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SRF Basics
 Part
I : Superconductivity Basics
 Part II : SRF Cavity Peculiarities
Cavity Design & Constraints
 Part III : ILC BCD Cavity
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
1
Part I
Superconductivity Basics
 Evolution
of Understanding
 Phenomenology & Theory
 Approximation Condition
 Predictions
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
2
Evolution of Understanding
Phenomena
Parameter
Zero DC Resistance
Tc
Phase Transition
Hc, T Dependence
Phenomenology
Theory
Onnes 1911
Specific Heat
Meissner Effect
B = 0 Below Hc
Meissner,Ochsenfeld 1933
Two Fluid Model
λ
Impurity Effect
ξ
Gorter, Casimir 1934
London 1935
Pippard
Modification of London
Δ
Phase Transition
Order Parameter
Ginzburug-Landau 1950
κ
Boundary Energy
Type I, First Order Transition
Intermediate State
Fluxoid
TypeII, Second Order Transition
Mixed State
Hc1
Abrikosov 1957
Hc2
Hc3
Electron-Phonon
Isotope Effect
Frolich 1950
BCS 1957
Vogoliubov
GL Equation from BCS Theory
Gorkov 1959
Strong Coupling
Makmilan
RF Surface Resistance
Mattis Bardeen, Miller
Josephson Effect
Josephson
High Tc Material
Bednorz,Muller 1986
Shuichi Noguchi
Josephson 1962
Hayama ILC Lecture, 2006.5.23
3
Phase Transition
Hc
Electron
Specific Heat
Normal State
Hc(0)
Hc(T)
B=0
Maisner State
= Hc(0) {1-t2}
gT
Tc
Shuichi Noguchi
T
Hayama ILC Lecture, 2006.5.23
Tc
T
4
Phase Transition
Ｆｒｅｅ Ｅｎｅｒｇｙ

1
GS  T , H   Gn  T , H    0 H C2  H 2
2
 G 
S 
 ,

S   
C  T 
 T 
 T 
 HC
S S  S n  0 H C
T
  H
C
CS  Cn   0 T 
  T
Shuichi Noguchi
Gn  T , H 

GS  T , H 
1
0 H 2
2
2

2HC
  H C
2

T




Hayama ILC Lecture, 2006.5.23
GS  T ,0
Ｔ
Tc
5
Two Fluid Model by Gorter & Casimir
3
Specific Heat
T 
C      Celectron
 
2 2
Cne   N   F  k B2 T
3
nS
x
n
n = n n + nS
1
1
G  G0  g T 2  1  x  2   0 H C2 x
2
2
1
T
CS e  3 g TC 
 TC
Shuichi Noguchi
3

 , H C  T   H C  0

12  4 N k B

5
n k B2
g
F
T
 x 1  
 TC
dG
0
dx
 T
1  
  TC



2



Hayama ILC Lecture, 2006.5.23
4

 , TC 

2 0 H C2
g
S
t2

 n 1 t 4
6
Two Fluid Model London
J  JS  Jn , Jn   E
n = ns + nn

m  JS
E
,
2
nS e  t

m
B
rot J S
2
nS e
+ Maxwell Equation
0 nS e2
rot rot B  

m
Shuichi Noguchi

2


m


B

B
B
  0 0
2
 nS e2  t 
t


Hayama ILC Lecture, 2006.5.23
7
B at Surface of Supercondu ctor
 B
2
L 
0 nS e2
m

m
 0 nS e  2
 x 
B , B x   B 0 exp   
 L 
  T
  1  
  TC



4




1
2

 1 t 4
B  B 0 exp  i t  kx   k 
2
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
1

2
L



1
2
2i

2
8
Pippard modified London Eq.
2
nS e
J S r   
A r 
mS
Impurity Effect
 A  r '   R  R


 R 3 '
4  0    
A r 
exp    d r
4

3
R
 0 

1
1
1
1
 
 0  l
Shuichi Noguchi
 0  2
    L ,    London Limit
 

1
3
 3

  
 0 2L  ,
 2

Hayama ILC Lecture, 2006.5.23
   Pippard Limit

9
Success of Two Fluid Model

Specific Heat

Maisner Effect

T Dependence of Hc

Ｔ Dependence of λ

Impurity Effect
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
10
Ginzburg – Landau
Phenomenology
  nS
2

Not Uniform
 Not Independent on B
 Near Tc Ａｐｐｒｏｘｉｍａｔｉｏｎ ;
 No Dynamics



 << 1
2


 4 1
1
2
GS  Gn  0      
rot A    j  e A 
2
20
2m
2
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
11
2
Gintzburg – Landau Equation
Stable State is Free Energy minimum


1

2
     
 i   e A   0

2m
2



2
i e
e
2


J
       A

2m
m
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
12
Outputs from GL Equation Step I
A  0 ,   0

2
    ,

2
> Tc
= Tc
G
< Tc
2
GS  Gn 
,
2
2
2
 0 H C

   ' T  TC 
Shuichi Noguchi


Hayama ILC Lecture, 2006.5.23
2
13
2
Step II
A  0 ,   0
2
e
2
2
J S     A ,   nS London Eq.
m

2
4 3
2 4
e  0 H C L
e  0 H C L
 
, 

2
m
m

Shuichi Noguchi

Hayama ILC Lecture, 2006.5.23
14
Step III
A  0 ,   0 ,


f


2
1

 f f 
 i   e A f  0

2m
2

2  f
3

 f  f  0,  
1
2

x
2m  2
3

f  x   tanh
Shuichi Noguchi


x
2 x

 1  exp  


2


Hayama ILC Lecture, 2006.5.23
15
Coherence Length

 T   0.74

2 e 0 L H C

 0
; l   0; Pure Limit
1  t  2
1

 0l  2
 0.85
1
1  t  2
Shuichi Noguchi
1
; l   0; Dirty Limit
Hayama ILC Lecture, 2006.5.23
16
Appearance of Superconductivity
 2 d 2  e2 02
2 2
2 2
 

H
x

2
H
x
C L   0
2

2m d x
2m


Minimum Eigen Value
2e 0 H C2 2L
0
H x  HC2 
 2  HC 
; Bulk
2

20
  

H x  H C 3  
 2
Shuichi Noguchi
1
2

e 0

2
1.66 H C 2 ; Parallel to Surface
Hayama ILC Lecture, 2006.5.23
17
Boundary Energy
g 






0
0
 GS  H   Gn H C  dx


2
0
 4 1
2

2

        i  e A     H  H C   dx
2
2m
2m


  4 0
2
    H  H C   dx
2
 2

0 H   
H
 1 



0
2
HC
 
2
C


  f 4  dx


2

B
ξ
0 H C
λ
From Step III
Normal
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
x
Super
18
Boundary Energy Results
4 2 0 H
g

3
2
8
g 
3


2  1 L
1
g 0,  
2
1

;
2
Shuichi Noguchi
L
0,  
1

2
C
Type I ,
0 H
2
C
2
 0 ,  1
 Abrikosov
1

;
2
Hayama ILC Lecture, 2006.5.23
Type II
19
Ａｂｒｉｋｏｓｏｖ
Boundary Energy of Fluxoid
Cylindrical Coordinate, Bulk
Br  

0 
 L 
Log

0
.
116
 
 ;   r  L ,    1
2 
2  
 r 

1
 r 
 0   L  2
   : r  L



exp
2 

2   2 r 
 L 
r
r2 
B r  0  

     1  2  1 
  8 
 0 H C 2 
20Hc1

B
r
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
20
Fluxoid Energy
κ>>1


2
0


Log


0
.
081
2
4   0 L

HC
 Log  0.081
H C1 

0
2
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
21
Fluxoid Parallel to the Surface
Bean – Livingston, de Gennes
κ>>1
0
HC
HS 

; Bean  Livingston
4  0  
2
 HC
; de Gennes
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
22
Phase Diagram of Type II Superconductor
Bulk
Parallel to Surface
HC3
Normal State
HC2
HC
Mixed State
HC
HC1
HS
Meisnner State
T
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
T
23
Ｇｏｒｋｏｖ
showed using BCS Theory
T ~ TC
7  3 n
2
 

2
8  k B TC 
2
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
24
Typical Superconductor
（Ｋ）
Nb
Pb
Nb3Sn
9.25
7.2
18.3
Critical Temperature
TC
Energy Gap
2Δ(0)/kBTc
3.5
3.6
4.5
Penetration Depth
λ（０）
（ｎｍ）
32
28
170
Coherence Length
ξ（０）
（ｎｍ）
39
110
3
Ginzburg-Landau Parameter
κ（０）
0.82
0.25
56
Thermo dynamical Critical Magnetic Field
μ０ＨＣ
（Ｔ）
0.2
0.08
0.53
μ０ＨＣ１
（Ｔ）
0.06
0.01
μ０ＨＣ２
（Ｔ）
0.4
29
Fluxoid
Φ0
2.068 x 10-15 Wb
Boltzman Constant
kＢ
1.38 x 10-23 JK-1 = 8.617 x 10-5 eV K-1
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
25
Flohrich
Isotope Effect
V  q
2


  k  q     q 
2
k
k-q
k’+ q
k
Shuichi Noguchi
k’- q
k+q
Phonon
q
2
q
k’
k
Hayama ILC Lecture, 2006.5.23
k’
26
Fermi Distribution
1
f   
  F
exp 
 kB T
N()
εＦ

  1

 
Fermi Sphere
ε
εＦ
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
27
Cooper Pair Interaction
 k ,  k   





 '
' 
 k , k 


Amplitude
Occupied
vk
vk’
Empty
uk
uk’
vk2 + uk2 = 1
vk’2 + uk’2 = 1
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
28
Total Energy of the System
W  2  k v   Vk ,k ' vk uk vk ' uk '
2
k
k
k ,k '
2

vk 
2  k vk   uk    Vk ,k ' vk ' uk '  0
uk  k '


vk2 
2  k vk   uk    k  0
uk 

Shuichi Noguchi
 k   Vk ,k ' vk ' uk '
Hayama ILC Lecture, 2006.5.23
k'
29
k
1
v   1 
2
Ek

k 
1
2
 ; u k   1 
 ;
2
Ek 

k
vk u k 
;
2 Ek
Ek    
2
k
2
k
vk2
2
k
uk2
2Δｋ
k 0
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
30
Vk ,k '  V ;
 k  D ,
 D
V
1 1
1 
 V 
2 k ' E k ' 2   D
 0;
T=0
N ( )
 
2
 k  D
2
d  VN 0  sinh
1
 D


1 
 ; V N 0  1, N 0   N   0 
0   2 D exp  
 V N 0  

D
V
1 1
1 
 N 0  V 
2 k ' Ek ' 2
  D
Shuichi Noguchi

tanh 


 2  2 
2k B T 
 d ,
 2  2
Hayama ILC Lecture, 2006.5.23
T 0 
31
Prediction of BCS Theory

1 
 2 g e   D

TC  
exp  

   kB
 VN 0  
1
1
2
2
 0 H C 0   N 0 0 
2
2
 vF
 vF
 0  
 0.18
 0 
k B TC
Shuichi Noguchi

Hayama ILC Lecture, 2006.5.23
2 0 
 3.53
k B TC
32
Near Tc
T  
1
1
8 2
2
k B TC 1  t   3.06 k B TC 1  t  2
7  3
8g
H C T  
H C 0 1  t   1.74 H C 0 1  t 
7  3
2
e
1
 T  
4
Shuichi Noguchi
7  3 
1
1
2
 0 1  t   0.74  0 1  t  2
3 g
Hayama ILC Lecture, 2006.5.23
33
Ｔ～０
 g e2 2 
2


H C T   H C 0 1  t   H C 0 1 1.06 t
3 


C S  Cn
Cn
Shuichi Noguchi
T TC

12

 1.43
7  3
Hayama ILC Lecture, 2006.5.23
34
RF Surface Resistance
Z S  RS  i X S 
E
H
Surface

1
2


PLoss  Re E x H  RS H

 2
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
35
London
  
1
2
2 3
RS   n  0 
2
X S   0 
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
36
BCS Surface Resistance
  
1 2
 ;    k BT ,   l , T  TC
RBCS  1
exp  
 kB T 
3 T
3
  
TC
1  2
 ;    k BT , T 
RBCS  1
exp  
2
 kB T 
2 T
Shuichi Noguchi
Hayama ILC Lecture, 2006.5.23
37