h^2 \frac{\sqrt{2\pi} \sigma\_\wp}{r\_\c} \,. \end{array}$$

The transverse emittance growth times are similarly given by

$$\pi\_{\boldsymbol{x}}^{-1} = \frac{1}{2\sigma\_{\boldsymbol{x}}^{2}} \frac{\mathrm{d}\sigma\_{\boldsymbol{x}}^{2}}{\mathrm{d}t} = A \left[ f\left(\frac{1}{a}, \frac{b}{a}, \frac{c}{a}\right) + \frac{\eta^{2}\sigma\_{p}^{2}}{\sigma\_{\boldsymbol{x}}} f\left(a, b, c\right) \right],\tag{21.13}$$

and

$$
\pi\_\text{y}^{-1} = \frac{1}{2\sigma\_\text{y}^2} \frac{\text{d}\sigma\_\text{y}^2}{\text{d}t} = A f\left(\frac{1}{b}, \frac{a}{b}, \frac{c}{b}\right) \,. \tag{21.14}
$$

These expressions allow the calculations of the emittance growth rate, which for most electron accelerators is small compared to radiation damping but become significant in proton and ion storage rings where high particle densities and long storage times are desired. Progress in the design of modern synchrotron radiation facilities allow ever smaller emittances which have reached a level where intrabeam scattering is significant again. From the density factor *A* it is apparent that high particle density in six-dimensional phase space increases the growth rates while this effect is greatly reduced at higher beam energies.

#### **21.2 Collective Self Fields**

The electric charges of a particle beam can become a major contribution to the forces encountered by individual particles while travelling along a beam transport line or orbiting in a circular accelerator. These forces may act directly from beam to particle or may originate from electromagnetic fields being excited by the beam interaction with its surrounding vacuum chamber. In this section, we will derive expressions for the fields from a collection of particles and determine the force due to these fields on an individual test particle. We use the particle charge *q* rather than the elementary charge *e* to cover particles with multiple charges like ions for which *q* D *eZ*. For all cases to be correct, we should distinguish between the electrical charge of particles in the beam and that of the individual test particle. This, however, would significantly complicate the expressions and we use therefore the same charge for both the beam and test particle. In a particular situation whenever particles of different charges are considered, the sign and value of the charge factors in the formulas must be reconsidered.

Individual particles in an intense beam are under the influence of strong repelling electrostatic forces creating the possibility of severe stability problems. Particle beam transport over long distances could be greatly restricted unless these spacecharge forces can be kept under control. First, it is interesting to calculate the magnitude of the problem.

If all particles would be at rest within a small volume, we would clearly expect the particles to quickly diverge from the center of charge under the influence of the repelling forces from the other particles. This situation may be significantly different in a particle beam where all particles propagate in the same direction.

In Sect. 1.5.10 we obtained the encouraging result that at least relativistic particle beams become stable under the influence of their own fields. For lower particle energies, however, significant diverging forces must be expected and adequate focusing measures must be applied. The physics of such space charge dominated beams is beyond the scope of this book and is treated elsewhere, for example in considerable detail in [16].

#### *21.2.1 Self Field for Elliptical Particle Beams*

The self fields of a beam depend on beam parameters like particle type, particle distribution, bunching, and energy of the particle. Here, we will derive the nature and effect of these self fields in a more restricted way for common particle beam cases in accelerators.

To determine self fields, we consider a continuous beam of particles with a line charge , or a volume charge .*x*; *y*/. The electric fields within a beam are derived from a potential *V* defined by

$$
\Delta V = -\frac{1}{\epsilon\_0} \rho(\mathbf{x}, \mathbf{y})\,,\tag{21.15}
$$

where , being the electric charge density in the beam, is finite within and zero outside the beam. Similarly, the magnetic vector potential is defined by

$$
\Delta A = -\frac{1}{\epsilon\_0} \nu \rho(\mathbf{x}, \mathbf{y}) \,. \tag{21.16}
$$

For a particle beam, we may set v .0; 0; v/ and the vector potential therefore contains only a longitudinal component *A* D .0; 0; *Az*/.

In Sect. 1.5.10 we discussed the self fields of a round beam. Generally, however, particle beams have an elliptical cross section and the solution to (21.15) for such a beam with constant charge density . D const/ has been derived by Teng [17, 18]. Within the elliptical beam cross section, where *x a* and *y b*, the electric potential is

$$V(x,y) = -\frac{1}{2\epsilon\_0} \rho \frac{ab}{a+b} \left[ \frac{x^2}{a} + \frac{y^2}{b} \right] \tag{21.17}$$

and *a*; *b* are the horizontal and vertical half axis respectively. The vector potential for the magnetic field is from the discussions above

$$A\_{\varepsilon}(\mathbf{x}, \mathbf{y}) = -\frac{1}{2\epsilon\_0} \rho \frac{v}{c} \frac{ab}{a+b} \left[ \frac{\mathbf{x}^2}{a} + \frac{\mathbf{y}^2}{b} \right] \tag{21.18}$$

and both the electric and magnetic field can be derived by simple differentiations

$$\mathbf{E} = -\nabla V \qquad \text{and} \qquad \mathbf{B} = \nabla \times \mathbf{A} \tag{21.19}$$

for

$$E\_x = \frac{1}{4\pi\epsilon\_0} \frac{4q\lambda}{a(a+b)}\mathbf{x}, \qquad \qquad E\_\mathbf{y} = \frac{1}{4\pi\epsilon\_0} \frac{4q\lambda}{b(a+b)}\mathbf{y}, \tag{21.20}$$

and

$$B\_x = -\frac{c\mu\_0}{4\pi} \frac{4q\lambda\beta}{b(a+b)}\mathbf{y} \,, \qquad B\_\mathbf{y} = \frac{c\mu\_0}{4\pi} \frac{4q\lambda\beta}{a(a+b)}\mathbf{x} \,, \tag{21.21}$$

where ˇ D v=*c* and the linear charge density is defined by

$$
\lambda = \pi ab \,\rho(\mathbf{x}, \mathbf{y})\,. \tag{21.22}
$$

Comparing (21.20) and (21.21) reveals the relationship between electric and magnetic self fields of the beam to be (21.22)

$$cB\_{\mathbf{x}} = -\beta E\_{\mathbf{y}}\,, \qquad \qquad cB\_{\mathbf{y}} = +\beta E\_{\mathbf{x}}\,. \tag{21.23}$$

The electric as well as the magnetic field scales linearly with distance from the beam center and therefore both cause focusing and a tune shift in a circular accelerator.

In many applications it is not acceptable to assume a uniform transverse charge distribution. Most particle beams either have a bell shaped particle distribution or a Gaussian distribution as is specially the case for electrons in circular accelerators. We therefore use in the transverse plane a Gaussian charge distribution given by

$$\rho(\mathbf{x}, \mathbf{y}) = \frac{\lambda}{2\pi\sigma\_{\mathbf{x}}\sigma\_{\mathbf{y}}} \exp\left[-\frac{\mathbf{x}^2}{2\sigma\_{\mathbf{x}}^2} - \frac{\mathbf{y}^2}{2\sigma\_{\mathbf{y}}^2}\right],\tag{21.24}$$

which also well describes a beam with bell shaped distribution. Although many particle beams, but specifically electron beams, come in bunches with a Gaussian distribution in all degrees of freedom, we will only introduce a bunching factor for the longitudinal particle distribution and refer the interested reader for the study of a fully six dimensional Gaussian charge distribution to reference [19].

The potential for a transverse bi-Gaussian charge distribution (21.24) can be expressed by [18]

$$V(\mathbf{x}, \mathbf{y}) = -\frac{e}{4\pi\epsilon\_0} \lambda \int\_0^\infty \frac{1 - \exp\left[-\frac{\mathbf{x}^2}{2(\sigma\_x^2 + t)} - \frac{\mathbf{y}^2}{2(\sigma\_y^2 + t)}\right]}{\sqrt{(\sigma\_x^2 + t)(\sigma\_y^2 + t)}} \mathrm{d}t \tag{21.25}$$

Equation (21.25) can be verified by back insertion into (21.15). From this potential we obtain for example the vertical electric field component by differentiation

$$E\_{\mathbf{y}} = -\frac{\partial V(\mathbf{x}, \mathbf{y})}{\partial \mathbf{y}} = \frac{e}{4\pi\epsilon\_0} \lambda \operatorname{y} \int\_0^\infty \frac{\exp\left[-\frac{\mathbf{x}^2}{2(\sigma\_x^2 + t)} - \frac{\mathbf{y}^2}{2(\sigma\_y^2 + t)}\right]}{(\sigma\_y^2 + t)\sqrt{(\sigma\_x^2 + t)(\sigma\_y^2 + t)}} \operatorname{dt.} \tag{21.26}$$

No closed analytical expression exists for these integrals unless we restrict ourselves to a symmetry plane with *x* D 0 or *y* D 0 and small amplitudes *y y* or *x x* respectively. These assumptions are appropriate for most space-charge effects and the potential in the vertical midplane becomes

$$V(\mathbf{x} = 0, \mathbf{y} \ll \sigma\_{\mathbf{y}}) = -\frac{1}{4\pi\epsilon\_0} \frac{\lambda}{\sigma\_{\mathbf{y}}(\sigma\_{\mathbf{x}} + \sigma\_{\mathbf{y}})} \mathbf{y}^2. \tag{21.27}$$

For reasons of symmetry a similar expression can be derived for the horizontal mid plane by merely interchanging *x* and *y* in (21.27). The associated electric fields are for *x* D 0 and *y y*

$$E\_x = \frac{1}{4\pi\epsilon\_0} \frac{2\lambda}{\sigma\_\text{x}(\sigma\_\text{x} + \sigma\_\text{y})} \text{x}, \qquad \qquad E\_\text{y} = \frac{1}{4\pi\epsilon\_0} \frac{2\lambda}{\sigma\_\text{y}(\sigma\_\text{x} + \sigma\_\text{y})} \text{y}, \tag{21.28}$$

and the magnetic fields according to (21.23) are from (21.28)

$$B\_{\rm x} = -\frac{c\mu\_0}{4\pi} \frac{2\lambda\beta}{\sigma\_\circ(\sigma\_\times + \sigma\_\circ)} \text{y}, \qquad \qquad B\_{\rm y} = +\frac{c\mu\_0}{4\pi} \frac{2\lambda\beta}{\sigma\_\circ(\sigma\_\times + \sigma\_\circ)} \text{x}.\tag{21.29}$$

All fields increase linearly with amplitude and we note that the field components in the horizontal midplane are generally much smaller compared to those in the vertical midplane because most particle beams in circular accelerators are flat and *y x*.

#### **Forces from Space-Charge Fields**

The electromagnetic self fields generated by the collection of all particles within a beam exert forces on individual particles of the same beam or of another beam. The Lorentz force due to these fields can be expressed by

$$F = eEf\_{\mathfrak{e}} + e\left[\mathfrak{v} \times \mathbf{B}\right]f\_{\mathfrak{e}}f\_{\mathfrak{v}}\,,\tag{21.30}$$

where we have added to the usual expression for the Lorentz force the factors *f*e and *f*v. Because the fields act differently depending on the relative directions and charge of beam and individual particle distinct combinations occur. We set *f*e D 1 if both the beam particles and the test particle have the same sign of their charge and

*f*e D 1 if their charges are of opposite sign. Similarly we set *f*v D 1 or *f*v D 1 depending on whether the beam and test particle have the same or opposite direction of movement with respect to each other.

The vertical force from the self field, for example, of a proton beam on an individual proton within the same beam moving with the same velocity is from (21.30)

$$F\_{\mathbf{y}}(\gamma\_{\uparrow\uparrow,++}) = +e(1-\beta^2)E\_{\mathbf{y}}\,. \tag{21.31}$$

An antiproton moving in the opposite direction through a proton beam would feel the vertical force

$$F\_{\mathbf{y}}(\boldsymbol{\gamma}\_{\cdot \downarrow , + \cdot}) = -e(1 + \boldsymbol{\beta}^{\cdot 2})E\_{\mathbf{y}}\,. \tag{21.32}$$

Expansion to other combinations of particles and directions of velocities are straightforward. For ions the charge multiplicity *Z* must be added to the fields or the individual particle or both depending on the case. The possible combinations of the force factors ˙.1 ˙ ˇ2/ are summarized in Table 21.1.

The ˙-signs in Table 21.1 indicate the charge polarity of beam and test particle and the arrows the relative direction. We note a great difference between the case, where particles move in the same direction, and the case of beams colliding head on.

#### *21.2.2 Beam–Beam Effect*

In colliding beam facilities two counter rotating beams within one storage ring or counter rotating beams from two intersecting storage rings are brought into collision to create a high center of mass energy at the collision point which transforms into known or unknown particles to be studied by high energy experimentalists. The event rate is given by the product of the cross section for the particular event and the luminosity which is determined by storage ring operating conditions. By definition, the luminosity is the density of collision centers in the target multiplied by the number of particles colliding with this target per unit time. In the case of a colliding beam facility a bunch of one beam is the target for the other beam. For simplicity we assume here that both beams have the same cross section. We also assume that each beam consists of *n*b bunches. In this case the luminosity is

$$\mathcal{L} = \frac{N\_1}{n\_\text{b}A} N\_2 \upsilon\_{\text{rev}}\,,\tag{21.33}$$

where *N*1 and *N*2 are the total number of particles in each beam, *A* the cross section of the beams, and rev the revolution frequency in the storage ring. In most storage rings the transverse particle distribution is Gaussian or bell shaped and since only the core of the beam contributes significantly to the luminosity we may define standard beam sizes for all kinds of particles. For a Gaussian particle distribution the effective beam cross section is

$$A\_{\mathfrak{g}} = 4\pi\sigma\_{\mathfrak{x}}\sigma\_{\mathfrak{y}}\tag{21.34}$$

and the luminosity

$$\mathcal{L} = \frac{N\_1}{4\pi\sigma\_\text{x}\sigma\_\text{y}B} N\_2 \upsilon\_{\text{rev}} \,. \tag{21.35}$$

The recipe for high luminosity is clearly to maximize the beam intensity and to minimize the beam cross section. This approach, however, fails because of the beam-beam effect which, due to electromagnetic fields created by the beams themselves, causes a tune shift and therefore limits the amount of beam that can be brought into collision in a storage ring. The beam-beam effect has first been recognized and analyzed by Amman and Ritson [20].

In case of counter rotating beams colliding at particular interaction points in a colliding-beam facility, we always have *f*v D 1 but the colliding particles still may be of equal or opposite charge. In addition, there is no contribution from magnetic image fields since collisions do not occur within magnets. Even image fields from vacuum chambers are neglected because the beam-beam interaction happens only over a very short distance. A particle in one beam will feel the field from the other beam only during the time it travels through the other beam which is equal to the time it takes the particle to travel half the effective length of the oncoming bunch. With these considerations in mind, we obtain for the beam-beam tune shift in the vertical plane from (21.63) with *f*corr D 1 and assuming head on collisions of particle-antiparticle beams (*f*e D 1)

$$
\Delta\nu\_{\rm y,bb} = \frac{r\_{\rm c}}{2\pi B\gamma} \frac{\beta\_{\rm y}^{\star}}{\sigma\_{\rm y}^{\star}(\sigma\_{\rm x}^{\star} + \sigma\_{\rm y}^{\star})} \tag{21.36}
$$

and in the horizontal plane

$$
\Delta\nu\_{\rm x,bb} = \frac{r\_{\rm c}N\_{\rm tot}}{2\pi B\gamma} \frac{\beta\_{\rm x}^{\*}}{\sigma\_{\rm x}^{\*}(\sigma\_{\rm x}^{\*} + \sigma\_{\rm y}^{\*})} \,, \tag{21.37}
$$

where indicates that the quantities be taken at the interaction point. In cases where other particle combinations are brought into collision or when both beams cross under an angle these equations must be appropriately modified to accurately describe the actual situation.

From (21.32) and (21.28)we find for two counter rotating beams of particle and antiparticle a vertical beam-beam force of

$$F\_{\rm y} = -\frac{1}{4\pi\epsilon\_0} \frac{e(1+\beta^2)2\lambda}{\sigma\_{\rm y}(\sigma\_{\rm x}+\sigma\_{\rm y})} \rm y. \tag{21.38}$$

This force is attractive and therefore focusing, equivalent to that of a quadrupole of strength

$$k = -\frac{F\_\text{y}/\text{y}}{c^2 \beta^2 \gamma m} \tag{21.39}$$

causing a vertical tune shift of

$$
\delta\nu\_\circ = \frac{1}{4\pi} \int\_{\text{coll}} \beta\_\circ k \,\text{d}z \,. \tag{21.40}
$$

Integrating over the collision length which is equal to half the bunch length ` because colliding beams move in opposite directions, we note that the linear charge density is D *eN*= .*B*`/, where *N* is the total number of particles per beam and *B* the number of bunches per beam. With these replacements the beam tune shift becomes finally

$$\delta\nu\_{\rm y} = \frac{r\_{\rm c}N\beta\_{\rm y}}{2\pi B\chi\sigma\_{\rm y}(\sigma\_{\rm x} + \sigma\_{\rm y})} \,, \tag{21.41}$$

where *r*c is the classical particle radius of the particle which is being disturbed. Obviously, the tune shift scales linear with particle intensity or particle beam current and inversely with the beam cross section. Upon discovery of this effect it was thought that the particle beam intensity is limited when the tune shift is of the order of 0:15 – 0:2 which is the typical distance to the next resonance. Experimentally, however, it was found that the limit is much more restrictive with maximum tune shift values of 0:04 – 0:06 for electrons [20–23] and less for proton beams [24].

A definitive quantitative description of the actual beam-beam effect has not been possible yet due to its highly nonlinear nature. Only particles with very small betatron oscillation amplitudes will experience the linear tune shift derived above. For betatron oscillations larger than one , however, the field becomes very nonlinear turning over to the well known 1=*r*-law at large distances from the beam center.

In spite of the inability to quantitatively describe the beam-beam effect by the linear tune shift it is generally accepted practice to quantify the beam-beam limit by the value of the linear tune shift. This is justified since the nonlinear fields of a particle beam are strictly proportional to the linear field and therefore the linear tune shift is a good measure for the amount of nonlinear fields involved.

#### *21.2.3 Transverse Self Fields*

Expressions for space-charge fields originating from a beam of charged particles have been derived earlier and we obtained for a Gaussian transverse distribution of particles with charge *q* the electric fields in (21.28) and the magnetic fields in (21.29).

The local linear particle density is defined by

$$
\lambda(\mathbf{z}) = \int \int \rho(\mathbf{x}, \mathbf{y}, \mathbf{z}) d\mathbf{x} \,\mathrm{d}\mathbf{y}\,\mathrm{d}\mathbf{y}\,,\tag{21.42}
$$

where .*x*; *y*;*z*/ is the local particle density normalized to the total number of particles in the beam R 1 1 .*z*/d*z* D *N*p. With these fields and the Lorentz equation, we formulate the transverse force acting on a single particle within the same particle beam. Since both expressions for the electrical and magnetic field differ only by the factor ˇ we may, for example, derive from the Lorentz equation the vertical force on a particle with charge *q*

$$F\_{\rm y} = q(1 - \beta^2)E\_{\rm y} = \frac{1}{4\pi\epsilon\_0} \frac{2q\lambda}{\mathcal{V}^2 \sigma\_{\rm y} (\sigma\_{\rm x} + \sigma\_{\rm y})} \text{y}.\tag{21.43}$$

The space-charge force appears at its strongest for nonrelativistic particles and diminishes quickly like 1=2 for relativistic particles. In accelerator physics, however, particle beams are carried from low to high energies and therefore spacecharge effects may become important during some or all phases of acceleration. This is specifically true for heavy particles like protons and ions for which the relativistic parameter  is rather low for most any practically achievable particle energies.

#### *21.2.4 Fields from Image Charges*

Discussing space charges, we ignored so far the effect of metallic and magnetic surfaces close to the beam. The electromagnetic self fields of the beam circulating in a metallic vacuum chamber and between ferromagnetic poles of magnets must meet certain boundary conditions on such surfaces. Laslett [25] derived appropriate corrections to free space electromagnetic fields by adding the electromagnetic fields from all image charges to the fields of the particle beam itself.

Following his reasoning, we consider a particle beam with metallic and ferromagnetic boundaries as shown in Fig. 21.2. For full generality, let the elliptical particle beam be displaced in the vertical plane by *y*N from the midplane, the metallic vacuum chamber and magnet pole are simulated as pairs of infinitely wide parallel surfaces at ˙*b* and ˙*g*, respectively, and the observation point of the fields be at *y*. The linear particle density is

$$
\lambda = \frac{N\_{\text{tot}}}{n\_{\text{b}} \ell\_{\text{b}}} = \frac{N\_{\text{tot}}}{n\_{\text{b}} \sqrt{2\pi} \sigma\_{\ell}},\tag{21.44}
$$

where *N*tot is the total number of particle in the circulating beam, *n*b the number of bunches, `<sup>b</sup> <sup>D</sup> <sup>p</sup>2-` the effective bunch length and ` the standard bunch length for a Gaussian distribution.

The locations and strength of the electrical images of a line current in the configuration of Fig. 21.2 are shown in Fig. 21.3. The boundary condition for electric fields is *Ez*.*b*/ D 0 on the surface of the metallic vacuum chamber and is satisfied if the image charges change sign from image to image. To calculate the

**Fig. 21.3** Location and source of image fields

electrical field *Ey*.*y*/, with (21.28) we add the contributions from all image fields in the infinite series

$$\begin{split} E\_{\text{y,image}}(\mathbf{y}) &= \frac{1}{4\pi\epsilon\_0} 2\lambda \\ &\times \left( \frac{1}{2b - \bar{\mathbf{y}} - \mathbf{y}} - \frac{1}{2b + \bar{\mathbf{y}} + \mathbf{y}} - \frac{1}{4b + \bar{\mathbf{y}} - \mathbf{y}} + \frac{1}{4b - \bar{\mathbf{y}} + \mathbf{y}} \right. \\ &+ \frac{1}{6b - \bar{\mathbf{y}} - \mathbf{y}} - \frac{1}{6b + \bar{\mathbf{y}} + \mathbf{y}} - \frac{1}{8b + \bar{\mathbf{y}} - \mathbf{y}} + \frac{1}{8b - \bar{\mathbf{y}} + \mathbf{y}} \\ &+ \frac{1}{10b - \bar{\mathbf{y}} - \mathbf{y}} - \frac{1}{10b + \bar{\mathbf{y}} + \mathbf{y}} - \dots \right). \end{split}$$

These image fields must be added to the direct field of the line charge to meet the boundary condition that the electric field enter metallic surfaces perpendicular. Equation (21.45) can be split into two series with factors .*y*N C *y*/ and .*y*N *y*/ in the numerator. We get after some manipulations with *y*N C *y b* and *y*N *y b*

$$E\_{\rm y,image}(\mathbf{y}) = \frac{1}{4\pi\epsilon\_0} \frac{\lambda}{b^2} \left[ \sum\_{m=1}^{\infty} \frac{\bar{\mathbf{y}} + \mathbf{y}}{(2m-1)^2} + \sum\_{m=1}^{\infty} \frac{\bar{\mathbf{y}} - \mathbf{y}}{4m^2} \right],\tag{21.46}$$

$$= \frac{1}{4\pi\epsilon\_0} \frac{\lambda}{b^2} \left[ (\bar{\mathbf{y}} + \mathbf{y}) \frac{\pi^2}{8} + (\bar{\mathbf{y}} - \mathbf{y}) \frac{\pi^2}{24} \right],$$

$$= \frac{1}{4\pi\epsilon\_0} \frac{\lambda}{b^2} \frac{\pi^2}{12} (2\bar{\mathbf{y}} + \mathbf{y}) = \frac{1}{4\pi\epsilon\_0} \frac{4q\lambda}{b^2} \epsilon\_1 (2\bar{\mathbf{y}} + \mathbf{y}).$$

The electric image fields depend linearly on the deviations *y*N and *y* from the axis of bunch center and test particle, respectively, and act therefore like a quadrupole causing a tune shift.

A similar derivation is used to get the magnetic image fields due to ferromagnetic surfaces at ˙*g* above and below the midplane. The magnetic field lines must enter the magnetic pole faces perpendicular and the image currents therefore flow in the same direction as the line current causing a magnetic force on the test particle which is opposed to that by the magnetic field of the beam itself.

Bunched beams generate high frequency electromagnetic fields which do not reach ferromagnetic surfaces because of eddy current shielding by the metallic vacuum chamber. For magnetic image fields we distinguish therefore between dc and ac image fields. The dc Fourier component of a bunched beam current is equal to twice the average beam current *c*ˇ*B*, where the Laslett bunching factor *B* is the bunch occupation along the ring circumference defined by

$$B = \frac{\overline{\lambda}}{\lambda} = \frac{n\_{\text{b}} \ell\_{\text{b}}}{2\pi R}. \tag{21.47}$$

The dc magnetic image fields are derived similar to electric image fields with *B*' D 2ˇ=*r* from (21.29) and are with (21.47)

$$\begin{split} B\_{\text{x,image,dc}}(\mathbf{y}) &= \frac{c\mu\_0}{4\pi} \frac{2\lambda\beta}{g^2} B \left[ \sum\_{m=1}^{\infty} \frac{\bar{\mathbf{y}} + \mathbf{y}}{(2m-1)^2} + \sum\_{m=1}^{\infty} \frac{\bar{\mathbf{y}} - \mathbf{y}}{4m^2} \right] \\ &= \frac{c\mu\_0}{4\pi} \frac{\lambda\beta}{g^2} B \left[ (\bar{\mathbf{y}} + \mathbf{y}) \frac{\pi^2}{8} + (\bar{\mathbf{y}} - \mathbf{y}) \frac{\pi^2}{24} \right] \\ &= \frac{c\mu\_0}{4\pi} \frac{4\lambda\beta}{g^2} B \epsilon\_2 (2\bar{\mathbf{y}} + \mathbf{y}). \end{split} \tag{21.48}$$

The magnetic image fields must penetrate the metallic vacuum chamber to reach ferromagnetic poles. This is no problem for dc or low frequency field components but in case of bunched beams relevant frequencies are rather high and eddy current shielding of the vacuum chamber for ac magnetic fields must be taken into account. In most cases we may assume that they do not penetrate the thick metallic vacuum chamber. Consequently, we ignore here the effect of ferromagnetic poles and consider only the contribution of magnetic ac image fields due to eddy currents in vacuum chamber walls. Similar to electric image fields, the magnetic image fields are in analogy to (21.46)

$$B\_{\rm x,image,ac}(\mathbf{y}) = -\frac{c\mu\_0}{4\pi} \frac{\lambda\beta}{b^2} (1 - B) \frac{\pi^2}{12} (2\bar{\mathbf{y}} + \mathbf{y}),$$

$$= -\frac{c\mu\_0}{4\pi} \frac{4\lambda\beta}{b^2} (1 - B) \epsilon\_1 (2\bar{\mathbf{y}} + \mathbf{y}),\tag{21.49}$$

where the factor .1 *B*/ accounts for the subtraction of the dc component ˇ*B*. Similar to the electric image fields, the magnetic image fields must be added to the direct magnet fields (21.29) from the beam current to meet the boundary condition of normal field components at ferromagnetic surfaces. The coefficients 1 and 2 are the Laslett form factors which are for infinite parallel plate vacuum chambers and magnetic poles

$$
\epsilon\_1 = \frac{\pi^2}{48} \qquad \text{and} \qquad \epsilon\_2 = \frac{\pi^2}{24}. \tag{21.50}
$$

The vacuum chamber and ferromagnetic poles are similar to infinitely wide surfaces. While this is a sufficiently accurate approximation for the magnet poles, corrections must be applied for circular or elliptical vacuum chambers. Laslett [25] has derived what we call now Laslett form factors for vacuum chambers with elliptical cross sections and variable aspect ratios which are compiled in Table 21.2.

All relevant field components have been identified and we collect these fields first for *y*N D 0 and obtain from (21.28), (21.46) for the electric field in the vertical mid plane

$$E\_{\rm y}(\mathbf{y}) = \frac{c^2 \mu\_0}{4\pi} \frac{2\lambda}{\sigma\_{\rm y}(\sigma\_{\rm x} + \sigma\_{\rm y})} \left[ 1 + \frac{2\sigma\_{\rm y}(\sigma\_{\rm x} + \sigma\_{\rm y})}{b^2} \epsilon\_1 \right] \mathbf{y}. \tag{21.51}$$

From (21.29), (21.48) the dc magnetic field is

$$B\_{\rm x,dc} = -\frac{c\mu\_0}{4\pi} \frac{2\lambda\beta B}{\sigma\_\circ(\sigma\_\rm x + \sigma\_\circ)} \left[1 - \frac{2\sigma\_\circ(\sigma\_\rm x + \sigma\_\circ)}{g^2} \,\epsilon\_2\right] \mathbf{y} \tag{21.52}$$

**Table 21.2** Laslett incoherent tune shift form factors for elliptical vacuum chambers

a*a* is the horizontal and *b* the vertical half-axis of an elliptical vacuum chamber

and from (21.49) the ac magnetic field

$$B\_{\rm x,ac} = -\frac{c\mu\_0}{4\pi} \frac{2\lambda}{\sigma\_\mathrm{y}(\sigma\_\mathrm{x} + \sigma\_\mathrm{y})} \left[1 + \frac{2\sigma\_\mathrm{y}(\sigma\_\mathrm{x} + \sigma\_\mathrm{y})}{b^2} \epsilon\_1\right] (1 - B) \,\mathrm{y}.\tag{21.53}$$

Tacitly, we have assumed that the transverse particle distribution is Gaussian which is a true representation of an electron beam but may not be correct for proton or ion beams. The standard deviations of a Gaussian distribution are very well defined and can therefore be replaced by other quantities like the full-width half maximum or as the particle distribution may require.

The electromagnetic force due to space charge on individual particles in a beam has been derived and it became obvious that image field effects can play a significant role in the perturbation of the beam. The fields scale linear with amplitude for very small amplitudes and act therefore like focusing quadrupoles. At larger amplitudes, however, the fields reach a maximum and then evanesce like 1=*r*. Consequently, the field gradient is negative decaying quickly with amplitude.

A complete set of direct and image fields have been derived which must be considered to account for space-charge effects. Similar derivations lead to other field components necessary to determine horizontal space-charge forces. In most accelerators, however, the beam cross section is flat and so is the vacuum chamber and the magnet pole aperture. As a consequence, we expect the space-charge forces to be larger in the vertical plane than in the horizontal plane.

#### *21.2.5 Space-Charge Effects*

The Lorentz force on individual particles can be calculated from the space-charge fields and we get

$$F\_{\rm y} = \frac{1}{4\pi\epsilon\_0} \frac{2f\_{\rm p}\lambda(1-\beta^2f\_{\rm v})}{\sigma\_{\rm y}(\sigma\_{\rm x}+\sigma\_{\rm y})} f\_{\rm corr} \,\mathrm{y} = q\mathcal{F}\mathrm{y},\tag{21.54}$$

where the correction factor due to image fields is with ˇ22 D 2 1;

$$f\_{\rm corr} = 1 + \frac{2\sigma\_{\rm y}(\sigma\_{\rm x} + \sigma\_{\rm y})}{b^2} \epsilon\_1 [1 + (\chi^2 - 1)B] + \epsilon\_2 (\chi^2 - 1) \frac{b^2}{g^2} B \tag{21.55}$$

and

$$\mathcal{F} = \frac{1}{4\pi\epsilon\_0} \frac{2f\_\text{p}\lambda\left(1 - \beta^2 f\_\text{\textdegree}\right)}{\sigma\_\text{\textdegree}\left(\sigma\_\text{x} + \sigma\_\text{y}\right)} f\_\text{corr} \,, \tag{21.56}$$

The factors *f*p and *f*v determine signs depending on the kind of particles interacting and the direction of travel with respect to each other. Specifically, *f*p D sign.*q q*b/ where *q* is the charge of a test particle and *q*b the charge of the field creating particles, e.g. the charge of a bunch. Similarly, *f*v D sign.vvb/ where v is the direction of travel for the test particle and **v**b the direction of travel of the bunch. To calculate the space-charge force of head-on colliding proton and antiproton beams, for example, we would set *f*p D 1 and *f*v D 1.

There is a significant cancellation of two strong terms, the repulsive electrical field and the focusing magnetic field, expressed by the factor 1 ˇ2 for spacecharge forces within a highly relativistic beam. This cancellation can be greatly upset if particle beams become partially neutralized by collecting other particles of opposite charge within the beams potential well. For example, proton beams can trap electrons in the positive potential well as can electron beams trap positive ions in the negative potential well. To avoid such partial neutralization and appearance of unnecessarily strong space-charge effects, clearing electrodes must be installed over much of the ring circumference to extract with electrostatic fields low energy electrons or ions from the particle beam.

The electromagnetic space-charge force on an individual particle within a particle beam increases linearly with its distance from the axis. A similar force occurs for the horizontal plane and both fields therefore act like a quadrupole causing a tune shift. This has been recognized and analyzed early by Kerst [26] and Blewett [27]. A complete treatment of space charge dominated beams can be found in [16]. The equation of motion under the influence of space charge forces can be written in the form

$$m\mathbf{y}\ \ddot{\boldsymbol{u}} + D\boldsymbol{u} = \frac{\partial F\_{\boldsymbol{u}}}{\partial \boldsymbol{u}} \boldsymbol{u} \quad \text{with} \quad \boldsymbol{u} = \left(\mathbf{x}, \mathbf{y}\right). \tag{21.57}$$