February 5, 2004 S. A. Pande - CAT-KEK School on SNS1 Radio Frequency Quadrupole (RFQ) S. A. Pande

February 5, 2004 S. A. Pande - CAT-KEK School on SNS

1

Radio Frequency Quadrupole

(RFQ)

S. A. Pande


2

Introduction The first linac was built in 1928 by

Widröe~

1MHz25 kV

d=/2

50 keV

K+ ionsK+ ions


3

The Sloan Lawrence Structure E. O. Lawrence in association with Sloan

built an improved version of Widröe’s linac

They used an array of 30 DTs excited by a 42 kV, 7 MHz oscillator to accelerate Hg ions to 1.26 MeV.

RFQ is also a Sloan-Lawrence kind of accelerator in which the successive accelerating gaps are /2 apart.


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The RFQ It was first proposed by I. Kapchinskii

and V. Teplyakov from ITEP Moscow for heavy ions.

The first RFQ was built and tested at LANL to get 2 MeV protons.

Though invented in the last, the RFQ forms the first accelerator in a chain of heavy ion (including proton) accelerators in recent times.


5

Before 80s almost all of the accelerator facilities for protons and heavy ions, invariably used DC accelerators from few 100 keVs to few MeVs as injectors for linear accelerators which in turn formed the main injectors for the bigger circular machines or acted as sources of charged particle beams.

The DC accelerators have certain inherent limitations and difficulties associated with handling of high voltages.

The beam has to be bunched before injecting into the linac in order to avoid energy spread in the out coming beam and also to avoid the loss of particles.


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There was another severe problem associated with the focusing of the beams. The defocusing due to space charge is more severe in the low energy beams.

The invention of RFQ, the low energy high current accelerator, helped in overcoming all the difficulties we have seen above.

The RFQ simultaneously• Focuses• Bunches and • Accelerates the beam

This avoided the need for large DC accelerators and avoided the problems to great extent.

Almost all of the DC accelerators were later replaced by RFQ after its invention.


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Principle of OperationAs its name suggests, the RFQ provides electric quadrupole focusing with the electric field oscillating at Radio Frequency

-1/2Vcos(t)

-1/2Vcos(t)

1/2Vcos(t) 1/2Vcos(t)

Four equispaced conducting electrodes with alternating polarity as we move from one electrode to the next forms the electric quadrupole.

Voltage 1/2V0cos(t) is applied in quadru-polar symmetry

The electric Quadrupole


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The off axis particles will experience a transverse force which is alternating in time and this transverse force provides ‘Alternating Gradient’ focusing.

The advantage of RFQ is that it provides electric focusing for low velocity particles which is stronger than conventional magnetic focusing.

A structure with uniform electrodes along its length will have no component of electric field along the axis and thus will not work as an accelerator.

To generate an axial electric field component, the quadrupole electrodes are modulated longitudinally. One pair of electrodes is shifted longitudinally wrt the other pair by 180 so that when the distance from the axis of vertical vanes is at its minimum ‘a’, the horizontal vanes will be maximum apart at ‘ma’.


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Modulation

Modulation of electrodes to generate longitudinal field component

ama

A´

A

Cross section through AA´

/2One unit

cell

ama

x

z

Beam axis

m 1


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The axial electric field component is generated due to the potential difference between the point of minimum separation from axis of vertical vanes (or horizontal vanes) and the point of minimum separation from the axis of the horizontal vane (or vertical vane).

In RFQ, the field in successive gaps is in opposite direction and therefore when it is accelerating in one cell, it is decelerating in the next.

There are two unit cells per structure period. At a given time every alternate cell will have a particle bunch.


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The general potential function In RFQ the electrodes in the form of ‘rods’ or

‘vanes’ are placed in cavity resonators to prevent the RF fields from radiating.

The issues related to the electrodynamics are distinct from those associated with the beam dynamics. The beam dynamics is confined to a region of small radius near axis as compared to the cavity radius which is proportional to the wavelength.

Due to the symmetry property the magnetic field is zero on the axis and also for the region r<<.


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The consequences are- The wave equation in this region can be replaced

by Laplace equation The vanes present well defined boundaries with

a potential from which we can analytically derive the fields or

We can ask for specific fields and then determine the corresponding vane boundaries.

Starting with the Laplace equation in cylin. Coordinates

Where U(r,,z) electric field potential.

011

),,(2

2

2

2

22

z

UU

rr

Ur

rrzrU


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Solving the above equation by the method of separation of variables, we obtain

This is the general K-T potential function a doubly infinite terms.K-T considered only the lowest order terms and proposed to construct the electrode shapes that conform to the resulting equipotential surface.Retaining only s=0 from the first and s=0, n=1 terms from the second summation, we have

)sin()2cos()(

])12(2cos[),,(

201

)12(2

0

nkzsnkrIA

srAzrU

snssn

ss

s


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The two term potential

Retaining only s=0 from the first and s=0, n=1 terms from the second summation, we have

where k=2/; =velocity of synchronous particleand I is the modified Bessel function.

The potential given by this equation is known as ‘Two Term Potential’ and the dynamics in the RFQ is studied with this potential function.

kzkrIArAzrU cos)(2cos),,( 0102

0


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By assuming the horizontal and vertical vanes at +V0/2 and –V0/2 respectively and putting the boundary conditions at the vane tips, we have

We define two dimensionless quantities

)()(

)()(

2 002

0020

0 mkaIkaIm

mkaIkaI

a

VA

)()(

1

2 002

20

10 mkaIkaIm

mVA

)()(

)()(

002

00

mkaIkaIm

mkaIkaIX

)()(

1

002

2

mkaIkaIm

mA


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With these two dimensionless quantities, A0=XV0/2a2 and A10=AV0/2, the two term time dependent potential is written as

---------------- ---------------I II

The first term gives the potential of an electric quadrupole and the second term gives the accelerating potential.The quantities X and A are known as focusing parameter and acceleration parameter respectively.From the defining equations of X and A we can write

X = 1 – AI0(ka)

)sin(]cos)(2cos)/([2

),,( 020 tkzkrAIarX

VzrU


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By rearranging the last equation, we can write XV + AI0(ka)V = V

This tells us that the inter-vane voltage V is composed of a part required for focusing (XV) and another required for acceleration (AI0(ka))

Similarly, if we put m=1 in the last equation, the vanes are unmodulated and the acceleration parameter goes to zero.

A = 0 for m = 1The RFQ will be just a focusing device.As m increases the acceleration parameter increases and the focusing parameter X decreases.


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The Field ComponentsThe field components are derived from the potential functionEr = - U/r = -V0/2[2(X/a2)rcos2-kAI1(kr)coskz]

E = -(1/r) U/=(XV/a2)rsin2Ez = - U/z=(kAV/2)I0(kr)sinkz

I1 is the modified Bessel function of first orderThe first term in Er and E is the quadrupole focusing fieldThe second term of Er is the gap defocusing term which applies a radial defocusing impulseSince I1(kr)kr/2, the radial impulse is proportional to the displacement from the axis.


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Voltage and energy gain across a unit cell

The voltage across a unit cell can be calculated by

where we have used Ez as defined earlier and Lc=/2The energy gain is given by

W=qeAVTcoss

For RFQ the transit time factor isT=/4

Lc

zcell AVdzEV0


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The Vane tip profiles

With time dependent voltages on horizontal & vertical electrodes as +V/2sin(t+) and –V/2sin(t+) and expressing the two term potential in cartesian coordinates by substituting x=rcos and y=rsin, we have

U(x,y,z,t)=V0/2[X/a2(x2-y2)+AI0(kr)coskz]with U=V/2, we have for the geometry of the vane surface

1=X/a2(x2-y2)+AI0(kr)coskz

Or x2-y2=a2/X(1-AI0(kr)coskz)The transverse cross sections are hyperbolas


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The ideal vane tip profile

The hyperbolic vane tip profiles


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But for the ease of machining, and also to control the peak surface electric field, the electrode contours deviate from the ideal hyperbolas.A combination of circular arcs and straight lines is used At the cell centre, i.e. at z=/4The RFQ has exact quadrupolar symmetryThe x and y tips of the electrode are equidistant from the axis (or have radius r0) given by .

r02=a2/X

r0=aX-1/2

This is known as the average radius of the RFQ.The focusing strength of a modulated structure is equivalent to that of an unmodulated structure with radius r0.


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The actual vane tip profiles

The vertical vane

The horizontal vane

One quadrant of RFQ

r0


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Characteristics of RFQ Adiabatic Capture and Bunching

Ion source provides a DC beam and thus is injected uniformly from - to over one period.

W=~0 and =360The RFQ can capture almost all the beam injected and bunch it slowly.In the initial part of RFQ there is no acceleration.The longitudinal electric field which is proportional to AV, is slowly increased by increasing m – the modulation parameter. This provides bunching.


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Characteristics of RFQ (Contd.)

Many cells are devoted to this part in an RFQ. This will not be economical in other linac structures. In RFQ, the cells are very short and many cells can be accommodated in a relatively shorter length. Thus RFQ provides adiabatic capture and bunching.

The synchronous phase is kept initially at -90 where we have maximum longitudinal focusing and no acceleration (i.e. the synchronous particle will have no acceleration).Once some rough bunching is achieved, the synchronous phase (s) and m are slowly increased further to impart energy and the bunch slowly becomes well defined.


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The complete RFQThe first RFQ was built at LANL. They divided the whole RFQ in 4 parts.1. Radial Matching Section (RMS)2. Shaper (Sh)3. Gentle Buncher (GB) and4. Accelerator (Acc)1. Radial Matching Section (RMS)Matches the input DC beam to the strong transverse focusing structure of the RF quadrupole. In this section m=1, no Ez no acceleration, few cells ~5.


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2. Shaper (Sh)This is a short section which starts the bunching process. This section smoothly joins the RMS where A=0 and s=-90 to the gentle buncher where A>0 and s>-90. This initiates the bunching process. 3. Gentle Buncher (GB)The GB adiabatically bunches the beam and also slowly accelerates to some intermediate energy. Being adiabatic, it forms the major part of the RFQ structure. s and m are increased ultimately to match those in the accelerator part.4. Accelerator (Acc)In this part the major emphasis is on the acceleration at a faster rate. s and m reach their ultimate values. s ~ -30 and m ~ 1.5 – 2.5.


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Vane Tip profile for first 50 cm

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 5 10 15 20 25 30 35 40 45 50

axial distance (cm)

van

e ti

p d

ista

nce

fro

m a

xis

(cm

)

RMS SHAPER


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Vane Tip profile for last 50 cm

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

495 505 515 525 535 545

axial distance (cm)

van

e ti

p d

ista

nce

fro

m a

xis

(cm

)

Longitudinal profile of the vane tip in 4.5 MeV 50 mA RFQ

Accelerator


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The RFQ Cavity or ResonatorWhatever we discussed was the story in the vicinity of the axis where the beam passes through.Let us see how we can generate these fields electro-magnetically.Two types of structures are most commonly used1. The four rod structure and2. The four vane structure3. Split Co-axial cavity is used at few places for heavy ion acceleration.We will study the first two.


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The four vane structure

TE21 mode in circular cylindrical waveguide

We introduce the vanes

The quadrupole field

concentrates near the vane

tipsVanes divide the waveguide in 4

quadrants


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On Quadrant of the RFQ showing electric field lines of quadrupole mode


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• The vanes concentrate the electric field near the axis providing strong quadrupole focusing field

• Magnetic field which is longitudinal is localized in outer part of the quadrant

• The vane to vane capacitance reduces the cutoff frequency of the waveguide or the resonant frequency of the cavity. To compensate this the waveguide diameter can be reduced

• The four vane cavity is obtained by shorting the two ends by conducting plates

• The boundary condition on each conducting end plate is Etangential=0

• This shows a true TE210 mode cannot exist in cylindrical cavity with metallic end walls. Instead the mode will be TE211


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TE211 and TE210 Modes Resulting

Field due to TE211 mode the last subscript denotes the no. of half wavelength variations in z direction

Desired

Field due to TE210 mode zero last subscript denotes that there is no variation in the longitudinal direction.E

Z


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Therefore gaps are provided between the end wall and the vane endsThis produces longitudinally uniform field throughout the interior of the cavityEtransverse is localized near the vane tips

Hlongitudinal is localized in outer part of the quadrants

vane

VANE

Side viewTop viewCross section through

RFQ at an end


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Eigen modes of a 4 vane cavity There is one more important mode in the 4 vane cavity slightly below in frequency of the quad mode.This is the dipole mode denoted by TE11nThe field pattern for quad and dipole modes are shown below

x

x x x

Quadrupole Dipole-1 Dipole-2


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Dipole modes are degenerate modesWhen a dipole mode is excited, a small potential difference appears across the the opposite vanes where as for the quad mode the opposite modes are exactly at the same potential. If these modes are close to the quadrupole mode, the transverse as well as longitudinal field will be perturbed and the performance will be affected. Therefore the dipole modes should be tuned away from the quadrupole mode.It may happen that the frequency of a higher order dipole mode may fall very close to the quad mode.


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335

340

345

350

355

360

365

370

375

0 1 2 3 4 5 6 7

Longitudinal mode no. n

Fre

qu

en

cy

(MH

z)

Dipole Quad

The longitudinal mode spectrum of a 4 vane RFQ cavity


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The field stabilization The perturbation caused due to the dipole or

othe modes can result in unflat field distribution along the RFQ structure.

Due to highly sensitive nature of the RFQ cavity, the machining and tuning errors can also result in dipole mode excitation.

Many proposals have been made at many places. Most successful are the ‘Vane Coupling Rings’ introduced at LBNL and Pi mode Stabilizing Loops (PISL) proposed at KEK.


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Vane Coupling Rings (VCR) The opposite vanes

are shorted together forcing them to the same potential.

The dipole modes are shifted away.

3 pairs of VCRs are used in structures of 1-2 m in length

Difficult to mount and cooling is a problem


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PISLPrinciple

The total magnetic flux normal to the surface surrounded by a closed conducting loop is zero.The dipole mode fields will be perturbed more and thus shifted away.

Dipole mode

x

Quad mode


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February 5, 2004 S. A. Pande - CAT-KEK School on SNS1 Radio Frequency Quadrupole (RFQ) S. A. Pande