high power proton linac basics - … · aurélien ponton hpp linacs. introduction beam emittance...

IntroductionBeam emittance

Beam powerAcceleration

FocusingConclusion

High Power Proton Linac Basics

Aurélien Ponton

European Spallation SourceAccelerator Division

Euroschool on Exotic Beams, August 21th-26th 2011, Jyväskylä, Finland

1/53Aurélien Ponton HPP Linacs



FocusingConclusion

Outline

1 Introduction

2 Beam emittanceLiouville’s theoremAccelerator referenceStatisical view

3 Beam powerExpression of the beam powerBeam pulse and beam bunch

4 AccelerationRF cavitiesLongitudinal dynamicsAttractiveness of RF superconductivity

5 FocusingSources of defocusingMagnetic quadrupoles

6 Conclusion




FocusingConclusion

Outline

1 Introduction





6 Conclusion




FocusingConclusion

High power proton beams applications

Nowadays there is a strong demand for proton beam sources in theMW range in order to cover numerous scientific aspects:

Fundamental laws of matter:neutrino factoriesmuon colliders

Nuclear science: investigation of the nuclear landscape with highintensity isotope beamsEnergy production and nuclear waste transmutation: AcceleratorDriven Subcritical Systems (ADS)Material science: neutron sources. . .




FocusingConclusion

High power proton beam applications

Most of the modern projects which aims at using high power protonbeams are linac-based. Linear accelerators advantages are:

acceleration of high intensity beams: not limited by resonanceshigh repetition rate: linacs can operate easily up to CWupgradability: extra modules can be addedmodularity: repetition of identical modulesreliabilityextraction without loss at any locationSRF technological advances




FocusingConclusion

Scope of the lecture

The following lecture aims at indroducing you to linac basics:beam emittanceRF accelerationfocusing

In addition we will try to identify the challenges required in designinghigh power proton linacs.

This lecture will also give you elements to understand the linacdesign strategy of ESS (presented in the next talk).




FocusingConclusion

Liouville’s theoremAccelerator referenceStatisical view

Outline

1 Introduction





6 Conclusion




FocusingConclusion


Exposition

Let’s consider a system of N particles characterized by their generalized coordinates qiand associated canonical momenta pi (i = 1, . . . ,N). For each particle we canassociate a point (qi , pi ) in a 6D phase space. The volume occupied in phase spaceby the N representative points is given, at a given time t by:

τ =

∫∫d3qi d3pi (1)

where the summation on the N particles is implicit.At a time t + δt , we have:

τ ′ =

∫∫d3q′i d3p′i (2)

Liouville’s theorem

τ = τ ′ (3)

The volume occupied by a given number of particles in phase space remains constant.




FocusingConclusion


Domain of validity

1 Volume conservation of space phase is a property of Hamitonian systems calledalso conservative systems.

2 Dissipative forces (synchrotron radiations, . . . ) or particle lost invalidate theLiouville’s theorem.

3 The theorem applies stricly for non-interacting particles but remains valid whileinteractions between particles are weak. Otherwise generalization of the theoremin a 6N-dimension hyper-space is necessary.

4 In electro-magnetic (EM) fields, mechanical, P, and conjugated, p, momenta aredifferent. Conjugated momentum for a charged particle in EM fields is given by:

p = P + qA(q, t) (4)

where q is the charge and A is the potential vector.We can show that the Liouville’s theorem is valid even if mechanical momenta areconsidered, i.e. in (q,P) "mechanical" phase space:∫∫

d3qi d3Pi = Const. (5)

Exercise 1: Liouville’s theorem in mechanical phase space

Show that∫∫

d3qi d3pi =∫∫

d3qi d3Pi .Clue: Start by writing down the density functions and differentiate.




FocusingConclusion


Projection of the volume onto 2D phase space

Neglecting the coupling between directions, the Hamiltonian can be written as:

H(q, p; t) = Hx (x , px ; t) +Hy (y , py ; t) +Hz (z, pz ; t) (6)

In this case the Liouville’s theorem applies in each 2D (sub-)phase space: the area(projection of the 6D volume) is conserved.The area enclosing all the particles is the beam emittance. For (x ,Px ) for example, wecan write the emittance as:

Ax,Px =∫∫

dxi dPx,i (7)

Stricly speaking the 6D volume is the emittance and we should precise "the 2Dprojection of the beam emittance on (x ,Px )".

Remark: We will see that ε =Aπ

is also referred as the beam emittance.




FocusingConclusion


New independant variable

Magnets, RF cavities and other components are defined at given locations alongthe reference trajectory.

We prefer thus make a change in independent variable from time t to path length.

Let’s choose our coordinates so that the z axis defines the reference trajectory.

Changing the variable of integration from time t to path length z, the canonicalcoordinates in the longitudinal plane become: t and −E (total energy)

Remarks:

In linacs the reference trajectory is a staight line defined by the symetry axis of theaccelerator elements.

In magnetic dipole fields the reference trajectory in no more a straigth line (circularmachines): we shall use of the Frenet’s frame and take into account the curvatureof the reference path.

In the ideal case the beam centroid lies on the reference trajectory. However thebeam may be off-centered (field errors, misalignment, off-axis injection).




FocusingConclusion


Transverse planes (1)

We can express the derivatives with respect to z (our new independant variable) of thetransverse coordinates as follows:

x ′ =dxdz

=xz

(8)

y ′ =dydz

=yz

(9)

(10)

where the prime denotes the derivative with respect to z.In general the longitudinal velocity in the laboratory frame is very high compared to thetransverse velocities:

vz vx and vz vy (11)

therefore:v = (vx + vy + vz )1/2 ≈ vz (12)




FocusingConclusion


Transverse planes (2)Taking into account the paraxial approximation we can rewrite the right-hand side ofEqs. 8 and 9 as:

xz

= tan θx ≈ θx andxz

= tan θy ≈ θy (13)

for θx 1 and θy 1.θx and θy are the angles between the velocity vz and the x-component respectively they-component of the particle velocity vector (vx or vy). Therefore x ′ (or y ′) gives directaccess to the convergence or divergence behavior of the beam in the transverseplanes.The transverse emittance is expressed in mm.mrad in the (x , x ′) or (y , y ′) tracespaces.

Figure: Diverging. Figure: Parallel. Figure: Converging. 13/53Aurélien Ponton HPP Linacs



FocusingConclusion


Longitudinal plane (1)

In the longitudinal plane, we can express the emittance using various parameters.A reference particle of longitudinal position z0 and velocity v0 is always introduced.All particle parameters are expressed with respect to the reference parameters.

We have:

∆t = t − t0 (14)

∆z = z − z0 (15)

∆φ = φ− φ0 = φ− φS (16)

and:

∆W = W −W0 (17)

δ =P − P0

P(18)

∆z′ =v − v0

v(19)

Remark: ∆φ = φ− φS is the phase difference (relative to the RF cavity phase)between one particle (φ) and the synchronous particle (φS). We will introduce thesynchronism concept in Section 4.




FocusingConclusion


Longitudinal plane (2)

Following table summarizes some of the longitudinal emittances and their usual units(in the proton linac world!) you may deal with in accelerator physics.

∆t ∆φ ∆z∆W ns.MeV deg.MeV mm.MeVδ ns.% deg.% mm.%

∆z′ ns.mrad deg.mrad mm.mrad

Table: Some useful longitudinal emittances and their units

Remark:

In the following we will stick to the emittance defined in the (∆z,∆z′) trace spacealways referred as (z, z′).

The reference particle is generally the beam centroid.




FocusingConclusion


Normalization

Normalization is necessary to observe (or not!) the Liouvillian behavior of the beam.In the following normalizations we assume that all particles move with a velocity veryclose to the reference.

Transverse

Let’s rewrite Eqs. 8 and 9:

x ′ ≈Px

P0and y ′ ≈

Py

P0(20)

Since P0 = β0γ0m0c2 with β0, γ0 andm0c2 repectively the reduced velocity,the reduced energy and the massenergy, we can define the tranversenormalized emittances as:

A(x,x′),n = β0γ0Ax,x′ (21)

A(y,y′),n = β0γ0Ay,y′ (22)

Longitudinal

Let’s remind that t and −E are thecanonical variables. We have:

∆t ≈1β0c

∆z (23)

∆W ≈ β20γ

30m0c2∆z′ (24)

Therefore we can express thenormalized emittance in the (z, z′)plane as:

A(z,z′),n = β0γ30Az,z′ (25)




FocusingConclusion


Describing the beam quality (1)Let’s observe on the following figures the beam phase portrait evolution.

Linear force

Figure: z = z1.

Nonlinear force

Figure: z = z1.17/53

Aurélien Ponton HPP Linacs



FocusingConclusion



Linear force

Figure: z = z2.

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z3.

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z4..

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z5.

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z6.

Nonlinear force

Figure: s = z6.17/53




FocusingConclusion



Linear force

Figure: z = z7.

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z8.

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z9.

Nonlinear force





FocusingConclusion



Linear force

Figure: z = z10.

Nonlinear force

Figure: z = z10.17/53




FocusingConclusion



Linear force

Figure: z = z11.

Nonlinear force

Figure: z = z11.17/53




FocusingConclusion



Linear force

Figure: z = z12.

Nonlinear force

Figure: z = z12.17/53




FocusingConclusion


Descibing the beam quality (2)

Figure: Beam in linear forces. Figure: Beam in nonlinear forces.

1 For linear forces the angular velocity in phase space or phase advance, σ,(generally expressed in deg.m−1) is constant: the shape is conserved.

2 For nonlinear forces σ is a function of the particle position. The area is conservedbut the beam has experienced filamentation. For beam densities decreasing withthe radial position, the rotation in phase space is reduced for particles in the core:it is called tune depression. For high intensity beams, non linearities comeprincipally from the self-force or space charge forces (see Section 5).

The definition of the normalized emittances of Eqs. 21, 22 and 25 does not give accessto the beam behavior under nonlinear forces.

−→We need another definition of the beam emittance.18/53




FocusingConclusion


RMS emittanceWe thus introduce a statistical definition of the emittance based on the momenta of thesecond order of the beam distribution function. If for a system of N particles theaverage value of X is X = 1

N∑N

i=1 Xi where Xi is associated to the particle i , thesecond order of the distribution function are (with w = x , y or ∆z):

w = (w − w)21/2(26)

w ′ = (w ′ − w ′)21/2

(27)

ww ′ = (w − w) · (w ′ − w ′). (28)

The (Root Mean Square) RMS emittance is then:

εw,w′ =(w2 · w ′

2− ww ′

2)1/2 (29)

Remarks:The same normalization factors as defined in Eqs. 21, 22 and 25 are valid for theRMS emittance.Equation 29 simplifies when the beam is centered (w = 0 and w ′ = 0):

εw,w′ =(w2 · w ′2 − ww ′2

)1/2 (30)

Evolution of the RMS quantities are quasi independant of the beam distribution. 19/53Aurélien Ponton HPP Linacs



FocusingConclusion


Emittance ellipse (1)

Figure: Concentration ellipses in (x, x ′)plane.

From the RMS quantities, we can define theTwiss parameters as:

αw = −ww ′

εw,w′(31)

βw =w2

εw,w′(32)

γw =w ′2

εw,w′(33)

verifying βwγw − α2w = 1.

We can obtain an ellipse of equation:

γw w2 + 2αw ww ′ + βw w ′2 = εww′ (34)

whoose area is:

Aw,w′ = πεw,w′ (35)Remark: An ellipse ramains an ellipse under linear transformations. 20/53




FocusingConclusion


Emittance ellipse (2)Let’s rewrite Eq. 35, we have obtained:

Aw,w′ = πεw,w′ (36)

We have defined the area in trace space as:

Aw,w′ =

∫∫dw dw ′ (37)

We can find a factor η so that ηA encloses all the particles. However we will alwayshave:

ηA ≥ A (38)

ηA = A if the beam distribution in trace space is an ellipseηA A if the beam portrait is filamented (distorded) by nonlinear forces(statistical view of the beam quality)

Remark on unities: We have seen that A or ε are referred as the beam emittance. Bothof them can be used if we are careful about the unities. Let’s consider an ellipse ofmajor and minor-axis a and b, we can write:

ε = a · b [π.mm.mrad] (39)

A = π a · b [mm.mrad] (40)

with π either number or unity.21/53




FocusingConclusion


Beam envelopeWe have seen in Eq. 32 that the beam extension can be expressed as:

x =√βx ε (41)

If we know the evolution of the betatron function βx (s) we thus have access to thebeam envelope.

Figure: x envelope in a FDO channel (50 mAproton beam, 500MeV).

The figure on the left shows the matched case:the Twiss parameters have been optimized sothat period after period the beam parametersremain identical.

If the beam was not matched to the focusingchannel the beam envelope would have beenirregular. Mismatching leads to envelopeinstabilities which is a major source of haloproduction: particles moving far from the coremay result in hazardous losses at high energy.




FocusingConclusion


In a nutshell!

We have defined the RMS normalized beam emittances in the 3 trace spaces:

in (x , x ′):

εx,n = β0γ0(x2 · x ′2 − xx ′2

)1/2 (42)

in (y , y ′):

εy,n = β0γ0(y2 · y ′2 − yy ′2

)1/2 (43)

in (z, z′):

εz,n = β0γ30

(∆z2 ·∆z′2 −∆z∆z′2

)1/2 (44)

The RMS normalized emittances are a measure of the beam quality.Minimizing emittance growth while the beam is propagating through theaccelerator is a major challenge for very intense beams.

Note that the subscripts x ′, y ′ and z′ generally disappear.




FocusingConclusion

Expression of the beam powerBeam pulse and beam bunch

Outline

1 Introduction





6 Conclusion




FocusingConclusion


Beam powerThe beam power, P, is given by:

P [MW] = Ipeak [mA]×W [GeV]× ηbeam [%] (45)

where Ipeak is the peak current, W is the beam energy and ηbeam is the beam duty cycle.The beam duty cycle can be expressed with the repetition rate, frep, and the beampulse length, Tbeam, by: ηbeam = frep Tbeam

Power is limited by thermo-mechanical stress and cooling issues in the target:survivability of the target intercepting energetic, high intensity proton bunches.Irradiation and integration has also to be taken into account.Intensity is limited by space charge effects at low energy (see Section 5): IFMIFlinac: 125 mA CW deuteron beams! But beam funneling may be an option toincrease intensity.Repetition rate may be governed by time of flight-type measurements. Highrepetition rate linacs are required for multi-physics projects (SPL). For injection inrings frep must match the revolution frequency.Maximal energy is chosen in order to have the highest production yield ofsecondary particles or it is given by the injection energy required in synchrotrons.Pulse length depends on the experiment requirements: CW for ADS and in the nsscale range for neutrino factories (with accumulator and compressor rings).




FocusingConclusion


Beam pulse and beam bunch

Figure: Beam pulses. Figure: Beam bunches.

Look at the different time scale!Beam pulse is a macro parameter of the beam time structure.Beam bunch is a micro time parameter related to the frequency of the resonantcavities.

In the upper example, Tbeam = 2.86 ms, ηbeam = 4 % and the cavity frequency is352.21 MHz. If all the RF periods are filled, there are about 1 · 106 bunches per pulse.




FocusingConclusion


Exercise

Exercise 2: Beam power and beam bunch

A proton linac has the following parameters:

duty cycle: ηbeam = 4 %

beam pulse length: Tbeam = 2.86 ms

frequency of the resonant cavities: fRF = 352.21 MHz

intensity: Ipeak = 50 mA

1 What is the kinetic energy of the beam in order to reach a power of 5 MW?2 How many protons in a bunch?3 What is the mass of protons produced in a year?




FocusingConclusion

RF cavitiesLongitudinal dynamicsAttractiveness of RF superconductivity

Outline

1 Introduction





6 Conclusion




FocusingConclusion


Energy gain in EM fields

A particle of charge q with velocity v propagating in EM fields (E,H) will change itsenergy W according to:

dWdt

= q (v.E) (46)

Equation 46 expresses the need to produce an electric field in the direction ofpropagation in order to accelerate (stricly speaking longitudinaly) the beam.Other combination of EM fields are used:

to focalize the beam

to guide, extract, deviate the particles

Radio-Frequency (RF) resonant cavities are used to produce high accelerating fields.

Remark: G. Ising established the RF acceleration principle in 1924!




FocusingConclusion


Fields in a cavity (single gap)Maxwell’s equations combine to yield to the wave equation in vacuum:(

∇2 −1c2

∂2

∂t2

)EH

= 0 (47)

Near a perfect conductor, the bundary conditions are:

n × E = 0 and n · H = 0 (48)

with n is the unity vector normal to the surface.

Figure: Pill-box cavity!.

Applying these conditions to the pill-box cavity,we find a discret set of standing wave modes:

Transverse Electric (TM)

Transverse Magnetic (TE)

Only TM have a non zero longitudinal electriccomponent on axis. In general the TM010 isused in most of the cavities.The fondamental frequency (TM010) for apill-box cavity of radius R is given by:

fRF =2.405 c

2πR(49)




FocusingConclusion



∇2 −1c2

∂2

∂t2

)EH

= 0 (47)


n × E = 0 and n · H = 0 (48)


Figure: Electric field configuration inthe pill-box cavity (TM010).





fRF =2.405 c

2πR(49)




FocusingConclusion



∇2 −1c2

∂2

∂t2

)EH

= 0 (47)


n × E = 0 and n · H = 0 (48)


Figure: Magnetic field configurationin the pill-box cavity (TM010).





fRF =2.405 c

2πR(49)




FocusingConclusion


Fields in a cavity (multiple gaps)In a multi-cell structure the coupling between the cells causes each mode to split into anumber of modes equal to the number of cells. If N is the number of cells we have thusN modes TM010:

TM010, nπN

for n = 1, . . . ,N (50)

Example of a 3-cell cavity:

mode π/3

Symmetryaxis

E

RF coupler

Figure: Electric field.1 2 3

Cell #

1.0

0.5

0.0

0.5

1.0

Long.

ele

ctri

c field

on a

xis

[u.a

.]

Figure: Electric field distribition. 31/53Aurélien Ponton HPP Linacs



FocusingConclusion



TM010, nπN

for n = 1, . . . ,N (50)


mode 2π/3

Symmetryaxis

E

RF coupler


Cell #

1.0

0.5

0.0

0.5

1.0

Long.

ele

ctri

c field

on a

xis

[u.a

.]

Figure: Electric field distribition. 31/53Aurélien Ponton HPP Linacs



FocusingConclusion



TM010, nπN

for n = 1, . . . ,N (50)


mode π

Symmetryaxis

E

RF coupler


Cell #

1.0

0.5

0.0

0.5

1.0

Long.

ele

ctri

c field

on a

xis

[u.a

.]

Figure: Electric field distribition.31/53




FocusingConclusion


Synchronism

Mode TM010,π is prefered for acceleration:

better acceleration

lower RF losses

better field distribution (lowering the peak field is mandatory for superconductingcavities)

Symmetryaxis

RF coupler

Beam bunch

Figure: Synchronism condition.

Synchronism condition

The beam bunch has to cross the cavity gap oflength Lgap in half of an RF period to see anaccelerating field:

Lgap =β0λRF

2(51)

with λRF the RF wave length.




FocusingConclusion


Practical designLet’s rewrite the synchronism condition of Eq. 51:

Lgap =β0λRF

2(52)

What are the consequences on the cavity design?The best will be to increase the gap length as the particle velocity increases:

very efficient at low energy where β0 increases a lotbut the phase is fixed (coupled cavities)only particles with the same mass over charge ratio (q/A) can be accelerated

Linac is composed generally of a few families of identical cavities which areefficient in a certain range of energy.

How to choose the frequency of the cavities?At low energy the beam moves slowly: low frequencies are required.High frequency for low energy beam leads to mechanical stability issues: shortgap and short radius (fRF ∝ 1/R see Eq. 49).At high frequency (always a multiple of the lowest frequency) we can beneficiatefrom high accelerating fields available in elliptical cavities (tens of MV/m).For protons, we have in general:

300− 500 MHz for β0 < 0.5600− 1000 MHz for β0 > 0.5' 1300 MHz for β0 close to 1 (ILC technology)

In Europe, the reference frequency is the LEP frequency: 352.21 MHz.33/53




FocusingConclusion


Energy gain in RF cavities

The beam bunch average energy gain, ∆W , in a cavity is given by:

∆W = qVacc cosφS (53)

where Vacc is the accelerating voltage and φS is the synchronous phase.

Writing the electric field developed on axis as:

Ez (ρ = 0, z, t) = Ez0(ρ = 0, z)ejωRFt (54)

where ωRF is the angular frequency, we can evaluate the accelerating voltage as:

Vacc(β) =∣∣∣ ∫ z2

z1

Ez0ej(ωRFzβ(z)c

)dz∣∣∣ (55)

with z1 and z2 the start and the end of the field respectively.Vacc is not the voltage developed in the cavity but the voltage the beam undergoes.

The synchronous phase is the average (over the cavity length) phase differencebetween the reference particle phase and the RF phase developed in the cavity.




FocusingConclusion


Phase stability (1)

For the sake of simplicity, let’s consider an RF cavity whose length is null reaching amaximum at t = 0.

1.0 0.5 0.0 0.5 1.0ωRFt/π

1.0

0.5

0.0

0.5

1.0

Long.

ele

ctri

c field

on a

xis

Deceleration Deceleration

Acceleration

M

M1

M2

P

P1

P2

StableUnstable

Figure: Phase stability.

1 Acceleration condition:

φS ∈[−π

2;π

2

](56)

2 Stability condition:

φS ∈[−π

2; 0]

(57)

Remark: Bunching cavities useφS = −π2 (no acceleration)




FocusingConclusion


Phase stability (2)

1.0 0.5 0.0 0.5 1.0ωRFt/π

1.0

0.5

0.0

0.5

1.0

Long.

ele

ctri

c field

on a

xis

Deceleration Deceleration

Acceleration

M

M1

M2

P

P1

P2

StableUnstable

Figure: Phase stability.

Phase synchronous law, φS(s), has to becarefully chosen in order not to looseparticle which migth be out of the stabilityphase.

If Φ0 is the beam phase length at a givenz1, Φ is given at any z > z1 by:

Φ(z)

Φ(z1)=

[β0(z1)γ0(z1)

β0(z)γ0(z)

]3/4

(58)

This phenomenon is called adiabatic phasedamping of the oscillations. We can thuschoose a higher synchronous phase whilethe energy increases (but losses might bemore dangerous!).

Remark: Cavities not only accelerate the beam but provide also longitudinal focusing.Synchrotron oscillations around the synchronous particle!




FocusingConclusion


Exercice

Exercise 3: Acceptable beam losses

We have already identified sources likely to induce beam losses: mismatching, fielderrors, alignment, bad phase synchronous law.Minimization of beam losses is mandatory for maintainability of the machine. Beamlosses should not surpass 1 W/m.

1 Take the answers you have found in exercise 2 and express the fraction of thebeam that can be lost at the maximal energy within the given tolerances.

2 What is your advice if one wants to simulate carefully the beam losses in the linac?




FocusingConclusion


Transit time factorThe transit time factor, TTF, gives the ratio of the voltage seen by the beam, Vacc, to themaximal voltage available in the cavity, Vmax. By definition we have:

TTF (β) =Vacc(β)

Vmax=

∣∣∣ ∫ z2z1

Ez0ej(ωRFzβ(z)c

)dz∣∣∣∫ z2

z1|Ez0|dz

< 1 (59)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TTF

β

1 cellule2 cellules3 cellules4 cellules5 cellules

Figure: Transit time factor.

We can define the optimal velocity, βopt, by:

TTF (βopt) = Max(TTF (β)) (60)

Cavities are efficient only on a certainvelocity range.

At low energy, we are in a range where thevelocity invreases a lot with energy:cavities with a few cells are required (onlyfor cavities with constant gap lengths).




FocusingConclusion


Accelerating field

RF cavities are always characterized by their accelerating field, Eacc. The latter is givenby:

Eacc =Vacc(βopt)

L(61)

where L is a normalization length.Unfortunately there is not any widely used definition of L and it should always beprecised when dealing with accelerating gradients.One definition independant of the cavity length is the following (for π modes):

L =ngap

2βoptλRF (62)

where ngap is the number of accelerating cells.

Remark: Real estate gradient (per meter) could be prefered.




FocusingConclusion


Surface resistance

Superconducting

For SC material, we have:

Rs = RBCS + R0 (63)

where RBCS is the BCS resistance given, forfRF in GHz, for Niobium by:

RBCS = 2 · 10−4 1T

(fRF

1.5

)2exp

(−

17.67T

)(64)

In the range 1− 1000 MHz, with T = 2 K,we find:

RBCS = 1− 100 nΩ (65)

R0 is the residual resistance depending onimpurities, trapped magnetic flux, surfacetreatment, . . . For well prepared cavities, wehave R0 = 1− 20 nΩ.

Normal conducting

Surface resistance, Rs , is expressed as:

Rs =

√µ0ωRF

2σ(66)

where µ0 = 4π · 10−7 H.m−1 is thepermeability constant.For Copper the electric conductivity:σ = 5.96 · 107 S.m−1 at room temperature.Considering the same frequency range, wefind:

Rs = 1− 10 mΩ (67)

Tipically 5 order of magnitude lower surfaceresistance in the SC case: voltage 100 000times less power will be lost in the cavity

walls.→ All RF power available goes to thebeam.




FocusingConclusion


Cryogenic needs

1 2 3 4 5 6 7 8 9 10Temperature [K]

10-8

10-7

10-6

Surf

ace

resi

stance

[Ω

]

704 MHz

352 MHz

Figure: Surface resistance as afunction of temperature.

102 103

Frequency [MHz]

10-8

10-7

10-6

Surf

ace

resi

stance

[Ω

]

2 K

4 K

Figure: Surface resistance as afunction of frequency.

Superconducting cavities have to be maintained at cryogenic temperatures with liquidhelium in cryostats.

For frequency greater than ∼ 400 MHz superfluid helium (T < 1.9 K) is required.The total refrigerator efficiency, ηref, has to be taken into account. It is the productof the technological efficiency, ηtec, typically 20− 30 % and the theoritical Carnotefficiency:

ηref = ηtec ×T

300− T(68)




FocusingConclusion


Cryostat (courtesy P. Pierini INFN)

“Cartoon” view of the system

All “spurious” sources of heat losses to the 2 K circuits need to be properly managed and intercepted at higher temperatures (e.g. conduction from penetration and supports, thermal)

Cold mass

To He production and distribution

system

2 K

T1

T2

Sup

port

s

Penetrations




FocusingConclusion


Comparison NC vs. SC

0 1 2 3 4 5

Accelerating voltage [MV]

0

5

10

15

20

25

30

35

Equiv

ale

nt

RF

loss

es

at

300

K [

kW]

ηbeam =4 %

SCNC

Figure: RF losses as a function of theaccelerating voltage.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Beam duty cycle [%]

0

2

4

6

8

10

12

Equiv

ale

nt

RF

loss

es

at

300

K [

kW]

Vacc =1 MV

SCNC

Figure: RF losses as a function of the dutycycle.

Choosing SRF technology shows significant power saving for high voltage andhigh duty cycle.SRF cavities are intrinsically more stable and can have large beam tubeapertures: required for high intensity low loss linacs.They are short independantely phase structures. 43/53




FocusingConclusion


Dynamic and static lossesIn addition to dynamic losses (when the beam is on), static losses to maintain the cavityat cryogenic temperatures play also an important role in the total losses contribution.

0 20 40 60 80 100Beam duty cycle [%]

0.0

0.2

0.4

0.6

0.8

1.0

Contr

ibuti

on t

o t

he t

ota

l lo

sses

Vacc =5 MV

StaticDynamic

Figure: Contribution to the total losses.

0 20 40 60 80 100Beam duty cycle [%]

0.0

0.2

0.4

0.6

0.8

1.0

Contr

ibuti

on t

o t

he t

ota

l lo

sses

Vacc =20 MV

StaticDynamic

Figure: Contribution to the total losses.

Static losses are generally the main contribution to the total losses.For continuous beam accelerating in high fileds dynamic losses becomesignificant.Cryostating strategy is one of the main challenge in designing high power protonlinacs: short modular cryomodule or long cryo-line to reduce static losses. 44/53




FocusingConclusion


In a nutshell!

The energy gain in a cavity is given by:

∆W = qVacc cosφS (69)

The linac is composed of a few families of cavities effective in a certain range ofenergy.

Modern linacs (high repetition rate and high accelerating gradients) are almostfully superconducting.

Cavities not only accelerate but also provide longitidinal focusing.




FocusingConclusion

Sources of defocusingMagnetic quadrupoles

Outline

1 Introduction





6 Conclusion




FocusingConclusion


RF defocusing

Using Gauss’law:∇ · E = 0 (70)

we can show that any change in the longidutinal field results in a radial field.Because the field in rising in time in a proton linac, RF defocusing occurs. Radial kickin a cavity can be expressed by the average change in radial mechanical momentum,∆Pr as:

∆Pr = −πqVaccr sinφS

m0c2β20γ

20λRF

(71)

Note that the change is positive since φs < 0 for acceleration.Transverse RF defocusing disappears at relativistic velocities.




FocusingConclusion


Space charge forceUsing the Lorentz force acting on a particle of charge q and velocity v:

F = q(E + v× B) (72)

we can show that:dvdt

=qEγ3m0

(73)

which becomes (with ′ = d/dz):

x′′ =qE

β2γ3m0c2(74)

In absence of external fields, the electric field seen by a particule j at position xj is thecontribution of all the electric fields created by the particles in the bunch:

E(xj) =N∑

i=1

Ei→j (xj) i 6= j (75)

The self-force or space charge force is significant for high intensity low energybeams.Attractive magnetic forces cancel the electric defocusing forces at retativisticvelocities.




FocusingConclusion


Tune depression

Figure: Beam in non linear forces.

We have already seen in Section 2 that nonlinear forcescause the beam to undergo filamentation since theangular velocity in phase space, phase advance, is afunction of the particle position. For any particle, itsphase advance σp must obey:

σcore ≤ σp ≤ σ0 (76)

with σcore and σ0 the phase advance in the beam coreand the phase advance at zero current (for a particle farfrom the core experiencing only external fields)respectively.Tune depression, η (RMS), expressed by:

η =σ

σ0< 1 (77)

is then a tool to measure the space charge strength.It can be as low as 0.5 for very intense beams.




FocusingConclusion


1 RF cavities provide longitudinal focusing.2 We need to produce a radial force to conteract transverse defocusing forces and

maintain the beam particles close to the reference axis.

Design

Due to the velocity dependance of thedefocusing forces, the length of the focusingperiod changes along the linac. Shortperiods at the beginning to longer periods athigh energy.

How to provide the transverse focusing?

Electric focusing: only for sourceextraction and possible up to a fewMeV.

Magnetic solenoid: for β < 0.4 (exceptSC solenoids).

Focusing by magnetic quadrupole is byfar the most common.

Nota: Electric field between electrodes are limited technologically to 10 MV/m(discharge arcs) while electromagnets can produce 2T before iron saturation. Sincethe efficiency of the electric to the magnetic force is vB/E , we need 60 MV/m to equalthe magnetic force at β = 0.1 for protons!




FocusingConclusion


Forces in magnetic quadrupoles

Figure: Magnetic qudrupoles at J-PARClinac.

In an ideal quadrupole lense, magneticgradient, G, is constant. At first order, wehave:

G =∂B∂x

=∂B∂y

=Bpole

R(78)

with Bpole the field on pole tip and R the poletip radius. This gives rise to the followingforces:

Fx = −q vo G x (79)

Fy = q vo G y (80)

linear in x and y .If G > 0 the quadrupole is focusing in thehorizontal plane and defocusing in thevertical plane. Opposite polarities wouldhave given G < 0.Quadrupole lenses are arranged in doubletto produce overall focusing.




FocusingConclusion

Outline

1 Introduction





6 Conclusion




FocusingConclusion

Conclusion

We have introduced some useful tools and concepts to describe thebeam dynamics:

RMS emittancesynchronous phase and synchronismphase advance and tune depression

We have also shown that SRF technology can provide high efficiencypower transmission to the beam.Nevertheless there are issues concerning the design of high intensityproton beams with linacs for high energy applications. Thechallenges are:

space charge at low energyminimization of beam losses for maintainabilitychoice of cryostating strategycost


high power proton linac basics - … · aurélien ponton hpp linacs. introduction beam emittance...

Documents