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19. April 2023
Mit
glie
d d
er
Helm
holt
z-G
em
ein
sch
aft
Yu. Senichev Electrostatic lattice with alternating
spin aberration
19. April 2023 Folie 2
“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:
mced 4/
,2
2
21
12
gG
BEEGBGm
e
Sdt
Sd
1522
312
102sec/9979.2sec10582.6
272.9381044
mMeVe
MeVme
ce
mcd
EDM
19. April 2023 Folie 3
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 i-th element if Si+1-Si<10-16.
-the required revolution time simulation is ~109
- high accuracy of simulation
z
x
y Sp
p
S
y
~ 10 rad per e lem ent-18
Therefore we take an approach, where the EDM signal is not implemented, and only the induced error signal is studied.
19. April 2023 Folie 4
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.
19. April 2023 Folie 5
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.
19. April 2023 Folie 6
Symplectic Runge-Kutta integrating (A. Ivanov, St.PbSU)
The integrating Runge-Kutta program is intended to model the spin-orbital 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 Runge-Kutta scheme was implemented from W. Oevel, M. Sofroniou,( Symplectic Runge- Kutta Schemes II: Classification of Symmetric Methods, preprint, University of Paderborn, Germany, 1996.)
19. April 2023 Folie 7
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.
ES
19. April 2023 Folie 8
In resonant method* the spin frequency is parameterized :
using RF flipper .In case of parametric resonance when we shall observe the resonant build up:
Advantage: the method can be realized in COSY ringDisadvantage: the high requirement
to stability of frf
B
B
RF field
z
x
y
y
y
Ex
S
p
BEEGBG
m
e
21
12
,...2,1,0 ; kGkf
f
rev
rf
nnhnh
nS ss
sssez
2cos2
2sin)2()(
2
T - period of envelope
Sz
max
T -peiod of fundam ental oscillation
e
f
signalEDM ~2 ciry
rfrf
LB
lEh
trevkfrffieE
)(~
Gs
*A.Lehrach, B.Lorentz, W.Morse, N.Nikolaev and F.Rathmann
19. April 2023 Folie 9
“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 :
,2
2
21
12
gG
BEEGBGm
e
Sdt
Sd
External fields EDM
01
12
Gmag
19. April 2023 Folie 10
“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
B =0electrostatic field E
z
x
y
y
x
Sp
p
SED M
AR C 1
AR C 2
y
dtSEm
eSd
2
19. April 2023 Folie 11
Spin tune aberration in purely electrostatic ringIn 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.
At “magic” energy it is no precession of spin. For no “magic” energy γmag±Δγ
SEnGcm
e
dt
Sd
1
12
0
vs energy vs field distribution
),(0 r
Spin tune aberration
19. April 2023 Folie 12
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.
z
x
y Sp
p
S
y
t
19. April 2023 Folie 13
Spin tune aberration due to energy spread
Longitudinal component
rev
xxorbz
zxorbx
Tdtd
SGmc
EeL
d
dS
SGmc
EeL
d
dS
/2
1
1
2
1
1
2
22
22
G
cm
EeL xorbs
1
1
2 220
...31
21
12
2
2
2
p
pG
p
pGG
19. April 2023 Folie 14
RF cavity as first step to increase SCT
RF off:
for Δp/p=10-4 SCT=6300 turns, which is ~ 1 msec.
RF on: 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) ].
The spin swing in a rapidly oscillating field with RF frequency and it is bounded within a very narrow angle ~10-6 dependent on
zpppp cos// max
0cos22
2
max2
02
2
zz
cirxz Sp
pG
cm
LEe
d
Sd
2max /~ zs
022
2
20
2
2
z
cirxz Sp
pG
cm
LEe
d
Sd
19. April 2023 Folie 15
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
At (Δp/p)max=10-4 and an axial
particle the number of turns for
SCT is ~6 107 turns, that is ~180 sec.
The code COSY infinity simulation
2
max2
2
20
22
max2
2
20 2
31
2cos
31cos2
2
p
pG
cm
LEe
p
pG
p
pG
cm
LEe cirxzz
cirxsz
spin drift term
spin oscilla tion max
19. April 2023 Folie 16
Off-axial particle: Longitudinal-transverse coupling in electrostatic storage ring
The electrical deflector has the central field symmetry:
where angular momentum . Due to this fact the total energy can be represented as the function of the coordinates :
The radial motion is one dimensional motion in the field with the effective potential, and the equilibrium radius:
nrE
rmr
Mr
mrrW
2
22
22),(
neq MR
constrmvM
19. April 2023 Folie 17
Off-axial particle: influence on spin motionThe 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:
p
pG
cm
LEe cirxsz
2
0
19. April 2023 Folie 18
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.
19. April 2023 Folie 19
Spin tune aberration vs momentum and axial deviation
Assuming violation “magic” condition for non-reference particle the spin tune aberration is defined:
with
x
x
orb
orbxorbs E
E
L
LELG
mc
e
11
1
2 22
...31
21
12
2
2
2
p
pG
p
pGG
...2
21
p
p
p
p
L
L
orb
orb
...2
21
R
xk
R
xk
E
E
x
x
19. April 2023 Folie 20
Spin tune aberration vs momentum
Grouping the coefficients of powers up to second order, we obtain an equation having a parabolic form :
with coefficients having a parabolic dependence on axial deviation
p
p
R
xkkF
p
p
R
xkkF
cm
GEeL xorbs ,,2,,,
2211
2
211220
2
21211
12
2
12
22
22
2
2112
,,
2133131
,,,
R
xk
R
xk
R
xkkF
R
xk
R
xk
R
xkkF
19. April 2023 Folie 21
Spin tune aberration vs axial deviation
Grouping the coefficients of powers up to second order, we obtain an equation having a parabolic form :
with coefficients having a parabolic dependence on momentum
p
pF
R
x
p
pkF
R
x
p
pkF
cm
GEeL xorbs ,
~,
~2,
~
21011
2
2220
p
p
p
p
p
pF
p
pk
p
pk
p
pkF
p
pk
p
pk
p
pkF
2213
,~
31,
~
231
,~
2
12
2
10
1
2
12
2
11
2
2
22
2
22
19. April 2023 Folie 22
Spin tune aberration vs and
Convex surface, or concave surface depends on the sign of
s
x
p
p x
2,1F
p
p
R
xkkF
p
p
R
xkkF
,,2,,, 211
2
2112
+
19. April 2023 Folie 23
Two steps to minimize the spin aberrations
- The lattice with a compensation of the mutual influence of deflector parameters and lattice parameters
- The lattice must provide the alternate change of the spin aberration surface from concave to convex and vice versa
21,kk1
19. April 2023 Folie 24
First step
We first investigate the structure with deflector having a purely
cylindrical electrodes:
Figure: Maximum spin deflection angle after billion turns versus k2
1,1 21 kk
19. April 2023 Folie 26
Second step: alternating spin aberration method
The ring is equipped with two types of deflector with and 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 therequired coefficients k1 and k2. It allows to adjust the spin ofaberration to minimum.
kkkk av 21; kkkk av 21;
19. April 2023 Folie 27
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.
Such plates may be the additional corrective elements placed on the sides of the main deflector
1
n
1 n
or
19. April 2023 Folie 28
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
19. April 2023 Folie 29
Electrostatic lattice consisting of electrostatic deflectors and electrostatic quadrupoles
Figure: Twiss functions of electrostatic ring for ring and one cell
19. April 2023 Folie 31
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.
19. April 2023 Folie 32
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
19. April 2023 Folie 33
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.