WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS M. Ferrario, M. Migliorati and L. Palumbo INFN-LNF and Universita’ di Roma ‘La Sapienza’ Abstract A charged particle beam travelling across perfectly conducting structures whose boundaries do not have constant cross section, such as an RF cavity or bellows, generates longitudinal and transverse wake fields. We discuss in this lecture the general features of wake fields, and show a few simple examples in cylindrical geometry: perfectly conducting pipe and the resonant modes of an RF cavity. We then study the effect of wake fields on the dynamics of a beam in a linac, such as beam break-up instabilities and how to cure them.

1

INTRODUCTION

Parasitic forces, also called wake fields, are generated by a charged particle beam interacting with the vacuum chamber components. These components may have a complex geometry: kickers, bellows, RF cavity, diagnostics components, special devices, etc. To solve Maxwell’s equations in a given structure with the beam current as the source of fields, a study of the field is required. For this complicated task, dedicated computer codes were developed to solve the electromagnetic problem in the frequency or in the time domain. There are several useful codes for the design of accelerator devices, like MAFIA, ABCI, URMEL, etc., as reported in Ref. [1]. In this lecture we discuss the general features of the parasitic fields [2-10], and then show a few simple examples of them in cylindrical geometry: a perfectly conducting pipe and the resonant modes of an RF cavity. Although the space charge forces have been studied separately [11], they can be seen as a particular case of wake fields, see Appendix B for a simple example [12]. We then study the effect of the wake fields on the dynamics of a beam in a linac such as beam break-up instabilities, assuming a high-energy beam where the motion is ‘frozen’ in longitudinal space, and the way to cure it [13].

2

WAKE FIELDS AND POTENTIALS

2.1

Wake potentials

Parasitic fields depend on the particular charge distribution of the beam. It is therefore desirable to know what is the effect of a single charge (i.e. find the Green function) in order to reconstruct the fields produced by any charge distribution. The electromagnetic fields created by a point charge act back on the charge itself and on any other charge of the beam. We therefore focus our attention on the source charge q0, and on the test charge q, assuming that both are moving with the same constant velocity v = βc on trajectories parallel to the axis. Let E and B be the fields generated by q0 inside the structure, (s0 = vt, r0) be the position of the source charge, and (s = s0 + z , r) be the position of the test charge q. Since the velocity of both charges is along z, the Lorentz force has the following components:

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M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

F = q ⎡⎣ Ez ˆz + ( Ex - vBy ) xˆ + ( E y + vBx ) yˆ ⎤⎦ ≡ F& + F⊥ .

(1)

Thus, there can be two effects on the test charge: a longitudinal force which changes its energy, and a transverse force which deflects its trajectory. If we consider a device of length L, the energy gain [J] is: L

U = ∫ Fz ds

(2)

0

and the transverse deflecting kick [Nm] is : L

M = ∫ F⊥ ds .

(3)

0

Note that the integration is performed over a given path of the trajectory. These quantities, normalized to the charges, are called wake potentials (volt/coulomb) and are both functions of the distance z:

U . q0 q

(4)

1 M . r0 q0 q

(5)

Longitudinal wake potential [ V C]: w& = − Transverse wake potential [ V Cm ]: w⊥ =

The minus sign in the longitudinal wake-potential means that the test charge loses energy when the wake is positive. Positive transverse wake means that the transverse force is defocusing. As a first example, consider the longitudinal wake-potential of ‘space charge’. The longitudinal force inside a relativistic cylinder of radius a travelling inside a cylindrical pipe of radius b is given by [11]:

Fs (r ,z ) =

⎛ r2 b ⎞ ∂ λ(z ) 1 − 2 + 2 ln ⎟ . 2 ⎜ 4πε 0γ ⎝ a a⎠ ∂ z −q

(6)

Note that since the space charge forces move together with the beam, and the electric field is constant vs. z, we can derive the wake potential per unit length (volt/coulomb metre). To get the wake potential of a piece of pipe, we just multiply by the pipe length. Assuming r → 0 and a charge line density given by λ ( z ) = q0δ ( z ) we obtain:

dw& (z ) ds

=

b⎞ ∂ ⎛ δ (z ) . ⎜1 + 2 ln ⎟ 4πε 0γ ⎝ a⎠∂ z 1

2

(7)

Another interesting case is the longitudinal wake-potential of a resonant higher order mode (HOM) in an RF cavity. When a charge crosses a resonant structure, it excites the fundamental and higher order modes. Each mode can be treated as an electric RLC circuit loaded by an impulsive current, as shown in Fig. 1.

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WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

Fig. 1: RF cavity and the equivalent RLC parallel circuit model driven by a current generator

Just after the charge passage, the capacitor is charged with a voltage V0 = Cq0 and the electric field is Es0 = V0/l0. The time evolution of the electric field is governed by the same differential equation of the voltage:

1  1 1 V + V+ V = I . RC LC C

(8)

The passage of the impulsive current charges only the capacitor, which changes its potential by an amount Vc(0). This potential will oscillate and decay producing a current flow in the resistor and inductance. For t > 0 the potential satisfies the following equation and boundary conditions:

1  1 V + V+ V =0 RC LC q V (t = 0+ ) = ≡ V0 C q I (0+ ) V0 +  V (t = 0 ) = = = C C RC

(9)

which has the following solution:

Γ ⎡ ⎤ V (t ) = V0 e- Γ t ⎢ cos(ωt ) − sin(ωt ) ⎥ ω ⎣ ⎦ 2 2 2 ω = ωr − Γ

(10)

where ωr2 = 1 / LC and Γ = 1 / 2RC. Putting z = ct (z is positive behind the charge) we obtain (see Fig 2.):

w& (z ) =

−V ( z ) Γ ⎡ ⎤ = w0 e − Γ z / c ⎢ cos(ωz/c) − sin(ωz/c) ⎥ . ω q0 ⎣ ⎦

345

(11)

M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

Fig. 2: Qualitative behaviour of a resonant-mode wake function

2.2

Loss factor

It is also useful to define the loss factor as the normalized energy lost by the source charge q0:

k =−

U (z = 0) . q02

(12)

Although in general the loss factor is given by the longitudinal wake at z = 0, for charges travelling with the velocity of light, the longitudinal wake potential is discontinuous at z = 0, see Fig. 3.

Fig. 3: Examples of longitudinal wake potentials: left β < 1, right β = 1

The exact relationship between k and w(z → 0) is given by the beam loading theorem [14]. 2.3

Beam loading theorem

When the source charge travels with the velocity of light v = c, the electromagnetic fields are left behind and called ‘wake fields’. Any electromagnetic perturbation produced by the charge cannot overtake the charge itself. This means that the longitudinal wake-potential vanishes in the region z < 0. This property is a consequence of the ‘causality principle’. Causality requires that the longitudinal wake-potential of a charge travelling with the velocity of light is discontinuous at the origin. The beam loading theorem states that:

k=

w& (z → 0) 2

.

(13)

For example, the beam loading theorem is fulfilled by the wake potential of the resonant mode. In fact the energy lost by the charge q0 loading the capacitor is:

CV02 q02 U= = 2 2C

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WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

giving k = 1 / 2C , to compare with:

w& ( z → 0 ) = 2.4

1 . C

Relationship between transverse and longitudinal forces

Another important feature worth mentioning is the differential relationship existing between longitudinal and transverse forces:

∂ F⊥ ∂z ∂ w⊥ ∇ ⊥ w& = ∂z

∇ ⊥ F& =

(14)

The above relations are known as the ‘Panofsky–Wenzel theorem’ [15]. 2.5

Coupling impedance

The wake potentials are used to study the beam dynamics in the time domain (s = vt). If we take the equation of motion in the frequency domain, we need the Fourier transform of the wake potentials. Since these quantities are in ohms they are called coupling impedances: ωz −i 1 Longitudinal impedance [Ω]: Z & (ω ) = ∫ w& ( z ) e v dz . v −∞

(15)

ωz −i i Transverse impedance [Ω/m]: Z ⊥ (ω ) = ∫ w⊥ ( z ) e v dz . v −∞

(16)

∞

∞

The coupling impedances (Ω/m) of the ‘space charge’ wake is:

∂ Z& (ω ) ∂s

=

∞ ωz −i 1 + 2 ln ( b a ) ∞ d 1 ∂ w& ( z ) −i ωvz v e dz = δ z e dz ( ) ∫ ∂s ∫ dz v −∞ v 4πε 0γ 2 −∞

(17)

∞

where:

∫ δ ′(z )f (z )dz = f ′(0) , so that:

−∞

∂ Z (ω ) −iω Z 0 ⎛ b⎞ = 1 + 2 ln ⎟ . 2 2 ⎜ ∂ s 4π cβ γ ⎝ a⎠ &

(18)

The longitudinal coupling impedance of a resonant HOM is given by:

Z & (ω ) =

Rs ⎛ω ω⎞ 1 − iQr ⎜ r − ⎟ ⎝ ω ωr ⎠

(19)

where Rs = w& / 2Γ is the shunt impedance and Qr = ωr / 2Γ is the quality factor, quantities that we can obtain with the computer codes. Note that the loss factor is also determined: k = ωr Rs / 2Qr . The transverse impedance is given by:

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M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

Z ⊥ (ω ) =

Rs ⊥ , ω ⎛ ωr ω ⎞ − ⎟ 1 − iQr ⎜ ⎝ ω ωr ⎠ c

(20)

with Rs ⊥ = Rs / b 2 being the transverse shunt impedance. 2.6

Wake potential and energy loss of a bunched distribution

When we have a bunch with density λ(z), we may wonder what is the amount of energy lost or gained by a single charge e in the beam. To this end we calculate the effect on the charge from the whole bunch by means of the convolution integral, see Fig. 4: ∞

U (z ) = −e ∫ w& ( z ′ − z ) λ(z ′)dz ′ .

(21)

−∞

Fig. 4: Convolution integral

This permits the definition of the wake potential of a distribution:

W& (z ) = −

U (z ) . qq0

(22)

The total energy lost by the bunch is computed summing up the loss of all particles: ∞

U bunch

1 = − ∫ U ( z ′ )λ ( z ′ ) dz ′ . e −∞

3

WAKE-FIELD EFFECTS IN LINEAR ACCELERATORS

3.1

Energy spread

(23)

The longitudinal wake forces change the energy of individual particles depending on their position in the beam. As a consequence, the wake can induce an energy spread in the beam. For example, the energy spread induced by the space charge force in a Gaussian bunch is given by:

dw& ( z ′ − z ) dU (z ) eq b ⎞ − (z 2 / 2σ z2 ) ⎛ . λ (z ′)dz ′ = − 1 2 ln = −e ∫ + ⎜ ⎟z e ds ds a⎠ 4πε 0γ 2 2πσ z3 ⎝ −∞ ∞

The bunch head gains energy, while the tail loses energy.

348

(24)

WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

In a similar way one can show that the energy loss induced by a resonant HOM on the charges inside a rectangular, uniform bunch is given by:

⎡ω ⎛ l ⎞⎤ sin ⎢ r ⎜ 0 + z ⎟ ⎥ −eqw& ⎠⎦ ⎣ c ⎝2 U (z ) = . l ω 2 ⎛ r0⎞ ⎜ 2c ⎟ ⎝ ⎠ 3.2

(25)

Beam break-up

A beam injected off-centre in a linac, because, for example, of focusing quadrupole misalignment, executes betatron oscillations. The displacement produces a transverse wake field in all the devices crossed during the flight, which deflects the trailing charges, see Fig. 5.

Fig. 5: Single-bunch beam break up originated by an injection error [7]

In order to understand the effect, consider a simple model with only two charges q1=Ne/2 (leading = half bunch) and q2=e (trailing = single charge). The leading charge executes free betatron oscillations:

⎛ ωy y1 (s ) = yˆ1 cos ⎜ ⎝ c

⎞ s⎟ . ⎠

(26)

The trailing charge, at a distance z behind, over a length Lw experiences a deflecting force proportional to the displacement y1, and dependent on the distance z:

Fyself (z , y1 ) =

Ne2 w⊥ (z )y1 (s ) . 2 Lw

(27)

Notice that Lw is the length of the device for which the transverse wake has been computed. For example, in the case of a cavity cell Lw is the length of the cell. This force drives the motion of the trailing charge: 2

⎛ω ⎞ ⎛ω Ne 2 w⊥ (z ) y2′′ + ⎜ y ⎟ y2 = yˆ1 cos ⎜ y 2 E0 Lw ⎝ c ⎠ ⎝ c

⎞ s⎟ . ⎠

(28)

This is the typical equation of a resonator driven at the resonant frequency. The solution is given by the superposition of the ‘free’ oscillation and a ‘forced’ oscillation which, being driven at the resonant frequency, grows linearly with s, as shown in Fig. 6.

⎛ ω y ⎞ forced y2 (s ) = yˆ 2 cos ⎜ s ⎟ + y2 ⎝ c ⎠

349

(29)

M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

y2forced =

⎛ ωy cNe 2 w⊥ (z )s ˆy1sin ⎜ 4ω y E0 Lw ⎝ c

⎞ s⎟ . ⎠

(30)

2 1.5

_[mm]

1 0.5 0 -0.5 -1 -1.5 -2 0

100

200

300

400 s_[m]

500

600

700

800

Fig. 6: HOMDYN [16] simulation of a typical BBU instability, 50 μm initial offset, no energy spread

At the end of the linac of length LL, the oscillation amplitude is grown by ( yˆ1 = yˆ 2 ):

⎛ Δˆy2 ⎞ ⎜ ⎟ = ⎝ yˆ 2 ⎠ max

cNew⊥ (z )LL . 4ω y (E0 /e)Lw

(31)

If the transverse wake is given per cell, the relative displacement of the tail with respect to the head of the bunch depends on the number of cells. It depends, of course, also on the focusing strength through the frequency ωy. To extend the analysis to a particle distribution, we write the transverse equation of motion of a single particle with the inclusion of the transverse wake field effects as [7]:

e2 N p ∂ ⎡ ∂y (z ,s ) ⎤ 2 s k s s y z s γ γ ( ) ( ) ( ) ( , ) + = − y m0 c 2 Lw ∂s ⎢⎣ ∂s ⎥⎦

∞

∫ y(s,z′)w

⊥

(z ′ − z )λ (z ′)dz ′

(32)

z

where γ ( s ) is the relativistic parameter, k y (s ) the beta function, Np the number of particles of the bunch, and λ (z ) the longitudinal bunch distribution. The solution of the equation in the general case is unknown. We can, however, apply a perturbation method to obtain the solution at any order in the wake-field intensity. Indeed we write:

y (z ,s ) = ∑ y (n ) (z ,s ) .

(33)

n

The first-order solution is found from the equation

∂ ⎡ ∂y (0) (z ,s ) ⎤ 2 (0) γ ( s ) ⎢ ⎥ + k y (s )γ(s)y (z ,s ) = 0 . ∂s ⎣ ∂s ⎦

350

(34)

WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

It is important to note that the above equation does not depend on z any more. This means that the bunch distribution remains constant along the structure. If the s-dependence of γ ( s ) and k y2 ( s )γ ( s ) is moderate, we can use the WKB approximation, and the solution of the above equation with the starting conditions y (0) = ym y′(0) = 0 is [2]

y (0) (s ) =

γ 0k y0 γ (s )k y (s)

ym cos [ ψ (s ) ]

(35)

where s

ψ (s) = ∫ k y (s′)ds′

(36)

0

represents the unperturbed transverse motion of the bunch. The second-order differential equation is obtained by substituting the first-order solution in the right-hand side of Eq. (31) giving

e2 N p ( 0) ∞ ∂ ⎡ ∂y (1) (z, s ) ⎤ 2 (1) y ( s )∫ w⊥ (z ′ − z )λ (z ′)dz ′ . ⎢γ (s ) ⎥ + k y (s )γ (s )y (z, s ) = − m0 c 2 Lw ∂s ⎣ ∂s ⎦ z

(37)

The forced solution of the above equation can be written in the form (1)

y ( z, s ) = − ym

e2 N p

γ 0k y0

m0 c 2 Lw

γ ( s )k y ( s )

∞

G( s )∫ w⊥ ( z ′ − z )λ ( z ′)dz ′

(38)

z

where s

1 sin [ψ ( s ) −ψ ( s′) ] cos [ψ ( s′) ] ds′ = ′ ′ ( ) ( ) γ s k s y 0

G(s) = ∫

1 sin [ψ ( s ) − 2ψ ( s′) ] 1 1 ds′ + sin [ψ ( s ) ] ∫ ds′ . = ∫ 20 2 γ ( s′)k y ( s′) γ ( s′)k y ( s′) 0 s

s

(39)

The first integral undergoes several oscillations with s and, if γ ( s ) and k y ( s ) do not vary much, we can write (1)

y ( z , s ) = − ym

e2 N p

γ 0k y0

2m0 c 2 Lw

γ ( s)k y ( s)

s

∞

ds′ w ( z ′ − z )λ ( z ′)dz ′ . γ ( s′)k y ( s′) ∫z ⊥ 0

sin [ψ ( s ) ] ∫

(40)

The last integral represents the transverse wake-potential produced by the whole bunch. This solution can then be substituted again in the right-hand side of Eq. (32) to obtain a second-order equation and so on. If we consider constant γ ( s ) and k y ( s ) , we obtain the same result of the two-particle model when we substitute λ ( z ) with 1/2 in the particle positions.

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M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

For example, by using the wake of LEP superconducting cavities, it is possible to find that for a Gaussian bunch, the wake potential is as given in Fig.7.

Fig.7: Transverse wake potential for LEP superconducting cavities

For a conservative estimation of the BBU effect, one should use the maximum value of the curve in this case, thus eliminating the z dependence. If the BBU effect is strong, it is necessary to include higher order terms in the perturbation expansion. Assuming: – rectangular bunch distribution, λ (z )=1 / l0 , −l0 / 2 < z < l0 / 2, l0 bunch length, – monoenergetic beam, – constant acceleration gradient dE0 /ds = const, – constant beta function, – linear wake function inside the bunch w⊥ ( z ) = w⊥0 z / l0 , the sum of Eq. (33) can be written in terms of powers of the dimensionless parameter η also called BBU strength

η=

e2 N p

w⊥0 ⎛ γ f ⎞ ln ⎜ ⎟ k y ( dE0 / ds ) Lw ⎝ γ i ⎠

(41)

with γ i and γ f respectively the initial and final relativistic parameter. By using the method of stepping descent [5], it is possible to obtain the asymptotic expression of y(z, s) finding at the end of the linac,

y( LL ) = ym

⎡ 3 3 1/ 3 ⎤ γ i −1 / 6 3 π⎤ ⎡ η exp ⎢ η ⎥ cos ⎢ k y LL − η 1 / 3 + ⎥ , 6πγ f 4 12 ⎦ ⎣ ⎢⎣ 4 ⎥⎦

(42)

the two-particle model is different from the first-order solution, and gives a tail displacement growing exponentially with η , resulting in better agreement with the simulation in Fig. 6. 3.3

BNS damping

The BBU instability is quite harmful and hard to get under control even at high energy with a strong focusing, and after careful injection and steering. A simple method to cure it has been proposed on the basis that the strong oscillation amplitude of the bunch tail is mainly due to ‘resonant’ driving. If the

352

WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

tail and the head move with a different frequency, this effect can be significantly reduced [13], compare Fig. 8 with Fig. 6. 0.15 0.1

_[mm]

0.05 0 -0.05 -0.1 -0.15

0

200

400

600

800

1000

1200

1400

s_[m]

Fig. 8: HOMDYN simulation of a typical BNS damping, 50 μm initial offset, 2% energy spread

Let us assume that the tail oscillates with a frequency ωy +Δωy , the two particle model equation of motion reads: 2

⎛ ω + Δω y ⎞ ⎛ω Ne 2 w⊥ ( z ) y2′′ + ⎜ y y yˆ1 cos ⎜ y = ⎟ 2 2 2β E0 Lw c ⎝ ⎠ ⎝ c

⎞ s⎟ ⎠

(43)

the solution of which is

⎛ ωy ⎞⎤ c 2 Ne2 w⊥ ( z ) ⎡ ⎛ ω y + Δω y ⎞ y2 ( s ) = yˆ1 ⎢ cos ⎜ s ⎟ − cos ⎜ s ⎟⎥ 4ω y Δω y E0 Lw ⎢⎣ ⎝ c ⎠ ⎝ c ⎠ ⎦⎥

(44)

where the amplitude of the oscillation is limited. Furthermore, by a suitable choice of Δωψ, it is possible to fully depress the oscillations of the tail. Setting

⎛ ω y + Δω y yˆ 2 cos ⎜ c ⎝

⎞ ⎛ ωy s ⎟ + y2 ( s ) = yˆ1 cos ⎜ ⎠ ⎝ c

⎞ s⎟ ⎠

(45)

we get

Δω y =

c 2 Ne 2 w⊥ ( z ) . 4ω y Eo Lw

(46)

The extra focusing at the tail can be obtained by using a RFQ, where head and tail see a different focusing strength; exploiting the energy spread across the bunch which, because of the chromaticity, induces a spread in the betatron frequency. An energy spread correlated with the position is attainable with the external accelerating voltage, or with the wake fields.

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M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

4

LANDAU DAMPING

There is a fortunate stabilizing effect against collective instabilities called ‘Landau Damping’. The basic mechanism relies on the fact that if the particles in the beam have a spread in their natural frequencies (synchrotron or betatron), their motion cannot be coherent for a long time. 4.1

Driven oscillators

In order to understand the physical nature of this effect, we consider a simple harmonic oscillator, at rest for t < 0, driven by an oscillatory force for t > 0.

d 2x + ω 2 x = A cos(Ωt ) . 2 dt

(47)

The general solution is given by the superposition of the free and forced solutions, see Appendix 1:

x(t ) =

A [cos(Ωt ) − cos(ω t )] . ω − Ω2 2

(48)

Let us assume that the external force is driving a particle population characterized by a spread of natural frequency of oscillation around a mean value ωx. Furthermore, leave the forcing frequency Ω inside the spectrum so that δ ≡ Ω − ω a

Er holds.

The electrostatic potential satisfying the boundary condition ϕ ( b ) = 0 is given by:

⎧ I ⎛ b r2 ⎞ + − ⎟ 1 2 ln ⎪ ⎜ b a a2 ⎠ ⎪ 4πε 0 v ⎝ ϕ ( r , z ) = ∫ Er ( r ′, z ) dr ′ = ⎨ r ⎪ I ln b ⎪⎩ 2πε 0 v r

for r ≤ a .

for a ≤ r ≤ b

How can a perturbation of the boundary conditions affect the beam dynamics? Consider the following example: a smooth transition of length L (taper) from a beam pipe of radius b to a larger beam pipe of radius d is experienced by the beam [6]. To satisfy the boundary condition of a perfectly conducting pipe in the tapered region, the field lines are bent as shown in Fig. B.2. Therefore there must be a longitudinal Ez(r,z) field component in the transition region. A test particle running outside the beam charge distribution along the transition of length L will experience a voltage given by [12]:

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M. F ERRARIO , M. M IGLIORATI AND L. PALUMBO

V =−

z+L

∫ z

Ez ( r, z ′ ) dz ′ = − ⎡⎣ϕ ( r, z + L ) − ϕ ( r, z ) ⎤⎦ = −

I 2πε 0 v

ln

d b

that is decelerating if d > b. The power lost by the beam in order to sustain the induced voltage is given by

Plost = VI =

I2 2πε 0 v

ln

d . b

(B.2)

Fig. B.2: Smooth transition of length L (taper) from a beam pipe of radius b to a larger beam pipe of radius d

It means that for d > b the power is deposited to the energy of the fields: moving from left to right of the transition, the beam induces the fields in the additional room around the bunch bunch (i.e. in the region b < r < d, 0 < z < l0) at the expense of the only available energy source, that is the kinetic energy of the beam itself.

Fig. B.3: During the beam propagation in the taper additional electromagnetic power flow is required to fill up the new available room

To verify this interpretation, compute the electromagnetic power radiated by the beam to fill up the additional room available, see Fig. B.3. Integrating the Poynting vector through the surface

(

)

ΔS = π d 2 − b 2 representing the additional power passing through the right part of the beam pipe [12], one obtains

d Er Bϑ ⎛ 1 G G⎞ I2 d ˆ =∫ Pem = ∫ ⎜ E × B ⎟ ⋅ ndS 2π rdr = ln μ μ 2πε 0 v b ⎠ ΔS ⎝ b

exactly the same expression of Eq. (B.2). Notice that if d < b the beam gains energy. If d → ∞ the power goes to infinity, an unphysical result like this is nevertheless consistent with the original assumption of an infinite energy beam ( γ → ∞ ).

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WAKE FIELDS AND INSTABILITIES IN LINEAR ACCELERATORS

References [1] Computer codes for particle accelerator design and analysis: a compendium, LAUR-90-1766 (1990). [2]

K. Yokoya, DESY 86-084 (1986).

[3]

K. Bane, SLAC-PUB-4169 (1986).

[4]

K. Bane and M. Sands, SLAC-PUB-4441 (1987).

[5]

A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley, New York, 1993).

[6]

L. Palumbo, V. G. Vaccaro and M. Zobov, Wake fields and impedance, CERN Accelerator School, CERN 95-06 (1995).

[7]

A. Mosnier, Instabilities in linacs, CERN Accelerator School, CERN 95-06 (1995), Vol. I, pp. 459–514.

[8]

B.W. Zotter and S.A. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientific, Singapore, 1998).

[9]

G. V. Stupakov, Wake and impedance, SLAC-PUB-8683 (2000).

[10]

K. Bane, Emittance control for very short bunches, Proc. of EPAC 2004, Lucerne, Switzerland.

[11]

M. Ferrario and L. Palumbo, Space charge effects, these proceedings.

[12]

N. A. Vinokurov, Space charge, in High Quality Beams, Eds. S.I. Kurokawa et al. (American Institute of Physics, 2001).

[13]

V. E. Balakin, A.V. Novokhatsky and V. P. Smirnov, Proc. 12th Int. Conf. on High Energy Accelerators, Batavia, 1983.

[14]

P.B. Wilson, SLAC PUB 2884 (1982).

[15]

W.K.H. Panofsky and W.A. Wenzel, Some considerations concerning the transverse deflection of charged particles in radio-frequency fields, Rev. Sci. Instrum. 27 (1956) 967.

[16]

M. Ferrario, V. Fusco, B. Spataro, M. Migliorati, and L. Palumbo, Wake field effects in the SPARC photoinjector, Proc. EPAC 2004, Lucerne, Switzerland.

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