USPAS Accelerator Physics 2017 University of...

1 T. Satogata / January 2017 USPAS Accelerator Physics

USPAS Accelerator Physics 2017 University of California, Davis

Chapter 9: RF Cavities and RF Linear Accelerators

Todd Satogata (Jefferson Lab) / [email protected] Randika Gamage (ODU) / [email protected]

http://www.toddsatogata.net/2017-USPAS

Happy Birthday to Hideki Yukawa, Paul Langevin, Rutger Hauer, and Gertrude Elion! Happy Rhubarb Pie Day, National Handwriting Day, and Measure Your Feet Day!


Vertical Dispersion Correction

§  The end of section 6.10 in CM notes how sextupoles at locations of horizontal dispersion compensate both horizontal and vertical chromaticity

§  Normal sextupoles: §  Horizontal dispersion:

§  First order in x, δ:

⇠x,N = � 1

4⇡QH

I�x(s)k0(s) ds CM 6.126

k(s) ⌘ B

0

(B⇢)�x

0 = [k(s)ds] x = (kL) x =x

f

⇡ [k0(s)ds] x(1� �)

b2(s) ⌘B

00

(B⇢)�x

0 ⇡ [b2(s)ds](x2 � y

2)(1� �)

x ! x+ ⌘

x

(s)�

�x

0 ⇡ b2 ds[x2 + 2⌘

x

(s)� x+ �

2](1� �) ⇡ 2[b2 ds ⌘x(s)�]x


Vertical Dispersion Correction

§  Vertically for quads, sign flips and x goes to y

§  Sextupole effect is coupled in vertical plane, with horz dispersion:

k(s) ⌘ B

0

(B⇢)�x

0 = [k(s)ds] x = (kL) x =x

f

⇡ [k0(s)ds] x(1� �)

⇠x,N = � 1

4⇡QH

I�x(s)k0(s) ds ⇠x(sext) =

+1

4⇡QH

I�x(s)b2(s)⌘x(s) ds

⇠y,N =+1

4⇡QV

I�y(s)k0(s) ds

Quad

Sext

CM 6.127b

�x

0 ⇡ b2 ds[x2 + 2⌘

x

(s)� x+ �

2](1� �) ⇡ 2[b2 ds ⌘x(s)�]x

�y

0 ⇡ [b2(s)ds](2xy)(1� �) = [b2(s)ds][2(x+ ⌘

x

(s)�)y](1� �)

⇠y

(sext) = � 1

4⇡QV

I�y

(s)[2b2(s)⌘x(s)] ds


RHIC FODO Cell

Horizontal

Vertical

Horizontal dispersion

dipole dipole

quadrupole

half quadrupole

half quadrupole

H sext

V sext


RF Concepts and Design

§  Much of RF is really a review of graduate-level E&M §  See, e.g., J.D. Jackson, “Classical Electrodynamics” §  The beginning of this lecture is hopefully review

•  But it’s still important so we’ll go through it •  Includes some comments about electromagnetic polarization

§  We’ll get to interesting applications later in the lecture •  (or tomorrow)

§  Particularly important are cylindrical waveguides and cylindrical RF cavities

•  Will find transverse boundary conditions are typically roots of Bessel functions

•  TM (transverse magnetic) and TE (transverse electric) modes •  RF concepts (shunt impedance, quality factor, resistive losses)


Electric charge density

Electric current density

9.1: Maxwell’s Equations


Constitutive Relations and Ohm’s Law

⇥D = � ⇥E = �r�0 ⇥E �0 = 8.85� 10�12 C

N ·m2

⇥B = µ ⇥H = µrµ0⇥H µ0 = 4� � 10�7 N · s2

C2

�0µ0 =1

c2

⇥J = � ⇥E

� : conductivity

~D : Electric flux density

~E : Electric field density

" : Permittivity

~H : Magnetic field density

~B : Magnetic flux density

µ : Permeability

µ0 = 4⇡ ⇥ 10�7N · s2

C2⇡ 1.257

cm ·GA


Boundary Conditions

§  Boundary conditions on fields at the surface between two media depend on the surface charge and current densities:

§  and are continuous

§  changes by the surface charge density (scalar)

§  changes by the surface current density

§  C&M Chapter 9: no dielectric or magnetic materials

Ek B?

D? ⇢s

Hk �Js

n⇥ ( �H1 � �H2) = �Js

µ = µ0 � = �0


Wave Equations and Symmetry

§  Taking the curl of each curl equation and using the identity

then gives us two identical wave equations These linear wave equations reflect the deep symmetry between electric and magnetic fields. Harmonic solutions:

�⇤⇥ (�⇤⇥ �E) = �⇤(�⇤ · �E)�⇤2 �E = �⇤2 �E

r2 ⌅E = µ⇥⇤ ⌅E

⇤t+ µ�

⇤2 ⌅E

⇤t2r2 ⌅H = µ⇥

⇤ ⌅H

⇤t+ µ�

⇤2 ⌅H

⇤t2

��(�r, t) = �(�r)e�i�t for �� = �E, �H

) r2⇥ = �2⇥ where �2 = (�µ�⇤2 + iµ⇥⇤)


Conductivity and Skin Depth

§  For high conductivity, , charges move freely enough to keep electric field lines perpendicular to surface (RF oscillations are adiabatic vs movement of charges) §  Copper: §  Conductivity condition holds for very high frequencies (1015 Hz)

§  For this condition

§  Inside the conductor, the fields drop off exponentially

� � !✏

� ⇡ 6⇥ 107 ��1 m�1

� ⇡r

µ⇥�

2(1 + i)

�(z) = �(0) e�z/�

skin depth � ⌘r

2

µ⇤⇥

e.g. for Copper


Surface Resistance and Power Losses

§  There is still non-zero power loss for finite resistivity §  Surface resistance: resistance to current flow per unit area

§  This isn’t just applicable to RF cavities, but to transmission lines, power lines, waveguides, etc – anywhere that electromagnetic fields are interacting with a resistive media

§  Deriving the power loss is part of your homework! §  We’ll talk more about transmission lines and waveguides

after one brief clarification…

surface resistance Rs ⌘r

µ⇤

2⇥=

1

⇥�

surface power loss �Ploss

⇥ = Rs

2

Z

S

|Hk|2 dS


Anomalous Skin Effect §  Conductivity really depends on the frequency and mean

time between electron interactions (or inverse temperature)

§  Classical limit is , adiabatic field wrt electron interactions §  For non-classical limit, no longer applies since electrons

see changing fields between single interactions

!⌧

�(⇤) =�0

(1 + i⇤⇥)!⌧ ! 0

⇥J = � ⇥E

Normal skin effect

Anomalous skin effect

Asymptotic value

Skin depth

Mean free path

Reuter and Sondheimer, Proc. Roy. Soc. A195 (1948), from Calatroni SRF’11 Electrodynamics Tutorial


Plane Wave Properties

The source-free divergence Maxwell equations imply

so the fields are both transverse to the direction (This will be the direction in a lot of what’s to come) Faraday’s law also implies that both fields are spatially transverse to each other

[r2 + µ�⇥2] ⇤E = 0 ) ⇤E(⇤x, t) = ⇤E0 ei⇥k·⇥x�i�t

[r2 + µ�⇥2] ⇤B = 0 ) ⇤B(⇤x, t) = ⇤B0 ei⇥k·⇥x�i�t

�k · �E0 = �k · �B0 = 0�k

⇥B0 =⇥k ⇥ ⇥E0

�=

nn⇥ ⇥E0

c

z

k =2⇥

�rf(wave number)

vph =

�

k(phase velocity)

vg =

d�

dk(group velocity)

n ⌘�k

kindex of refraction

Generally F (z, t) =

Z 1

�1A(�)ei(�t�k(�)z)d�


Standing Waves

Note that Maxwell’s equations are linear, so any linear combination of magnetic/electric fields is also a solution. Thus a standing wave solution is also acceptable, where there are two plane waves moving in opposite directions:

[r2 + µ�⇥2] ⇤E = 0 ) ⇤E(⇤x, t) = ⇤E0 ei⇥k·⇥x�i�t

[r2 + µ�⇥2] ⇤B = 0 ) ⇤B(⇤x, t) = ⇤B0 ei⇥k·⇥x�i�t

�E(�x, t) = �E0

⇣ei⇥k·⇥x�i�t + e�i⇥k·⇥x�i�t

⌘

�B(�x, t) = �B0

⇣ei⇥k·⇥x�i�t + e�i⇥k·⇥x�i�t

⌘


Polarization

§  As long as the and fields are transverse, they can still have different transverse components. §  So our description of these fields is also incomplete until we

specify the transverse components at all locations in space §  This is equivalent to an uncertainty in phase of rotation of

and around the wave vector . §  The identification of this transverse field coordinate basis

defines the polarization of the field.

[r2 + µ�⇥2] ⇤E = 0 ) ⇤E(⇤x, t) = ⇤E0 ei⇥k·⇥x�i�t

[r2 + µ�⇥2] ⇤B = 0 ) ⇤B(⇤x, t) = ⇤B0 ei⇥k·⇥x�i�t

�E �B

�B

�E�k


Linear Polarization

§  So, for example, if and transverse directions are constant and do not change through the plane wave, the wave is said to be linearly polarized.

[r2 + µ�⇥2] ⇤E = 0 ) ⇤E(⇤x, t) = ⇤E0 ei⇥k·⇥x�i�t

[r2 + µ�⇥2] ⇤B = 0 ) ⇤B(⇤x, t) = ⇤B0 ei⇥k·⇥x�i�t

�E �B

Fields each stay in one plane

E/B field oscillations are IN phase


Circular Polarization

§  If and transverse directions vary with time, they can appear as two plane waves traveling out of phase. This phase difference is 90 degrees for circular polarization.

�E �B

Two equal-amplitude linearly polarized plane waves 90 degrees out of phase

�E(�x, t) = �E0

⇣ei⇤k·⇤x�i⇥t + ei

⇤k·⇤x�i⇥t+�/2⌘

�B(�x, t) = �B0

⇣ei⇤k·⇤x�i⇥t + ei

⇤k·⇤x�i⇥t+�/2⌘


9.2: Cylindrical Waveguides

§  Consider a cylindrical waveguide, radius a, in z direction

§  can be imaginary for attenuation down the guide §  We’ll find constraints on this cutoff wave number §  Maxwell in cylindrical coordinates (math happens) gives

⇥E = ⇥E(r, �)ei(�t�kgz)

kg

k2c ⌘ k2 � k2g

k ⌘⇣�c

⌘(free space wave number)

(and the same for Hz)


Transverse Electromagnetic (TEM) Modes

§  Various subsets of solutions are interesting §  For example, the nontrivial solutions require

§  The wave number of the guide matches that of free space §  Wave propagation is similar to that of free space §  This is a TEM (transverse electromagnetic) mode

•  Requires multiple separate conductors for separate potentials or free space

•  Sometimes similar to polarization pictures we had before

Ez, Hz = 0

k2c = k2 � k2g = 0

Coaxial cable


Transverse Magnetic (TM) Modes

§  Maxwell’s equations are linear so superposition applies §  We can break all fields down to TE and TM modes §  TM: Transverse magnetic (Hz=0, Ez≠0) §  TE: Transverse electric (Ez=0, Hz≠0) §  TM provides particle energy gain or loss in z direction!

§  Maxwell’s equations in cylindrical coordinates give

Separate variables Ez(r, �) = R(r)�(�)

Boundary conditions Ez(r = a) = E�(r = a) = 0

d2R

dr2+

1

r

dR

dr+

✓k2c �

n2

r2

◆R = 0

d2�

d�2+ n2� = 0

Bessel equation

SHO equation => n integer, real


TM Mode Solutions

§  The radial equation has solutions of Bessel functions of first Jn(kcr) and second Nn(kcr) kind §  Toss Nn since they diverge at r=0 §  Boundary conditions at r=a require Jn(kca)=0 §  This gives a constraint on kc

§  This mode of this field is commonly known as the TMnj mode §  The first index corresponds to theta periodicity, while the

second corresponds to number of radial Bessel nodes §  TM01 is the usual fundamental accelerating mode

kc = Xnj/a where Xnj is the j

thnonzero root of Jn

) Ez(r, �, t) = (C1 cosn� + C2 sinn�)Jn

✓Xnj

ar

◆e

i(�t�kgz)


TM Mode Visualizations

§  Java app visualizations are also linked to the class website §  For example, http://www.falstad.com/emwave2 (TM visualization)

•  Scroll down to Circular Modes 1 or 2 in first pulldown

) Ez(r, �, t) = (C1 cosn� + C2 sinn�)Jn

✓Xnj

ar

◆e

i(�t�kgz)


Cavity Modes

Dave McGinnis, FNAL


TM020 Mode

R

L

Ez ⇠ J0⇣u02

r

R

⌘

Beam

B✓ ⇠ �J1⇣u02

r

R

⌘

Peggs/Satogata


TM110 Mode

L

R

Beam

(a) (b)

Tail, Q = Ne/2 Head, Q = Ne/2

~B

~F on tail

~B(r, ✓) excited by head

Ez interacting with head

Peggs/Satogata


Cutoff Frequency

§  It’s easy to see that there is a minimum frequency for our waveguide that obeys the boundary conditions §  Modes at lower frequencies will suffer resistance §  Dispersion relation in terms of radial boundary condition zero

§  A propagating wave must have real group velocity §  This gives the expected lower bound on the frequency

§  Wavelength of lowest cutoff frequency for j=1 is

k2 � k2g =⇣�c

⌘2� k2g = k2c =

✓Xnj

a

◆2

k2g > 0

�c =2⇥

kc⇡ 2.61a

�

c� Xnj

a) cuto� frequency �c =

cXnj

a


Dispersion or Brillouin Diagram

Propagating frequencies � > �c

�ph > 45�, vph > c

•  Circular waveguides above cutoff have phase velocity >c •  Cannot easily be used for particle acceleration over long distances


Iris-Loaded Waveguides

§  Solution: Add impedance by varying cylinder radius §  This changes the dispersion condition by loading the waveguide


Cylindrical RF Cavities

§  What happens if we close two ends of a waveguide? §  With the correct length corresponding to the longitudinal

wavelength, we can produce longitudinal standing waves §  Can produce long-standing waves over a full linac §  Modes have another index for longitudinal periodicity

Simplified “tuna fish can” RF cavity model

TM010 mode


Boundary Conditions

§  Additional boundary conditions at end-cap conductors at z=0 and z=l


TEnmj Modes

fnmj =c

2�

s✓X 0

nj

a

◆2

+⇣m�

l

⌘2

Longitudinal periodicity

X 0nj is j

throot of J 0

n

Ez = 0 everywhere


TMnmj Modes

Hz = 0 everywhere

Longitudinal periodicity

fnmj =c

2�

s✓Xnj

a

◆2

+⇣m�

l

⌘2

Jn(kca) = 0 Xnj is the j

throot of Jn

Ez(r, �, z) ⇠ Jn(kcr)(C1 cosn� + C2 sinn�) cos⇣m⇥z

l

⌘

TM010 is (again) the preferred acceleration mode


Equivalent Circuit

§  A TM010 cavity looks much like a lumped LRC circuit §  Ends are capacitive §  Stored magnetic energy is inductive §  Currents move over resistive walls

§  E, H fields 90 degrees out of phase

§  Stored energy

C&M 9.82


TM010 Average Power Loss

ends

sides

§  Compare to stored energy §  Both vary like square of field, square of Bessel root §  Dimensional area vs dimensional volume in stored energy

U =�02E2

0⇥a2l[J1(X01)]

2


Quality Factor

§  Ratio of average stored energy to average power lost (or energy dissipated) during one RF cycle §  How many cycles does it take to dissipate its energy? §  High Q: nondissipative resonator §  Copper RF: Q~103 to 106

§  SRF: Q up to 1011

Q =U�

hPloss

i

U(t) = U0e��0t/Q

Pillbox cavity Q =

al

�(a+ l)


Niobium Cavity SRF “Q Slope” Problem

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