Transcript Electrostatic lattice with alternating

Mitglied der Helmholtz-Gemeinschaft
Yu. Senichev
Electrostatic lattice with alternating
spin aberration
2. April 2016
“Tomas-Bargmann, Michel,Telegdi” equation with EDM term
The spin is a quantum value, but in the classical physics representation
the “spin” means an expectation value of a quantum mechanical spin
operator:
dS
 S
dt

 1

e 
    


   G B  2
G  E  E   B 
  1

m 
2



g 2
G
,
2

 
d  e / 4mc

EDM


4d  mc 2
4  10 31e  m  938 .272 MeV 
15



2

10
ec
e  6.582  10  22 MeV  sec  2.9979 m / sec

2. April 2016

Folie 2
Main problems of EDM simulation
-the arithmetic coprocessor has a mantissa length of
52 bits (minimum recorded figure~2*10-16) and can
make a mistake in calculating spin projections after ith element if Si+1-Si<10-16.
-the required revolution time simulation is
~109
S
y
p
z
y
x
S
~ 10-18 rad per element
p
- high accuracy of simulation
Therefore we take an approach, where the EDM signal is not
implemented, and only the induced error signal is studied.
2. April 2016
Folie 3
Methods of spin-orbital motion investigation
-COSY Infinity program;
-Symplectic Runge-Kutta integrating program;
-Direct numerical integration of T-BMT differential equations
by Fortran subroutine;
-Analytical approach.
2. April 2016
Folie 4
COSY Infinity program (M. Berz, MSU)
COSY Infinity is a program for the simulation, analysis and design of
particle optical systems, based on differential algebraic methods.
In the EDM search, it is the only program which allows the spin-orbit
motion of millions of particles to be simulated over a real time scale
experiment during n x1000 seconds.
At present, we use the MPI (Message Passing Interface) version of
the COSY Infinity program installed on a supercomputer with 3·105
processors.
At the initial stage of EDM research, we use COSY Infinity to study
the behaviour of the spin aberrations for a large number of particles
and a long-time calculation.
2. April 2016
Folie 5
Symplectic Runge-Kutta integrating (A. Ivanov,
St.PbSU)
The integrating Runge-Kutta program is intended to model the spinorbital motion with fringe fields in elements and including the EDM
signal directly in the simulation.
The algorithm used in the program is not as fast as COSY Infinity by
several orders of magnitude. Therefore, we use it mostly to
investigate a short-time phenomenon for single particle that does not
require long calculation periods.
As a basic method for the tracking program, a symplectic RungeKutta scheme was implemented from W. Oevel, M. Sofroniou,
( Symplectic Runge- Kutta Schemes II: Classification of Symmetric Methods, preprint, University of Paderborn,
Germany, 1996.)
2. April 2016
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SE
The EDM search methods in Storage Ring:
1. Resonant method with initial spin orientation in ring
S║B; S={0,Sy ,0} and B={0,By,0}
2. “Magic” method with initial spin orientation in ring
S║p; S┴E; S={0,0,Sz} and E={Ex,0,0}
in three electrostatic lattices based on:
- electrostatic deflectors with optimized 2D-profile with quadrupoles and
without sextupoles;
-
electrostatic deflectors with quadrupoles and optimized sextupoles;
-
electrostatic continuous deflector with optimized 2D-profile (YkS) without
quadrupoles and without sextupoles.
2. April 2016
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In resonant method* the spin
frequency is parameterized :
  
S
p

 1

e 
    
   G B   2
 G   E  E    B 
  1

m 
2



Ex
RF field
i ( f rf kfrev )t
h

2
2 s2
 
Erf lrf
B y Lcir
~ EDM signal
*A.Lehrach,
f rf
f rev
 k  G; k  0,1,2,...
we shall
Tf -peiod of fundamental oscillation
 hn  cos 2 s n
 s  G
Advantage: the method can be
realized in COSY ring
Disadvantage: the high requirement
to stability of frf
2. April 2016
By
Sz max
S z (n ) 
( 2 e s  h s )  sin 2 s n
z
x

using RF flipper E ~ e
.
In case of parametric resonance when
observe the resonant build up:
By
y
Te - period of envelope
B.Lorentz, W.Morse, N.Nikolaev and F.Rathmann
Folie 8
“Magic” method for purely electrostatic ring
In the “magic” method the beam is injected in the electrostatic ring with the spin
directed along momentum S║p and S┴E; S={0,0,Sz} and E={Ex,0,0}
at “magic” energy :
1
2
 mag
1
G  0
dS
 S
dt

 1

e 
    


   G B  2
G  E  E   B 


m 
2

  1

g 2
G
,
2

External fields
2. April 2016
 

EDM
Folie 9
“Magic” method in purely electrostatic ring
In purely electrostatic ring the spin of particle with “magic
energy” rotates with the same angular frequency as the
momentum and it tilts up in the YZ plane due to the EDM with
angular rate
 
e
dS  
E  S dt
2m
S
y
p
ARC2
z
y
x
By=0
electrostatic
field E x
S
 EDM
p
ARC1
2. April 2016
Folie 10
Spin tune aberration in purely electrostatic ring
In reality the beam has energy spread γmag±Δγ and all particles
move in different external field. Therefore the spin tune has the
aberrations dependent on energy γ and trajectory r(t) of particles.


dS
e  1

 G    n  E   S
 2
dt
m0c    1


vs energy
vs field distribution
At “magic” energy it is no precession of spin. For no “magic”
energy γmag±Δγ
  0  (  , r )
Spin tune aberration
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Spin Coherence Time is time when RMS spin orientation
of the bunch particles reaches one radian (YS,BNL),
and it has to be > 1000sec.
During SCT each particle performs
~109 turns in the storage ring
moving with different trajectories
through the optics elements.
At such conditions the spin-rotation
aberrations
associated
with
various types of space and the
time dependent nonlinearities start
to play a crucial role.
2. April 2016
S
y
p
z
y
x
S
 t
p
Folie 12
Spin tune aberration due to energy spread
Longitudinal component
dS x
eL E
  orb 2x
d
2mc
 1


 G   Sz
  2 1



eL E
dS z
  orb 2x
d
2mc
 1


  Sx

G
  2 1



 1



 s 



G

2m0c 2   2  1

eLorbE x
d  2dt / Trev
2
2
 1




p
1

3


p
  2
 G   2G

G 
  ...
2

p

 p 
  1

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Folie 13
RF cavity as first step to increase SCT
RF off:
2
d S z  e E x Lcir
p 

 2G
Sz  0
2
2


p 
d
 2m0 c 
2
for Δp/p=10-4 SCT=6300 turns, which is ~ 1 msec.
RF on: p / p   p / p max  cos z   
Idea of using the RF cavity was expressed long time ago by many
authors, for instance [ A.P. Lysenko et al., Part.Accel. 18, 215 (1986) ].
2

 p 
d S z  e E x Lcir


 2G 
 cos z    S z  0
2
2
d
 p  max
 2m0c 

2
The spin swing in a rapidly oscillating field with RF frequency and it is
bounded within a very narrow angle ~10-6 dependent on  max ~  s / z 2
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Folie 14
RF on: Second order approach of spin tune versus Δp/p
However, in the second approach versus momentum the average tune
spin is not zero
2
2

 eE x L
 p 
p
1  3 2
1  3 2 G  p 
2
cir


 sz 
 2G
 cos z    
 G  
 cos  z    

  
2
2
2
p
2  p  max
 2m0 c 2
2m0 c  


 p  max

e E x Lcir
At (Δp/p)max=10-4 and an axial
particle the number of turns for
SCT is ~6 107 turns, that is ~180 sec.
spin drift term
spin oscillation
2. April 2016
max
The code COSY infinity simulation
Folie 15
Off-axial particle: Longitudinal-transverse coupling in
electrostatic storage ring
The electrical deflector has the central field symmetry:

E n
r
where angular momentum M  mv r . const
Due to this fact the total energy can be represented as the function of
the coordinates :
2
W ( r, r) 
m 2 M
r 
  r 
2
2
2mr
The radial motion is one dimensional motion in the field with the
effective potential, and the equilibrium radius: Req  M n
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Off-axial particle: influence on spin motion
The particle with different momentum oscillates with respect to
different energy levels:
COSY infinity tracking results
for initial coordinates x=0, y=0 and x=3mm, y=0.
Thus, RF cavity will not be able to reduce the oscillation of the
spin for off-axial particles, since:
e E x Lcir
p
 sz 

G
p
m0 c 2
2. April 2016
Folie 17
Equilibrium energy level modulation as method to
increase SCT
Following physical logic the only solution to increase SCT is the modulation
of the energy level itself relative of the magic level.
For this purpose, we have increase coupling between longitudinal and
transverse motion that is, approach frequencies as close as possible to each
other. In result we have got SCT=400 sec
COSY infinity tracking results
for initial coordinates x=0, y=0 and x=3mm, y=0.
2. April 2016
Folie 18
Spin tune aberration vs momentum and axial
deviation
Assuming violation “magic” condition for non-reference particle the spin
tune aberration is defined:
 Lorb E x 
 1

e
 s 
   2
 G   Lorb E x 1 


2
2mc


1
L
E


orb
x 

with
2
2
 1



p 1  3
p

  ...
  2
 G   2G

G

2

p

 p 
  1

Lorb
Lorb
E x
2
 p 
p
  ...
 1 
 2  
p
 p 
2
x
x
 k1  k2    ...
Ex
R
R
2. April 2016
Folie 19
Spin tune aberration vs momentum
Grouping the coefficients of powers up to second order, we
obtain an equation having a parabolic form :
2
eLorbE xG  
x   p 
x  p 


  2 F1 k1, k2 ,  
 s 
 F  , k , k ,

2  2 1 1 2 R   p 
R p 

 

2m0c



with coefficients having a parabolic dependence on axial
deviation
2
x  1  3 2  x 
1  3 2
x 3 2  1

F2  1 , k1 , k 2 ,  
k2   
k1 
 21
2
2
2
R
R




R
x
x

x
F1  k1 , k 2 ,   k1  k 2  
R
R

R
2. April 2016
2
Folie 20
Spin tune aberration vs axial deviation
Grouping the coefficients of powers up to second order, we
obtain an equation having a parabolic form :
2
eLorbE xG  ~  p   x 
p 
~  p  x ~ 
     2 F1 k1,
   F0  1,

 s 
  F2  k2 ,
2
p
R
p
R
p
2m0c  
  




with coefficients having a parabolic dependence on
momentum
2
p  1  3 2  p 
p
~
 


F2  k 2 ,
k

2
k
2
2
 p 
p 
p
2



2
1  3 2  p 
p
~  p 
  


F1  k1 ,
k

k
1
1

p 
p
2

 p 

 p  2
p   3 2  1

p
~

  
  2
F0  1 ,
 21 
2


p   
p 

 p 


2. April 2016
Folie 21
x
Spin tune aberration  s
vs
p
p
and
Convex surface, or concave surface depends on the sign of
x
F1, 2
2
x   p 
x  p


F2  1 , k1 , k 2 ,      2 F1  k1 , k 2 ,  
R  p 
R p


+
2. April 2016
Folie 22
Two steps to minimize the spin aberrations
-
The lattice with a compensation of the mutual
influence of deflector parameters k1 , k2 and lattice
parameters 1
-
The lattice must provide the alternate change of the
spin aberration surface from concave to convex and
vice versa
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Folie 23
First step
We first investigate the structure with deflector having a purely
cylindrical electrodes: k1  1, k2  1
Figure: Maximum spin deflection angle after billion turns versus k2
2. April 2016
Folie 24
Method realization
Question is how to customize the required k1 , k2 ?
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Folie 25
Second step: alternating spin aberration method
The ring is equipped with two types of deflector with
k1 ; k 2  k av  k and k1 ; k 2  k av  k
changing from one deflector to another.
-
In such optics is easier to achieve minimum spin aberration
Raising the field strength between the plates in even deflectors
and reducing in the odd deflectors it effectively adjusts the
required coefficients k1 and k2. It allows to adjust the spin of
aberration to minimum.
2. April 2016
Folie 26
Alternating spin aberration method
Another possibility is the creation of the required potential distribution due
to potential changes in stripline deposited on the surface of the ceramic
plates.
1

n
or
1
n
Such plates may be the additional corrective elements placed on the
sides of the main deflector
2. April 2016
Folie 27
First step
The maximum flatness of spin aberration surface is reached
in the alternating spin aberration lattice
Figure: Maximum spin deflection angle after billion turns versus x deviation
2. April 2016
Folie 28
Electrostatic lattice consisting of electrostatic
deflectors and electrostatic quadrupoles
Figure: Twiss functions of electrostatic ring for ring and one cell
2. April 2016
Folie 29
Comparison with other lattices
Alternating spin aberration
2. April 2016
Folie 30
The limit capabilities of alternating spin
aberration method
The spread due to final Δp/p it is impossible to remove
completely using the correct k1 and k2 only. Nevertheless
the total spread of spin deflection angle does not exceed
±0.5 rad after billion turns, which one corresponds to a SCT
about 5000 seconds.
2. April 2016
Folie 31
Tracking results:
We used differential algebra methods realized in COSY Infinity and
integrating program with symplectic Runge-Kutta methods.
1. Cylindrical deflector: after 106 turns Sxrms≈0.002 that is SCT~500
sec
2. Alternating k2 deflector: after 106 turns Sxrms≈0.0002 that is
SCT~5000 sec
2. April 2016
Folie 32
Conclusion:
-we have studied the behavior of spin aberration in the structure and
developed techniques to minimize it;
-one of the most effective methods is the alternating spin aberration;
-the analytical model allows finding the general solution of the
retention of aberrations within the values allowed SCT to have about
5000 seconds confirmed by COSY-Infinity.
2. April 2016
Folie 33