RF FUNDAMENTALS CAVITY FUNDAMENTALS - …casa.jlab.org/publications/viewgraphs/USPAS2011/TAAD...

Jean Delayen

Center for Accelerator Science

Old Dominion University

and

Thomas Jefferson National Accelerator Facility

RF FUNDAMENTALS

CAVITY FUNDAMENTALS

ODU TAAD I Fall 2012

RF Cavity

• Mode transformer (TEM→TM)

• Impedance transformer (Low Z→High Z)

• Space enclosed by conducting walls that can sustain an

infinite number of resonant electromagnetic modes

• Shape is selected so that a particular mode can

efficiently transfer its energy to a charged particle

• An isolated mode can be modeled by an LRC circuit

RF Cavity

Lorentz force

An accelerating cavity needs to provide an electric field E

longitudinal with the velocity of the particle

Magnetic fields provide deflection but no acceleration

DC electric fields can provide energies of only a few MeV

Higher energies can be obtained only by transfer of energy from

traveling waves →resonant circuits

Transfer of energy from a wave to a particle is efficient only is

both propagate at the same velocity

( )F q E v B= + ´

Equivalent Circuit for an rf Cavity

Simple LC circuit representing an accelerating resonator

Metamorphosis of the LC circuit into an accelerating cavity

Chain of weakly coupled pillbox

cavities representing an accelerating module

Chain of coupled pendula as its mechanical analogue

Electromagnetic Modes

Electromagnetic modes satisfy Maxwell equations

With the boundary conditions (assuming the walls are

made of a material of low surface resistance)

no tangential electric field

no normal magnetic field

22

2 2

10

E

c t H

ì üæ ö¶Ñ - =í ýç ÷¶è ø î þ

0

0

n E

n H

´ =

=

Electromagnetic Modes

Assume everything

For a given cavity geometry, Maxwell equations have an infinite

number of solutions with a sinusoidal time dependence

For efficient acceleration, choose a cavity geometry and a mode

where:

Electric field is along particle trajectory

Magnetic field is 0 along particle trajectory

Velocity of the electromagnetic field is matched to particle velocity

22

20

E

c H

w ì üæ öÑ + =í ýç ÷è ø î þ

i te w-

Accelerating Field (gradient)

Voltage gained by a particle divided by a reference length

For velocity-of-light particles

For less-than-velocity-of-light cavities, there is no

universally adopted definition of the reference length

1( )cos( / )zE E z z c dz

Lw b= ò

2

NL

l=

Design Considerations

,max

,max

2

2

2

2

2

minimum critical field

minimum field emission

minimum shunt impedance, current losses

minimum dielectric losses

minimum control of microphonics

maximum

s

acc

s

acc

s

acc

s

acc

acc

H

E

E

E

H

E

E

E

U

E

< >

< >

voltage drop for high charge per bunch

Energy Content

Energy density in electromagnetic field:

Because of the sinusoidal time dependence and the 90º

phase shift, he energy oscillates back and forth between

the electric and magnetic field

Total energy content in the cavity:

( )2 2

0 0

1

2u e m= +E H

2 20 0

2 2V VU dV dV

e m= =ò òE H

Power Dissipation

Power dissipation per unit area

Total power dissipation in the cavity walls

2 20

4 2

sRdP

da

m wd= =H H

2

2

s

A

RP da= ò H

Quality Factor

Quality Factor Q0:

00

00 0

0

Energy stored in cavity

Energy dissipated in cavity walls per radian diss

UQ

P

w

ww t

w

º =

= =D

2

00 2

V

s

A

dVQ

R da

wm=

ò

ò

H

H

Geometrical Factor

Geometrical Factor QRs (Ω)

Product of the Quality Factor and the surface resistance

Independent of size and material

Depends only on shape of cavity and electromagnetic mode

2 2 2

00 2 2 2

0

1 22

377 Impedance of vacuum

V V Vs

A A A

dV dV dVG QR

da da da

m phwm p

e l l

h

= = = =

» W

ò ò ò

ò ò ò

H H H

H H H

Shunt Impedance, R/Q

Shunt impedance Rsh:

Vc = accelerating voltage

Note: Sometimes the shunt impedance is defined as

or quoted as impedance per unit length (ohm/m)

R/Q (in Ω)

2

in csh

diss

VR

Pº W

2

2c

diss

V

P

2 2 2R V P E L

Q P U Uw w= =

Q – Geometrical Factor (Q Rs)

32 20 0

0

0 0

2 2 2

0

0

0

2 2 1 1

2 2 2 2 6

1 1 1

2 2 2

2006

Energy contentQ:

Energy disspated during one radian

Rough estimate (factor of 2) for fundamental mode

is si

s s

s

s

U

P

c LU H dv H

L

P R H dA R H L

QR

G QR

ww wt

w

m mp p pw

l e m

p

mp

e

= = =D

= =

= =

~ = W

=

ò

ò

275

ze (frequency) and material independent.

It depends only on the mode geometry

It is independent of number of cells

For superconducting elliptical cavities sQR W

Shunt Impedance (Rsh), Rsh Rs, R/Q

( )

2 22

2 2

0

2

1 1

2 2

33,000

/ 100

/

In practice for elliptical cavities

per cell

per cell

and

Independent of size (frequency) and material

Depends on mode geometr

zsh

s

sh s

sh

sh s sh

E LVR

PR H L

R R

R Q

R R R Q

p

=

W

W

y

Proportional to number of cells

Power Dissipated per Unit Length or Unit Area

2

12

1

2

1

2

2

2

1

For normal conductors

For superconductors

S

S

S

S

E RP

RLQR

Q

R

P

L

P

A

R

P

L

P

A

w

w

w

w

w

w

w

-

µ

µ

µ

µ

µ

µ

µ

External Coupling

• Consider a cavity connected to an rf source

• A coaxial cable carries power from an rf source to the cavity

• The strength of the input coupler is adjusted by changing the penetration of the center conductor

• There is a fixed output coupler, the transmitted power probe, which picks up power transmitted through the cavity. This is usually very weakly coupled

Cavity with External Coupling

Consider the rf cavity after the rf is turned off. Stored energy U satisfies the equation: Total power being lost, Ptot, is: Pe is the power leaking back out the input coupler. Pt is the power coming out the transmitted power coupler. Typically Pt is very small Ptot Pdiss + Pe Recall Similarly define a “loaded” quality factor QL: Now energy in the cavity decays exponentially with time constant:

tot diss e tP P P P= + +

0L

tot

UQ

P

wº

tot

dUP

dt= -

00

diss

UQ

P

wº

0

00

L

t

Q

L

UdUU U e

dt Q

ww -

= - Þ =

0

LL

Qt

w=


Equation

suggests that we can assign a quality factor to each loss mechanism, such that

where, by definition,

Typical values for CEBAF 7-cell cavities: Q0=1x1010, Qe QL=2x107.

0 0

tot diss eP P P

U Uw w

+=

0

1 1 1

L eQ Q Q= +

0

e

e

UQ

P

wº


• Define “coupling parameter”:

therefore

is equal to:

• It tells us how strongly the couplers interact with the cavity.

Large implies that the power leaking out of the coupler is

large compared to the power dissipated in the cavity walls.

0

e

Q

Qb º

0

1 (1 )

LQ Q

b+=

e

diss

P

Pb =

Several Loss Mechanisms

(

1 1

L

i

-wall losses

-power absorbed by beam

-coupling to outside world

Associate Q will each loss mechanism

index 0 is reserved for wall losses)

Loaded Q: Q

Coupling co

i

i

i

i

L

P P

UQ

P

P

Q U Q

w

w

=

=

= =

å

åå

0

0

0

1

efficient: ii

i

L

i

Q P

Q P

QQ

b

b

= =

=+å

Equivalent Circuit for an rf Cavity

Simple LC circuit representing

an accelerating resonator

Metamorphosis of the LC circuit

into an accelerating cavity

Chain of weakly coupled pillbox

cavities representing an accelerating

cavity

Chain of coupled pendula as

its mechanical analogue

Parallel Circuit Model of an Electromagnetic Mode

• Power dissipated in resistor R:

• Shunt impedance:

• Quality factor of resonator:

21

2

cdiss

VP

R=

2

csh

diss

VR

Pº 2shR RÞ =

1/2

0

0 0

diss c

U R CQ CR R

P L L

ww

w

æ öº = = = ç ÷è ø

1

0

0

0

1Z R iQww

w w

-

é ùæ ö= + -ê úç ÷è øë û

1

0

0 0

0

1 2Z R iQw w

w ww

-

é ùæ ö-» » +ê úç ÷è øë û

,

1-Port System

2 2 00 0

00

0

1 21 2

g

g

kV RI V kV

R Qk Z R k Z iQi

www

w

= =æ ö

+ + + Dç ÷è ø+ D

2

00

0

1 2

Total impedance: R

k ZQ

i ww

+

+ D

1-Port System

( )

( )

2 20

22 20

22

2 4 2 2

0 0 0

0

2

0

2

0 0

0

2 22

00

0

1 1

2 2

1

24

8

4 1

1 21

1

Energy content

Incident power:

Define coupling coefficient:

g

g

inc

inc

QU CV V

R

Q Rk V

RR k Z k Z Q

VP

Z

R

k Z

QU

P Q

w

w w

w

b

b

w b w

b w

= =

=æ öD

+ + ç ÷è ø

=

=

=+ æ öæ ö D

+ ç ÷ ç ÷+è ø è ø

1-Port System

( )

( )

2 22

00

0

2

4 1

1 21

1

0, 1 :

4 11

1 21

Power dissipated

Optimal coupling: maximum or

critical coupling

Reflected power

diss inc

diss inc

inc

ref inc diss mc

UP P

Q Q

UP P

P

P P P P

w b

b w

b w

w b

b

b

= =+ æ öæ ö D

+ ç ÷ ç ÷+è ø è ø

=

Þ D = =

= - = -+

+

2

0

01

Q w

b w

é ùê úê úê ú

æ öDê úç ÷ê ú+è øë û

1-Port System

( )

( )

0

2

0

2

2

4

1

4

1

1

1

At resonance

inc

diss inc

ref inc

QU P

P P

P P

b

w b

b

b

b

b

=+

=+

æ ö-= ç ÷+è ø

Dissipated and Reflected Power

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Equivalent Circuit for a Cavity with Beam

• Beam in the rf cavity is represented by a current generator.

• Equivalent circuit:

(1 )

shL

RR

b=

+

0

0

tan -21

produces with phase (detuning angle)

produces with phase

b b

g g

c g b

i V

i V

V V V

Q

y

y

wy

b w

= -

D=

+


1/21/2

0

0

2( ) cos

1

cos2(1 )

sin22

2

: beam rf current

: beam dc current

: beam bunch length

g g sh

b shb

b

bb

b

b

V P R

i RV

i i

i

i

by

b

yb

q

q

q

=+

=+

=


( ) [ ]{ }2

2211 (1 ) tan tan

4

c

g

sh

VP b b

Rb b y f

b= + + + + -

0 cosPower absorbed by the beam =

Power dissipated in the cavitysh

c

R ib

V

f=

2

(1 ) tan tan

1

1 (1 )

2

opt opt

opt

opt cg

sh

b

b

b bVP

R

b y f

b

+ =

= +

+ + +=

Minimize Pg :

Cavity with Beam and Microphonics

• The detuning is now

0 0

0 0

0

0tan 2 tan 2

where is the static detuning (controllable)

and is the random dynamic detuning (uncontrollable)

m

L L

m

Q Qdw dw dw

y yw w

dw

dw

±= - = -

-10

-8

-6

-4

-2

0

2

4

6

8

10

90 95 100 105 110 115 120

Time (sec)

Fre

qu

en

cy

(H

z)

Probability Density

Medium CM Prototype, Cavity #2, CW @ 6MV/m

400000 samples

0

0.05

0.1

0.15

0.2

0.25

-8 -6 -4 -2 0 2 4 6 8

Peak Frequency Deviation (V)

Pro

ba

bilit

y D

en

sit

y

Qext Optimization with Microphonics

2

2

0

0

22

2

0

0

( 1) 2

( 1) ( 1) 22

mopt

opt c mg

sh

b Q

VP b b Q

R

dwb

w

dw

w

æ ö= + + ç ÷è ø

é ùæ ö

ê ú= + + + + ç ÷ê úè øë û

Condition for optimum coupling:

and

In the absence of beam (b=0):

and

2

0

0

22

0

0

1 2

1 1 22

If is very large

mopt

opt c mg

sh

m m

Q

VP Q

R

U

dwb

w

dw

w

dw dw

æ ö= + ç ÷è ø

é ùæ ö

ê ú= + + ç ÷ê úè øë û

Example

7-cell, 1500 MHz

0

2

4

6

8

10

12

14

16

18

20

1 10 100

Qext (106)

P (

kW

)

21.0 MV/m, 460 uA, 50 Hz 0 deg

21.0 MV/m, 460 uA, 38 Hz 0 deg

21.0 MV/m, 460 uA, 25 Hz 0 deg

21.0 MV/m, 460 uA, 13 Hz 0 deg

21.0 MV/m, 460 uA, 0 Hz 0 deg

Another Simple Model:

Coaxial Half-wave Resonator

2b

2a

L


Capacitance per unit length

Inductance per unit length

0 0

0

2 2

1ln ln

Cb

a

pe pe

r

= =æ ö æ öç ÷ ç ÷è ø è ø

0 0

0 0

1ln ln

2 2

bL

r

m m

p p r

æ ö æ ö= =ç ÷ ç ÷è ø è ø

2b

2a

L

Center conductor voltage

Center conductor current

Line impedance


0

2( ) cosV z V z

p

l

æ ö= ç ÷è ø

0

2( ) sinI z I z

p

l

æ ö= ç ÷è ø

0 0

0

0 0 0

1ln , 377

2

VZ

I

mhh

p r e

æ ö= = = Wç ÷è ø

2b

2a

L


d: coaxial cylinders

Vp : Voltage on center conductor

Outer conductor at ground

Ep: Peak field on center conductor

Peak Electric Field

Peak magnetic field


0

0

1ln

300

9

m, A/m

m, T

cm, G

: Voltage across loading capacitance

mT at 1 MV/m

p

p

HV

c Bb

B

V

B

h

rr

ì ü ì üæ öï ï ï ï

= í ý í ýç ÷è øï ï ï ïî þ î þ

2b

2a

L

Power dissipation (ignore losses in the shorting plate)


2 0

2 2

0

2 2

2

1 1/

4 ln

sp

s

RP V

b

RP E

rp l

h r

b lh

+=

µ

2b

2a

L

Energy content


( )2 0

0

2 2 3

0

1

4 ln 1/pU V

U E

pel

r

e b l

=

µ

2b

2a

L

Geometrical factor


( )0

0

ln 1/2

1 1/s

bG QR

G

rp h

l r

h b

= =+

µ

2b

2a

L

Shunt impedance


22

0

0

2

ln16

1 1/sh

s

sh s

bR

R

R R

rh

p l r

h b

=+

µ

( )24 /pV P

2b

2a

L

R/Q


( )02

8ln 1/sh

sh

R

Q

R

Q

h rp

h

=

µ

2b

2a

L

Some Real Geometries (l/4)

l/4 Resonant Lines

l/2 Resonant Lines

l/2 Resonant Lines – Single-Spoke

l/2 Resonant Lines – Double and Triple-Spoke

l/2 Resonant Lines – Multi-Spoke

1300 MHz 9-cell

Pill Box Cavity

Hollow right cylindrical enclosure

Operated in the TM010 mode

0zH =

2 2

2 2 2

1 1z z zE E E

r r r c t

¶ ¶ ¶+ =

¶ ¶ ¶0

2.405c

Rw =

0

0 0( , , ) 2.405i t

z

rE r z t E J e

R

w-æ ö= ç ÷è ø

001

0

( , , ) 2.405i tE r

H r z t i J ec R

w

jm

-æ ö= - ç ÷è ø

E

TM010 mode

Modes in Pill Box Cavity

• TM010

– Electric field is purely longitudinal

– Electric and magnetic fields have no angular

dependence

– Frequency depends only on radius, independent on

length

• TM0mn

– Monopoles modes that can couple to the beam and

exchange energy

• TM1mn

– Dipole modes that can deflect the beam

• TE modes

– No longitudinal E field

– Cannot couple to the beam

TM Modes in a Pill Box Cavity

0

2

2

0

0

sin cos

sin sin

sin cos

rl lm

lm

l lm

lm

zl lm

E n R r zJ x n l

E x L R L

E ln R r zJ x n l

E x rL R L

E r zJ x n l

E R L

j

pp j

pp j

p j

æ ö æ ö= - ¢ç ÷ ç ÷è ø è ø

æ ö æ ö= ç ÷ ç ÷è ø è ø

æ ö æ ö= ç ÷ ç ÷è ø è ø

2

2

0

0

0

cos sin

cos cos

0

rl lm

lm

l lm

lm

z

H l R r zi J x n l

E x r R L

H R r zi J x n l

E x R L

H

E

j

we p j

we p j

æ ö æ ö= - ç ÷ ç ÷è ø è ø

æ ö æ ö= - ¢ç ÷ ç ÷è ø è ø

=

is the mth root of ( )lm lx J x

2 2

lmlmn

x nc

R L

pw

æ ö æ ö= + ç ÷ç ÷ è øè ø

TM010 Mode in a Pill Box Cavity

0 0 01

0 1 01

01

01 01

01

0

0

2.405

0.3832

r z

r z

rE E E E J x

R

R rH H H i E J x

x R

cx x

R

xR

j

j we

w

l lp

æ ö= = = ç ÷è ø

æ ö= = = - ç ÷è ø

= =

= =


Energy content

Power dissipation

Geometrical factor

2 2 2

0 0 1 01( )2

U E J x LRp

e=

2 2

0 1 012( )( )sR

P E J x R L Rph

= +

01

2 ( )

x LG

R Lh=

+

01

1 01

2.40483

( ) 0.51915

x

J x

=

=


Energy Gain

Gradient

Shunt impedance

0 sinL

W El p

p lD =

2 22

3 2

1 01

1sin

( ) ( )sh

s

LR

R J x R R L

h l p

p l

æ ö= ç ÷è ø+

0

2sin

/ 2acc

W LE E

p

l p l

D= =

Real Cavities

Beam tubes reduce the electric field on axis

Gradient decreases

Peak fields increase

R/Q decreases

Real Cavities

Single Cell Cavities

Electric field high at iris

Magnetic field high at equator

Coupling between cells

+ +

+ -

Symmetry plane for

the H field

Symmetry plane for

the E field

which is an additional

solution

ωo

ωπ

0

0cck

2

p

p

w w

w w

-=

+

The normalized

difference between these

frequencies is a measure

of the energy flow via the

coupling region

Multi-Cell Cavities

Mode frequencies:

Voltages in cells:

2

2

0

2

1

0

1 2 1 cos

1 cos2

m

n n

mk

n

kk

n n

w p

w

w w p p

w-

æ ö= + -ç ÷è ø

- æ ö æ ö-ç ÷ ç ÷è ø è ø

2 1sin

2

m

j

jV m

np

-æ ö= ç ÷è ø

/ 2b k

k

Ck C C

C= =

Pass-Band Modes Frequencies

-1

0 1 2 3 4 5 6 7 8 9

9-cell cavity

0w

1/2

0 (1 4 )kw +

Cell Excitations in Pass-Band Modes

9 Cell, Mode 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 4

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 7

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

9 Cell, Mode 9

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

Jean Delayen

Center for Accelerator Science

Old Dominion University

and

Thomas Jefferson National Accelerator Facility

SRF FUNDAMENTALS

ODU TAAD I Fall 2012

Historical Overview

Perfect Conductivity

Kamerlingh Onnes and van der Waals

in Leiden with the helium 'liquefactor'

(1908)

Perfect Conductivity

Persistent current experiments on rings have measured

1510s

n

s

s>

Perfect conductivity is not superconductivity

Superconductivity is a phase transition

A perfect conductor has an infinite relaxation time L/R

Resistivity < 10-23 Ω.cm

Decay time > 105 years

Perfect Diamagnetism (Meissner & Ochsenfeld 1933)

0B

t

¶=

¶0B =

Perfect conductor Superconductor

Penetration Depth in Thin Films

Very thin films

Very thick films

Critical Field (Type I)

2

( ) (0) 1c c

c

TH T H

T

é ùæ öê ú- ç ÷è øê úë û

Superconductivity is destroyed by the application of a magnetic field

Type I or “soft” superconductors

Critical Field (Type II or “hard” superconductors)

Expulsion of the magnetic field is complete up to Hc1, and partial up to Hc2

Between Hc1 and Hc2 the field penetrates in the form if quantized vortices

or fluxoids

0e

pf =

Thermodynamic Properties

Entropy Specific Heat

Energy Free Energy


( ) 1

2

When phase transition at is of order latent heat

At transition is of order no latent heat

jump in specific heat

st

c c

nd

c

es

T T H H T

T T

C

< = Þ

= Þ

3

( ) 3 ( )

( )

( )

electronic specific heat

reasonable fit to experimental data

c en c

en

es

T C T

C T T

C T T

g

a

=

»


3 3

2 200

3

3

: ( ) ( )

(0) 0

33

( ) ( )

( ) ( )

At The entropy is continuous

Recall: and

For

c c

c s c n c

T T

es

c c

s n

c c

c s n

T S T S T

S CS

T T

T T Tdt dt C

T T T T

T TS T S T

T T

T T S T S T

a g ga g

g g

=

¶= =

¶

Þ = ® = =

= =

< <

òò

superconducting state is more ordered than normal state

A better fit for the electron specific heat in superconducting state is

9, 1.5 with for cbT

Tes c cC a T e a b T Tg

-

= » »

( )4 4 2 2

2

( ) ( )

3( ) ( ) ( ) ( )

4 2

Εnergy is continuous

c

n c s c

T

n s es en c cT

c

U T U T

U T U T C C dt T T T TT

g g

=

- = - = - - -ò

( ) ( )2

210 0 0

4 8at c

n s c

HT U U Tg

p= - = =

2

8is thecondensationenergycH

p

2

0,8

at is the free energy differencecHT

p¹

( ) ( )

22

22( ) 1

18 4

cn s n c c

c

H T TF U U T S S T

Tg

p

é ùæ öê ú= D = - - - = - ç ÷è øê úë û

( )21

2( ) 2 1c c

c

TH T T

Tpg

é ùæ öê ú= - ç ÷è øê úë û

The quadratic dependence of critical field on T is

related to the cubic dependence of specific heat

Energy Difference Between Normal and

Superconducting State

Isotope Effect (Maxwell 1950)

The critical temperature and the critical field at 0K are dependent

on the mass of the isotope

(0) with 0.5c cT H M a a-

Energy Gap (1950s)

At very low temperature the specific heat exhibits an exponential behavior

Electromagnetic absorption shows a threshold

Tunneling between 2 superconductors separated by a thin oxide film

shows the presence of a gap

/1.5 with cbT T

sc e b-

µ

Two Fundamental Lengths

• London penetration depth λ

– Distance over which magnetic fields decay in

superconductors

• Pippard coherence length ξ

– Distance over which the superconducting state decays

Two Types of Superconductors

• London superconductors (Type II)

– λ>> ξ

– Impure metals

– Alloys

– Local electrodynamics

• Pippard superconductors (Type I)

– ξ >> λ

– Pure metals

– Nonlocal electrodynamics

Material Parameters for Some Superconductors

Phenomenological Models (1930s to 1950s)

Phenomenological model:

Purely descriptive

Everything behaves as though…..

A finite fraction of the electrons form some kind of condensate

that behaves as a macroscopic system (similar to superfluidity)

At 0K, condensation is complete

At Tc the condensate disappears

Two Fluid Model – Gorter and Casimir

( )

( )

( )

1/2

2

2

(1 ) :

( ) = ( ) (1 ) ( )

1

2

1

4

gives

c

n s

n

s c

T T x

x

F T x f T x f T

f T T

f T T

F T

g

b g

< =

-

+ -

= -

= - =-

fractionof "normal"electrons

fractionof "condensed"electrons (zero entropy)

Assume: free energy

independent of temperature

Minimizationof

C

4

4

1/2

3

2

=

( ) ( ) (1 ) ( ) 1

T3

T

C

n s

C

es

Tx

T

TF T x f T x f T

T

C

b

g

æ öç ÷è ø

é ùæ öê úÞ = + - = - + ç ÷è øê úë û

Þ =

Two Fluid Model – Gorter and Casimir

4

1/2

2

2

2

( ) ( ) (1 ) ( ) 1

( ) ( ) 22

8

1

n s

C

n

C

c

TF T x f T x f T

T

TF T f T T

T

H

T

T

b

gb

p

b

é ùæ öê ú= + - = - + ç ÷è øê úë û

æ ö= = - = - ç ÷è ø

=

= -

Superconducting state:

Normal state:

Recall difference in free energy between normal and

superconducting state

22 2

( )1

(0)

c

C c C

H T T

H T

é ùæ ö æ öê ú Þ = -ç ÷ ç ÷è ø è øê úë û

The Gorter-Casimir model is an “ad hoc” model (there is no physical basis

for the assumed expression for the free energy) but provides a fairly

accurate representation of experimental results

Model of F & H London (1935)

Proposed a 2-fluid model with a normal fluid and superfluid components

ns : density of the superfluid component of velocity vs

nn : density of the normal component of velocity vn

2

superelectrons are accelerated by

superelectrons

normal electrons

s s

s s

n n

m eE Et

J en

J n eE

t m

J E

u

u

s

¶= -

¶

= -

¶=

¶

=


2

2 2

2

0 = Constant

= 0

Maxwell:

F&H London postulated:

s s

s s

s s

s

s

J n eE

t m

BE

t

m mJ B J B

t n e n e

mJ B

n e

¶=

¶

¶Ñ´ = -

¶

æ ö¶Þ Ñ´ + = Þ Ñ´ +ç ÷¶ è ø

Ñ´ +


combine with 0 sB = JmÑ´

( ) [ ]

22 0

1

2

2

0

- 0

exp /

s

o L

L

s

n eB B

m

B x B x

m

n e

m

l

lm

Ñ =

= -

é ù= ê úë û

The magnetic field, and the current, decay

exponentially over a distance λ (a few 10s of nm)

1

2

2

0

4

14 2

1

1( ) (0)

1

L

s

s

C

L L

C

m

n e

Tn

T

T

T

T

lm

l l

é ù= ê úë û

é ùæ öê úµ - ç ÷è øê úë û

=

é ùæ ö-ê úç ÷è øê úë û

From Gorter and Casimir two-fluid model



2

0

2

0, 0

1

London Equation:

choose on sample surface (London gauge)

Note: Local relationship between and

s

n

s

s

BJ H

A H

A A

J A

J A

lm

l

Ñ´ = - = -

Ñ´ =

Ñ = =

= -

Penetration Depth in Thin Films

Very thin films

Very thick films

Quantum Mechanical Basis for London Equation

2* * *

1

0

2

( ) ( ) ( )2

0 ( ) 0 ,

( )( ) ( )

( )

n n n n n

n

e eJ r A r r r dr dr

mi mc

A J r

r eJ r A r

m e

r n

y y y y y y d

y y

y

r

r

ì üé ù= Ñ - Ñ - - -í ýë û

î þ

= = =

= -

=

åò

In zero field

Assume is "rigid", ie the field has no effect on wave function

Pippard’s Extension of London’s Model

Observations:

-Penetration depth increased with reduced mean free path

- Hc and Tc did not change

-Need for a positive surface energy over 10-4 cm to explain

existence of normal and superconducting phase in

intermediate state

Non-local modification of London equation

4

0

0

1-

( )3( )

4

1 1 1

R

J Ac

R R A r eJ r d

c R

x

l

su

px l

x x

-

=

é ù¢ë û=-

= +

ò

Local:

Non local:

London and Pippard Kernels

Apply Fourier transform to relationship between

( ) ( ) : ( ) ( ) ( )4

cJ r A r J k K k A k

p= - and

2

2

2

( )( )ln 1

eff effo

o

dk

K kK k kdk

k

pl l

p

¥

¥= =

+ é ù+ê úë û

òò

Specular: Diffuse:

Effective penetration depth

London Electrodynamics

Linear London equations

together with Maxwell equations

describe the electrodynamics of superconductors at all T if:

– The superfluid density ns is spatially uniform

– The current density Js is small

2

2 2

0

10sJ E

H Ht l m l

¶= - Ñ - =

¶

0s

HH J E

tm

¶Ñ´ = Ñ´ = -

¶

Ginzburg-Landau Theory

• Many important phenomena in superconductivity occur

because ns is not uniform

– Interfaces between normal and superconductors

– Trapped flux

– Intermediate state

• London model does not provide an explanation for the

surface energy (which can be positive or negative)

• GL is a generalization of the London model but it still

retain the local approximation of the electrodynamics


• Ginzburg-Landau theory is a particular case of Landau’s theory of second order phase transition

• Formulated in 1950, before BCS

• Masterpiece of physical intuition

• Grounded in thermodynamics

• Even after BCS it still is very fruitful in analyzing the behavior of superconductors and is still one of the most widely used theory of superconductivity


• Theory of second order phase transition is based on an order parameter which is zero above the transition temperature and non-zero below

• For superconductors, GL use a complex order parameter Ψ(r) such that |Ψ(r)|2 represents the density of superelectrons

• The Ginzburg-Landau theory is valid close to Tc

Ginzburg-Landau Equation for Free Energy

• Assume that Ψ(r) is small and varies slowly in

space

• Expand the free energy in powers of Ψ(r) and its

derivative 2

22 4

0

1

2 2 8n

e hf f

m i c

ba y y y

p

*

*

æ ö= + + + Ñ- +ç ÷è ø

A

Field-Free Uniform Case

Near Tc we must have

At the minimum

2 4

02

nf fb

a y y- = +0

f fn

-0

f fn

-

yyy ¥

0a > 0a <

0 ( ) ( 1)t tb a a> = -¢

2 22

0 (1 )8 2

and cn c

Hf f H t

ay

p b- = - = - Þ µ -

2 ay

b¥ = -


At the minimum

2 4

02

nf fb

a y y- = +

20 ( ) ( 1) (1 ) t t tb a a y ¥> = - Þ µ -¢

22

02 8

(1 )

(definition of )cn c

c

Hf f H

H t

a

b p- = - = -

Þ µ -

2 ay

b¥ = -

It is consistent with correlating |Ψ(r)|2 with the density of superelectrons

2 (1 ) c near Tsn tl -µ µ -

which is consistent with 2

0(1 )c cH H t= -


Identify the order parameter with the density of superelectrons

222

22 2

22

2 22 42

2 4

( )(0)1 1 ( )

( ) ( ) (0)

( )1 ( )

2 8

( ) ( )( ) ( )( )

4 (0) 4 (0)

since

and

Ls

L L

c

c cL L

L L

T Tn

T T n

H TT

H T H TT Tn T n

l a

l l b

a

b p

l la b

p l p l

Y= Y Þ = = -

Y

=

= - =

Field-Free Nonuniform Case

Equation of motion in the absence of electromagnetic

field

221( ) 0

2T

my a y b y y

*- Ñ + + =

Look at solutions close to the constant one 2 ( )

whereTa

y y d yb

¥ ¥= + = -

To first order: 21

04 ( )m T

d da*

Ñ - =

Which leads to 2 / ( )r Te

xd

-»

Field-Free Nonuniform Case

2 / ( )

2

(0)1 2( )

( ) ( )2 ( ) where

r T L

c L

ne T

m H T Tm T

x lpd x

la

-

**» = =

is the Ginzburg-Landau coherence length.

It is different from, but related to, the Pippard coherence length. ( )

0

1/22

( )1

Tt

xx

-

GL parameter: ( )

( )( )

L TT

T

lk

x=

( ) ( )

( )

Both and diverge as but their ratio remains finite

is almost constant over the whole temperature range

L cT T T T

T

l x

k

®

2 Fundamental Lengths

London penetration depth: length over which magnetic field decay

Coherence length: scale of spatial variation of the order parameter

(superconducting electron density)

1/2

2( )

2

cL

c

TmT

e T T

bl

a

*æ ö= ç ÷ -¢è ø

1/22

( )4

c

c

TT

m T Tx

a*æ ö

= ç ÷ -¢è ø

The critical field is directly related to those 2 parameters

0( )2 2 ( ) ( )

c

L

H TT T

f

x l=

Surface Energy

2 2

2

2

1

8

:8

:8

Energy that can be gained by letting the fields penetrate

Energy lost by "damaging" superconductor

c

c

H H

H

H

s x lp

l

p

x

p

é ù-ë û

Surface Energy

2 2

c

Interface is stable if >0

If >0

Superconducting up to H where superconductivity is destroyed globally

If >> <0 for

Advantageous to create small areas of normal state with large

cH H

s

x l s

xl x s

l

>>

>

1:

2

1:

2

area to volume ratio

quantized fluxoids

More exact calculation (from Ginzburg-Landau):

= Type I

= Type II

lk

x

lk

x

®

<

>

2 21

8cH Hs x l

pé ù-ë û

Magnetization Curves

Intermediate State

Vortex lines in

Pb.98In.02 At the center of each vortex is a

normal region of flux h/2e

Critical Fields

2

2

1

2

2

1(ln .008)

2

Type I Thermodynamic critical field

Superheating critical field

Field at which surface energy is 0

Type II Thermodynamic critical field

(for 1)

c

csh

c

c c

cc

c

c

H

HH

H

H H

HH

H

H

k

k

k kk

=

+

Even though it is more energetically favorable for a type I superconductor

to revert to the normal state at Hc, the surface energy is still positive up to

a superheating field Hsh>Hc → metastable superheating region in which

the material may remain superconducting for short times.

Superheating Field

0.9

1

Ginsburg-Landau:

for <<1

1.2 for

0.75 for 1

csh

c

c

HH

H

H

kk

k

k >>

The exact nature of the rf critical

field of superconductors is still

an open question

Material Parameters for Some Superconductors

BCS

• What needed to be explained and what were the

clues?

– Energy gap (exponential dependence of specific heat)

– Isotope effect (the lattice is involved)

– Meissner effect

Cooper Pairs

Assumption: Phonon-mediated attraction between

electron of equal and opposite momenta located

within of Fermi surface

Moving electron distorts lattice and leaves behind a

trail of positive charge that attracts another electron

moving in opposite direction

Fermi ground state is unstable

Electron pairs can form bound

states of lower energy

Bose condensation of overlapping

Cooper pairs into a coherent

Superconducting state

Dw

Cooper Pairs

One electron moving through the lattice attracts the positive ions.

Because of their inertia the maximum displacement will take place

behind.

BCS

The size of the Cooper pairs is much larger than their spacing

They form a coherent state

BCS and BEC

BCS Theory

( )

0 , 1 ,-

, : ( ,- )

0 1

:states where pairs ( ) are unoccupied, occupied

probabilites that pair is unoccupied, occupied

BCS ground state

Assume interaction between pairs and

q q

q q

q qq qq

qk

q q

a b q q

a b

q k

V

Y = +

=

P

0

q k if and

otherwise

D DV x w x w- £ £

=

BCS

• Hamiltonian

• Ground state wave function

destroys an electron of momentum

creates an electron of momentum

number of electrons of momentum

k k qk q q k k

k qk

k

q

k k k

n V c c c c

c k

c k

n c c k

e * *

- -

*

*

= +

=

å åH

( ) 0q q q qq

a b c c f* *

-Y = +P

BCS

• The BCS model is an extremely simplified model of reality

– The Coulomb interaction between single electrons is ignored

– Only the term representing the scattering of pairs is retained

– The interaction term is assumed to be constant over a thin

layer at the Fermi surface and 0 everywhere else

– The Fermi surface is assumed to be spherical

• Nevertheless, the BCS results (which include only a very few

adjustable parameters) are amazingly close to the real world

BCS

2

2 2

0

1

(0)

0

0, 1 0

1, 0 0

2 1

21

sinh(0)

q q q

q q q

q

q

q

VDD

a b

a b

b

e

V

r

x

x

x

x

ww

r

-

= = <

= = >

= -+ D

D =é ùê úë û

Is there a state of lower energy than

the normal state?

for

for

yes:

where

BCS

( )

11.14 exp

( )

0 1.76

c D

F

c

kTVN E

kT

wé ù

= -ê úë û

D =

Critical temperature

Coherence length (the size of the Cooper pairs)

0 .18 F

ckT

ux =

BCS Condensation Energy

2

0

2

0 00

0

4

(0)

2

8

/ 10

/ 10

s n

F

VE E

HN

k K

k K

r

e p

e

D- = -

æ öD- D =ç ÷è ø

D

F

Condensation energy:

BCS Energy Gap

At finite temperature:

Implicit equation for the temperature dependence of the gap:

2 2 1 2

2 2 1 20

tanh ( ) / 21

(0) ( )

D kTd

V

w ee

r e

é ù+ Dë û=

+ Dò

BCS Excited States

0

2 22k

Energy of excited states:

ke x= + D

BCS Specific Heat

10

Specific heat

exp for < ces

TC T

kT

Dæ ö-ç ÷è ø

Electrodynamics and Surface Impedance

in BCS Model

0

4

( , )

[ ] ( , , )

ex

ex i i

ex

rf c

R

l

H H it

eH A r t p

mc

H

H H

R R A I R T eJ dr

R

ff f

w-

¶+ =

¶

=

<<

×µ

å

ò

There is, at present, no model for superconducting

surface resistanc

is treated as a smal

e

l perturbation

at high rf fie

simil

ld

ar to

( ) ( ) ( )4

(0) 0 :

cJ k K k A k

K

p= -

¹

Pippard's model

Meissner effect

Penetration Depth

2

4

2( )

( )

dkdk

K k k

TT

T

lp

l

=+

æ öç ÷è ø

ò

c

c

specular

1Represented accurately by near

1-

Surface Resistance

( )4

32 2

2

:(1 )

:2

exp

-kT

2

Temperature dependence

close to dominated by change in

for dominated by density of excited states e

Frequency dependence

is a good approximation

c

c

s

tT t

t

TT

AR

T kT

l

w

w

D

--

- <

Dæ ö-ç ÷è ø

Surface Resistance

Surface Impedance - Definitions

• The electromagnetic response of a metal,

whether normal or superconducting, is described

by a complex surface impedance, Z=R+iX

R : Surface resistance

X : Surface reactance

Both R and X are real

Definitions

For a semi- infinite slab:

0

0

0

(0)

( )

(0) (0)

(0) ( ) /

Definition

From Maxwell

x

x

x x

y x z

EZ

J z dz

E Ei

H E z zw m

+

¥

=

=

= =¶ ¶

ò

Definitions

The surface resistance is also related to the power flow

into the conductor

and to the power dissipated inside the conductor

212 (0 )P R H -=

( )0

0

0

1/2

0

(0 ) / (0 )

377 Impedance of vacuum

Poynting vector

Z Z S S

Z

S E H

m

e

+ -=

= W

= ´

Normal Conductors (local limit)

Maxwell equations are not sufficient to model the

behavior of electromagnetic fields in materials.

Need an additional equation to describe material

properties

( ) 0

14

1

3 10 secFor Cu at 300 K,

so for wavelengths longer than infrared

J JE

t i

J E

sss w

t t wt

t

s

-

¶+ = Þ =

¶ -

= ´

=

Normal Conductors (local limit)

In the local limit

The fields decay with a characteristic

length (skin depth)

( ) ( )J z E zs=

/ /

0

1/2

00

( ) (0)

(1 )( ) ( )

(0) (1 ) (1 )(1 )

(0) 2 2

z iz

x x

y x

x

y

E z E e e

iH z E z

E i iZ i

H

d d

m w d

m wm wd

s d s

- -=

-=

æ ö+ += = = = + ç ÷è ø

1/2

0

2d

m ws

æ ö= ç ÷è ø

Normal Conductors (anomalous limit)

• At low temperature, experiments show that the surface

resistance becomes independent of the conductivity

• As the temperature decreases, the conductivity s increases

– The skin depth decreases

– The skin depth (the distance over which fields vary) can

become less then the mean free path of the electrons (the

distance they travel before being scattered)

– The electrons do not experience a constant electric field

over a mean free path

– The local relationship between field and current is not

valid ( ) ( )J z E zs¹

1/2

0

2d

m ws

æ ö= ç ÷è ø


Introduce a new relationship where the current is related to

the electric field over a volume of the size of the mean

free path (l)

Specular reflection: Boundaries act as perfect mirrors

Diffuse reflection: Electrons forget everything

/

4

( , / )3( , )

4with

F R l

V

R R E r t R vJ r t dr e R r r

l R

s

p

-é ù× -¢ë û

= = -¢ ¢ò


In the extreme anomalous limit

( )1/3

2 2

091 08

31 3

16

1 :

: fraction of electrons specularly scattered at surface

fraction of electrons diffusively scattered

p p

lZ Z i

p

p

m w

p s= =

æ ö= = +ç ÷

è ø

-

2

2

31

2 cl

l

d

æ öç ÷è ø

1/3

5 2/3( ) 3.79 10l

R l ws

- æ ö®¥ = ´ ç ÷è ø

For Cu: 16/ 6.8 10 2ml s -= ´ W×

1/3

5 2/3

0

3.79 10(4.2 ,500 )

0.12(273 ,500 )

2

K MHz

K MHz

l

R

R

ws

m w

s

- æ ö´ ç ÷è ø

= »

Does not compensate for the Carnot efficiency


Surface Resistance of Superconductors

Superconductors are free of power dissipation in static fields.

In microwave fields, the time-dependent magnetic field in the

penetration depth will generate an electric field.

The electric field will induce oscillations in the normal

electrons, which will lead to power dissipation

BE

t

¶Ñ´ = -

¶

Surface Impedance in the Two-Fluid Model

In a superconductor, a time-dependent current will be carried

by the Copper pairs (superfluid component) and by the

unpaired electrons (normal component)

0

2

0 0

0

2

2

0

(

2( )

2 1

Ohm's law for normal electrons)

with

n s

i t

n n

i t i tcs e c

e

i t

cn s s

e L

J J J

J E e

n eJ i E e m v eE e

m

J E e

n ei

m

w

w w

w

s

w

s

s s s sw m l w

-

- -

-

= +

=

= = -

=

= + = =


For normal conductors 1sR

sd=

For superconductors

( ) 2 2 2

1 1 1n ns

L n s L n s L s

Ri

s s

l s s l s s l s

é ù= Â =ê ú

+ +ë û

The superconducting state surface resistance is proportional to the

normal state conductivity


2

2

2

0

3 2

1

( ) 1exp

( )exp

ns

L s

nn s

e F L

s L

R

n e l Tl

m v kT

TR l

kT

s

l s

s sm l w

l w

Dé ù= µ - =ê ú

ë û

Dé ùµ -ê ú

ë û

This assumes that the mean free path is much larger than the

coherence length


For niobium we need to replace the London penetration depth with

1 /L ll xL = +

As a result, the surface resistance shows a minimum when

lx »

Surface Resistance of Niobium

Surface Resistance - Nb - 1500 MHz

1

10

100

1000

10000

10 100 1000 10000

Mean Free Path (Angstrom)

nohm

2.0 K

4.0 K

4.2 K

3.5 K

3.0 K

2.5 K

1.8 K

Electrodynamics and Surface Impedance

in BCS Model

( )

[ ] ( )

0

4

,

, ,

There is, at present, no model for

superconducting sur

is treated as a small pe

face resistance at high rf

rturbation

similar to Pippar

field

ex

ex i i

ex rf c

R

l

H H it

eH A r t p

mc

H H H

R R A I R T eJ dr

R

ff f

w-

¶+ =

¶

=

<<

×µ

å

ò( ) ( ) ( )

( )4

0 0 :

d's model

Meissner effect

cJ k K k A k

K

p= -

¹


( )( )

4

32 2

2

:

1

:2

exp

-kT

2

Temperature dependence

close to

dominated by change in

for

dominated by density of excited states e

kT

Frequency dependence

is a good approximation

c

c

s

T

tt

t

TT

AR

T

l

w

w

D

-

-

- <

Dæ ö-ç ÷è ø

( )2

59 10 exp 1.83

A reasonable formula for the BCS surface resistance of niobium is

GHzc

BCS

f TR

T T

- æ ö= ´ -ç ÷è ø


• The surface resistance of superconductors depends on

the frequency, the temperature, and a few material

parameters

– Transition temperature

– Energy gap

– Coherence length

– Penetration depth

– Mean free path

• A good approximation for T<Tc/2 and ω<<Δ/h is

2 exps res

AR R

T kTw

Dæ ö- +ç ÷è ø


2 exps res

AR R

T kTw

Dæ ö- +ç ÷è ø

In the dirty limit

In the clean limit

Rres:

Residual surface resistance

No clear temperature dependence

No clear frequency dependence

Depends on trapped flux, impurities, grain boundaries, …

1/2

0 BCSl R lx -µ

0 BCSl R lx µ

Surface Resistance of Niobium

Super and Normal Conductors

• Normal Conductors

– Skin depth proportional to ω-1/2

– Surface resistance proportional to ω1/2 → 2/3

– Surface resistance independent of temperature (at low T)

– For Cu at 300K and 1 GHz, Rs=8.3 mΩ

• Superconductors

– Penetration depth independent of ω

– Surface resistance proportional to ω2

– Surface resistance strongly dependent of temperature

– For Nb at 2 K and 1 GHz, Rs≈7 nΩ

However: do not forget Carnot

The Real World

11

10

9

8 0 25 50 MV/m

Accelerating Field

Residual losses

Multipacting

Field emission

Thermal breakdown

Quench

Ideal

RF Processing

10

10

10

10

Q

Losses in SRF Cavities

• Different loss mechanism are associated with

different regions of the cavity surface

Electric field high at iris

Magnetic field high

at equator

Ep/Eacc ~ 2

Bp/Eacc ~ 4.2 mT/(MV/m)

Characteristics of Residual Surface Resistance

• No strong temperature dependence

• No clear frequency dependence

• Not uniformly distributed (can be localized)

• Not reproducible

• Can be as low as 1 nΩ

• Usually between 5 and 30 nΩ

• Often reduced by UHV heat treatment above 800C

Origin of Residual Surface Resistance

• Dielectric surface contaminants (gases, chemical

residues, dust, adsorbates)

• Normal conducting defects, inclusions

• Surface imperfections (cracks, scratches,

delaminations)

• Trapped magnetic flux

• Hydride precipitation

• Localized electron states in the oxide (photon

absorption)

Rres is typically 5-10 nW at 1-1.5 GHz

Trapped Magnetic Field

A parallel magnetic filed is expelled from a

superconductor.

What about a perpendicular magnetic field?

The magnetic field will be concentrated in normal cores where it is

equal to the critical field.


• Vortices are normal to the surface

• 100% flux trapping

• RF dissipation is due to the normal

conducting core, of resistance Rn

2

ires n

c

HR R

H Hi

= residual DC

magnetic field

• For Nb:

• While a cavity goes through the superconducting transition, the ambient

magnetic filed cannot be more than a few mG.

• The earth’s magnetic shield must be effectively shielded.

• Thermoelectric currents can cause trapped magnetic field, especially in

cavities made of composite materials.

0.3resR W to 1 n /mG around 1 GHzDepends on material treatment


A fraction of the material will be in the normal state.

This will lead to an effective surface resistance

For Nb:

While a cavity goes through the superconducting transition, the ambient

magnetic filed cannot be more than a few mG.

The earth’s magnetic shield must be effectively shielded.

In cavities made of composite materials, thermoelectric currents can cause

trapped magnetic field.

/ cH H

( )/n cH Hr

0.5 to 1 n /mG around 1 GHzeffr » W

Rres Due to Hydrides (Q-Disease)

• Cavities that remain at 70-150 K for several hours (or slow cool-down, < 1

K/min) experience a sharp increase of residual resistance

• More severe in cavities which have been heavily chemically etched

Multipacting

• No increase of Pt for increased Pi during MP

• Can induce quenches and trigger field emission

Multipacting

Multipacting is characterized by an exponential growth in the number of

electrons in a cavity

Common problems of RF structures (Power couplers, NC cavities…)

Multipacting requires 2 conditions:

• Electron motion is periodic (resonance condition)

• Impact energy is such that secondary emission coefficient is >1

One-Point Multipacting

One-point MP

Cyclotron frequency:

Resonance condition:

Cavity frequency (g) = n x cyclotron

frequency

Possible MP barriers given by

n: MP

order

The impact energy scales as 2 2

2

g

e EK

m

+ SEY, d(K), > 1 = MP

Empirical formula: 0

0.3Oe MHznH f

n

Two-Point Multipacting

Empirical formula:

0

0.6Oe MHz

2 1nH f

n

Secondary Emission in Niobium

Field Emission

• Characterized by an exponential drop of the Q0

• Associated with production of x-rays and emission of dark current

1.E+09

1.E+10

1.E+11

0 5 10 15 20

Qo

E (MV/m)

SNS HB54 Qo versus Eacc

0

0

0

0

1

10

100

1000

10000

0 5 10 15 20 25

mR

em

/hr)

E (MV/m)

SNS HTB 54 Radiation at top plate versus Eacc 5/16/08 cg

DC Field Emission from Ideal Surface

Fowler-Nordheim model

6 2 9 3/21.54 10 6.83 10exp

: )

:

:

2 Current density (A/m

Electric field (V/m)

Work function (eV)

EJ

E

J

E

- æ ö´ ´ F= -ç ÷F è ø

F

Field Emission in RF Cavities

Acceleration of

electrons drains cavity

energy

Impacting electrons produce:

• line heating detected by

thermometry

• bremsstrahlung X rays

Foreign particulate

found at emission

site

FE in cavities occurs at fields that

are up to 1000 times lower than

predicted…

5/26 9 3/21.54 10 ( ) 6.83 10exp

:

:

Enhancement factor (10s to 100s)

Effective emitting surface

kE

JE

k

b

b

b

- æ ö´ ´ F= -ç ÷F è ø

C, O, Na, In Al, Si

Stainless steel

Melted

Melted

Melted

Example of Field Emitters

Type of Emitters

Smooth nickel particles emit

less or emit at higher fields.

Ni

V

•Tip-on-tip model

explains why only 10%

of particles are emitters

for Epk < 200 MV/m.

Cures for Field Emission

• Prevention:

– Semiconductor grade acids and solvents

– High-Pressure Rinsing with ultra-pure water

– Clean-room assembly

– Simplified procedures and components for

assembly

– Clean vacuum systems (evacuation and venting

without re-contamination)

• Post-processing:

– Helium processing

– High Peak Power (HPP) processing

Helium Processing

• Helium gas is introduced in the cavity at a pressure just

below breakdown (~10-5 torr)

• Cavity is operating at the highest field possible (in heavy

field emission regime)

• Duty cycle is adjusted to remain thermally stable

• Field emitted electrons ionized helium gas

• Helium ions stream back to emitting site

– Cleans surface contamination

– Sputters sharp protrusions

Thermal Breakdown (Quench)

Localized heating

Hot area increases with field

At a certain field there is a thermal runaway, the field collapses

• sometimes displays a oscillator behavior

• sometimes settles at a lower value

• sometimes displays a hysteretic behavior

Thermal Breakdown

Thermal breakdown occurs when the heat generated at the hot spot is

larger than that can be transferred to the helium bath causing T > Tc:

“quench” of the superconducting state

RF FUNDAMENTALS CAVITY FUNDAMENTALS - …casa.jlab.org/publications/viewgraphs/USPAS2011/TAAD...

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