Electrostatic ring for the pEDM
madx, matrix and leapfrog tracking
Mario Conte
University of Genova and INFN, Italy
Alfredo U. Luccio, Nicholas D’Imperio
Brookhaven National Laboratory, Upton, New York
September 10, 2015
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EUCard Workshop, Mainz, Sept./10-11 2015
1
Synopsis
We designed and tracked an electrostatic FODO ring for pEDM.
1. Basic design criteria for an EDM electrostatic ring;
2. Find the regions of stability and parameters for the ring
through a csh script that runs a modified version of CERN
Madx with electric bends and quadrupoles;
3. Track orbits using (a)Madx matrices and then (b)to higher
order by differential equations kick integration with a symplectic code based on Leapfrog algoritm;
4. Track spin dynamics by integration of the Thomas-BMT
equation with the Spink algorithm[9]. Address the issue of
spin coherence;
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EUCard Workshop, Mainz, Sept./10-11 2015
Basic design and tracking criteria
1. Design a ring lattice on the footprint of the AGS (800 m),
who seemed, for awhile to be a suitable site. The design
can be scaled to any other site. A large footprint asks for
a small value of the electric bend field;
2. Choose a basic FODO structure where the vertical tune is
much smaller than the horizontal, to optimize the measurement of the vertical spin component, proportional to the
EDM. The FODO will not be symmetrical since cylindrical
electric bends produce an intrinsic horizontal focusing;
3. Design a lattice with a positive phase slip;
4. Adopt a convenient design package (say: madx) to find
tune islands of stability. Use a fast and symplectic tracking
code (say: leapfrog);
5. Recur to parallel computing to calculate and optimize spin
coherence for a full beam of representative particles.
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EUCard Workshop, Mainz, Sept./10-11 2015
Orbit/Spin tracking in an electric ring
Tracking of orbits and spin in an electrostatic storage ring for
the EDM is important and should be done by more than one
method to compare and benchmark.
Tracking should be symplectic, stable in the long range and
fast, because for EDM search ring turns will be in the billions.
Keywords to keep in mind in tracking are accuracy and long
range stability..
Codes proposed and used by various Authors for the tasks are
based on
1-Runge-Kutta integration of differential equations
for orbit (Lorentz) and spin (Thomas-BMT);
2-map description
of machine elements (Madx) or the whole lattice (Cosy-infinity);
3-discrete kicks symplectic integration for propagation by kicks
(Leapfrog, Teapot.)
In this contribution we will describe a madx matrix and a Leapfrog
orbit code of type 3, pllus the some features of a spin dynamics
code.
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EUCard Workshop, Mainz, Sept./10-11 2015
Example of a ring designed using Madx
The code Madx distributed and maintained by CERN is a package used by many accelerator designers, to optimize lattices.
Madx was designed for magnetic bends. We modified it using
matrices proposed by Mario Conte [1], to deal with electrostatic
bends and quadruoiles.
Once conceived the ring, we tested its stability by a Linux script
to run Madx (by Nick D’Imperio [2]), to find islands of stability in
tune space for different values of quadrupole strengths. An example table is shown. The values of βmax are growing at the edge
of each island while the values of the betatron tune decreases.
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EUCard Workshop, Mainz, Sept./10-11 2015
The magic condition
Spin dynamics is governed by the covariant Thomas-BargmanMichel-Telegdi T-BMT equation
ds
q
=−
f × s,
dt
mγ
(1)
where s is the real 3-dimensional spin vector of a 1/2-spin particle, and f is a function of the position and the momentum of
the particle and of the electric and magnetic field encountered
by the particle along its trajectory.
Spin is a passenger on the orbit, that has to be calculated first
at each step. In a pure electrostatic ring, e.g. with no magnets
or RF cavities, f reduces to
1
f =γ a− 2
γ −1
E×v
,
c2
(2)
√
with a the spin anomaly. At the magic momentum pc = mc2 / a
it is exactly f = 0 and the spin remains frozen in its direction at
injection (longitudinal) respect to the orbit.
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EUCard Workshop, Mainz, Sept./10-11 2015
Issues for electric accelerator lattices
An electrostatic lattice behaves differently than a classical magnetic lattice. In an e.element the kinetic energy of a particle is
modulated, while in a m.element it is not, since in the Lorentz
equation of motion
dp/dt = e(E + v × B)
(3)
only the vector term parallel to the momentum appears while in
a m.lattice it is the vector product term present, with the force
perpendicular to the momentum.
In the present design we adopted simple cylindrical electrodes for
the bends, that produce only a radial field far from edges. While
conventional no-gradient magnetic bends don’t focus the beam,
electric cylindrical bends produce a small horizontal focusing, so
that to produce a FODO condition the focusing and defocusing
quadrupoles should be slightly different.
Vertical focusing will be obtained with electrostatic quadrupoles.
An option would be to provide electric bends with also a vertical curvature of the electrodes. For the moment we are not
considering this option because it is more hard and expensive to
construct with the desired accuracy.
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EUCard Workshop, Mainz, Sept./10-11 2015
We considered a ring of 800 m length (taylored on the BNL-AGS
tunnel, that was for awhile proposed as a possible site for the
pEDM ring.) 72 bends of 9 m length. 80 FODO quadrupoles of
2×0.5 m length, 4 drifts of 2×9 m. Values of basic parameters
are listed below
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EUCard Workshop, Mainz, Sept./10-11 2015
F
D
F
D
D
F
F
D
D
F
D
F
D
F
D
D
F
F
D
D
F
Mario Conte
Universit‘a di Genova, Italy
Alfredo U Luccio
Brookhaven National Laboratory
February 2015
F
The ring, with 4×18 bends an 8 straights
F
D
D
F
F
D
D
F
F
’y2’ pEDM electrostatic ring
D
F
D
F
72 bends− 4 double drifts
each 9 meter
D
F
D
F
D
D
F
F
80 FODO quadrupoles
D
F
D
F
total ring length = 800 m
D
D
F
F
D
D
bend
drift
D
D
D
F
F
D
D
F
D
F
F
F
F
D
D
F
F
D
D
D
D
D
F
F
F
F
F
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EUCard Workshop, Mainz, Sept./10-11 2015
Electrostatic bend
El.static bend matrix
α=
p
M =
2 − β 2 , a1 = cos(αθ), a2 = (ρ/α) sin(αθ), a3 = −(α/ρ) sin(αθ),
a1
a3
0
0,
a2
a1
0
0
0
0
1
0
0
0
Lb
1
!
, Lb = length of bend
Other matrices, say: for drifts, are Madx’s
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EUCard Workshop, Mainz, Sept./10-11 2015
.
Electrostatic quadrupole
gap = 2a = 10cm, Lq = 0.5m(quad length)
e GE
a2 mc2 2
2V0
, V0 =
(β γ)k
El.gradient :GE = 2 , k =
a
mc2 β 2 γ
2 e
El.static quadrupole matrices
√
1
√ sin( kLq )
k √
cos( kLq )
!
MF =
√
cos( kLq )
√
√
− k sin( kLq )
√
1
√ sinh( kLq )
k √
cosh( kLq )
!
MD =
√
cosh( kLq )
√
√
k sinh( kLq )
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EUCard Workshop, Mainz, Sept./10-11 2015
Value (MKSA) and dimension of all ring parameters are
magic proton of
β
γ
β 2γ
k
V0
Eq in the quads
Eb in the bends
=
=
=
=
=
=
=
0.59837912.,
1.24810740,
0.44689430,
0.043,
1.42245166.105 V,
2.844903.106 V /m,
2.54972867.106 V /m
Compared with the Stability Table, the above shows that the
working γ of this particle is less than γT and the vertical betatron
tune can be much smaller than the horizontal, as we want, for
EDM measurements.
About the field in the quads, note that for the optics the quantity
of importance is
√
kLq
with Lq the length of the quadrupole. The field in the quadrupole
is proportional to k. Therefore increasing the length of√ the
quadrupole, but at the same time decreasing k and keeping kLq
constant, can effectively reduce the field.
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EUCard Workshop, Mainz, Sept./10-11 2015
The matrix driven tracking program
We started with a first order madx matrices tracking
MTRACK3
INIT
GLOBAL
¨uber alles
MTRACK
MATRICES
DBDR
loop
QUAD
loop
BCELL
DRIFT
RAYTRANSFER
(WRITE)
BEND
DBDR
ENERMOD
FINIS
flowchart
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EUCard Workshop, Mainz, Sept./10-11 2015
Betatron oscillations by matrix tracking
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EUCard Workshop, Mainz, Sept./10-11 2015
Orbit tracking by leapfrog
Going beyond first order orbit tracking, consider canonical integration of the Lorentz diff. equation of motion, Eq.(6) by
leapfrog or Verlet[4] kicks, method invented for astronomy by
Delambre[5] in 1792, and adapted for accelerators by Ronald
Ruth[6]. It is a kick integration method that interleaves drifts,
where only space coordinates are advanced, with symplectic
kick bends where the momentum components are advanced.
Leapfrog is an algorithm accurate to 2.nd order in time step.
Teapot, by R.Talman and L.Schachinger [6] it is similarly constructed. See also [7].
Other integration algorithms, like Runge-Kutta are accurate to
4.th order in time. However they were written with mathematical accuracy in mind, while the 2.nd order Leapfrog is exactly
symplectic, i.e. was written with physical accuracy in mind.
Symplectic Runge-Kutta has been discussed[8]. It makes a computer code slower to run, which defies our goal of short computer
time for tracking an EDM ring for so many turns.
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EUCard Workshop, Mainz, Sept./10-11 2015
Notes on the algorithm ’order’
For spin tracking, in particular to address the issue of spin coherence (see later sections), orbit tracking algorithms would be
of high order. Madx matrices are first order, leapfrog is of higher
order. How higher? Leapfrog works with calculation of the orbit
in the electric field of bends and quadrupoles, then we should
use high order power expansions for the electric field, in (x, y, z)
[11], and for the Hamiltonian, whose constant value should be
used for a steady check of the symplecticity of the algorithm.
The Hamiltonian contains the electric potential φ obtained as a
power expansion solution of the Laplace equation
∇2 φ = 0.
(4)
In the present simulation we expanded field and potential expression to the third order.
Also leapfrog integration should be done at high order, see
R.Ruth [5]. Of course all this will slow down orbit calculation
tracking and should be carefully considered.
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EUCard Workshop, Mainz, Sept./10-11 2015
Leapfrog coordinates
While for madx matrix tracking we used Frénet-Serret coordinates, for Leapfrog one uses Cartesian ”laboratory” coordinates
(x, z, y),with ŷ vertical axis, and time as the independent variable. Vertical electric field component is calculated by a power
expansion out of the ”horizontal” x, z plane of the ring
The circular (plus straights)
ring lattice shown is obtained
by tracking a ”reference particle” i.e at magic energy injected tangentially. Note that
while a matrix method tracks
trough a pre-designed lattice,
leapfrog ”designs” the lattice
by tracking a reference (magic)
particle.
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EUCard Workshop, Mainz, Sept./10-11 2015
Orbit Leapfrog formalism basics
Ménagerie of quantities for the game
ρ[m]
a
Uo [GeV ]
℘ ≡ pc[GeV ]
UT [GeV ]
γ
Bρ[V · s/m]
eE[eV /m]
=
=
=
=
=
=
=
=
radius of curvature
magnetic anomaly
mc2√
, mass − energy
U
/
po a, momentum
℘2 + Uo2 , total energy
/mc2 ,
p
UT
β=
1 − 1/γ 2
109 ℘/c, rigidity
(= ℘/r0 )βc Electric bend field
Leapfrog tracking conserves the value of the Hamiltonian, that
is being continuously recalculated by series expansion from the
Hamilton Equation ∇2 φ = 0 during runs.
H=
p
(℘ − eA)2 + (mc2 )2 + eφ.
(5)
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EUCard Workshop, Mainz, Sept./10-11 2015
Will work on the EDM ring described above,
Basic leapfrog cell for a bend is a sequence
drift + momentum kick + drift
Momentum kick are done by integration of the Lorentz equation,
for an electric bend:
dp
= eE, with E = −∇φ
dt
(6)
The potential, needed for the Hamiltonian, should obey the
Laplace equation
∇2 φ = 0.
(7)
Explicit expressions for φ and A are found by power expansion.
The reference particle, around which the whole beam oscillates,
is the magic particle whose spin would remain frozen in longitudinal position during the propagation.
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EUCard Workshop, Mainz, Sept./10-11 2015
Leapfrog cell-Electric bends
Let us discuss what happens to a reference particle confined to
the horizontal plane of the FODO. For simplicity we draw only
3 instead of 18 kick bends in a ring quadrant.
z
x
d
Fig.1
G
d
AOB → C→
BC,CD
drift,
kick-bend, LFdrifts
F
ρ
O
b
B
C, center of curvature
E
do
a
A
ell
u
θb
a
c
ble
D
150714−1
C
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EUCard Workshop, Mainz, Sept./10-11 2015
drift AOB
call ℘ = pc , in [GeV]. Start in A with initial coordinates
(A) x = −xo , z = −zo , ℘x = 0, ℘z = ℘.
where zo = −(d + ρ), see the figure.
Eq’s for the drift, with a time step dt for the drift A→B:
dx
℘x
=
c,
dt
Uo γ
dz
℘z
=
c, or
dt
Uo γ
x := x + ℘x /(Uo γ)c dt,
z := z + ℘z /(Uo γ)c dt
(8)
using the identity ℘ = Uo βγ, we obtain at the kick bend C the
new position
(B) x = ro , z = βc dt, ℘x = 0, ℘z = ℘.
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EUCard Workshop, Mainz, Sept./10-11 2015
kick in B
In B a kick is imparted to the momentum pc, using the Lorentz
Equation, with a time step δt, different from the dt of the drift.
℘x := ℘x − eEx c δt,
℘z := ℘z − eEz c δt
(9)
For cylindrical bend the field E is purely radial, with components
eEx = −eE (ro /r) cos θ
eEz = eE (ro /r) sin θ.
(10)
Now find the relation between dt and δt for leapfrog i.e:
1. Through the bend the value of the total momentum pc
must be conserved,
2. The trajectory in C should return tangent to the circle, as
in the figure. Namely:
arccos (p(A) · p(C))/p2 = 2θ
(11)
If both conditions hold, the basic trajectory will be a polygon
circumscribed to the circle. Other particles in the beam will
dance around it in betatron oscillations.
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EUCard Workshop, Mainz, Sept./10-11 2015
For condition (1): momentum conservation, combining the preceding equations
℘x = −℘/r cos θ βc δt, ℘z = ℘ (1 − (1/r) sin θ βc δt)
(12)
then after kick (C):
℘2x + ℘2z = (pc)2 1 + ((βc/r)δt)2 − (2/r) sin θ βc δt .
(13)
Since: cos θ = z/r, sin θ = x/r, taking the value of x from Eq.(8),
the term in [ ] in Eq.(13) above reduces to 1 when
δt = 2 dt
The time at the kick bend should be the same as for both
straights across it !
For condition (2): new angle, it is calculated from the scalar
product of the momentum before and after the kick
• (A) before kick: ℘x = 0, ℘z = ℘
• (C) after kick: ℘x = −(℘/r) cos θ βc δt, ℘z = ℘ 1 − 2 sin2 θ
angle = arccos
℘(A) · ℘(B)
2
=
arccos
1
−
2
sin
θ
=
(pc)2
2θ
q.e.d.
The constructed orbit remains tangent to the arc !
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EUCard Workshop, Mainz, Sept./10-11 2015
Reference Trajectory
Let us produce a reference trajectory on the horizontal plane
by leapfrog tracking a magic particle along a polygonal pattern
tangent to a structure made of straights (drifts) and circular arcs
(bends). The leapfrog orbit is slightly longer than the reference
orbit. The more kicks we put in a bend the lesser this difference
is.
For our structure with 8 bends and 8 drifts of circa 270 m of
total length, using 32 kicks in each bend of 103 m of radius, the
difference in effective radius between the geometrical base line
and the polygon is about 1 mm.
The step is much larger than the required step of a solution
by integration for similar accuracy, with a very large gain in
computing speed.
Tracking a reference particle will create a reference trajectory.
An example is shown in the following picture.
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EUCard Workshop, Mainz, Sept./10-11 2015
Reference Trajectory by tracking
Ring with eight bends and eight straights
Fig.2
32 kicks per bend
bend length=28.276 m
drift length 2 × 2.83 m
intra bend drift length = 0.44 m
nominal curvature radius = 36 m
Ecyl = −1.1647455107 V /m
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EUCard Workshop, Mainz, Sept./10-11 2015
Evaluation of the electric field
D d
Fig.3
d
In a general lattice the center
of curvature for the calculation
of the electric field
continuously changes
and has to be re-evaluated
every time
D
o
r
d
o
r
0
θ=
π/4
D
θ =0
π/4
d
(in the present case the center
of curvature changes from quadrant
to quadrant)
D
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EUCard Workshop, Mainz, Sept./10-11 2015
General tracking by leapfrog
The Leapfrog formalism extends to 3 dimensions and applies
unchanged to particles that don’t have a magic energy or are
injected in the lattice on a finite transverse emittance.
Eqs.(8) and .(9) in 3 dimensions are
x := x + ℘x /(Uo γ)c dt
℘x := ℘x − eEx 2c dt
y := y + ℘y /(Uo γ)c dt , ℘y := ℘y − eEy 2c dt .
℘z := ℘z − eEz 2c dt
z := z + ℘z /(Uo γ)c dt
(14)
However, In a general case the leapfrog conditions (1) for momentum and angle are not fully satisfied in a bend because, due
to transverse oscillations, the particle sees a tangential component of the electric field that modulates the energy.
During tracking the Hamiltonian is continuously calculated. It
conserves its initial vaiue.
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EUCard Workshop, Mainz, Sept./10-11 2015
Orbit coordinates
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EUCard Workshop, Mainz, Sept./10-11 2015
Fig.4 x,y betatron oscillations vs. turn #
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EUCard Workshop, Mainz, Sept./10-11 2015
Add a RF - Example of RF bucket
Fig.5 - Phase space of ∆ × pc for two particles, with
dp/p = 1.e−4 and 2.e−4 , with VRF = 1000V and h = 24. Number
of turns for a complete oscillations is 335, corresponding
synchrotron frequency νs = 0.002985 oscillations per turn
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EUCard Workshop, Mainz, Sept./10-11 2015
Code Spink for Spin Dynamics package
The package used for spin dynamics in our simulation is the
code Spink [9], developed and used for many years for the AGS
and RHIC at Brookhaven, that calculates the evolution of the
particle spin.
In Spink the spin in described as a 3-dimensional real vector
(sx , sy , sz ). The code uses a unitary matrix formalism, to integrate by kicks the T-BMT covariant differential equation,
Eq.(1), that we repeat here
ds
q
=−
f × s,
dt
mγ
(15)
Elements of the function (matrix) f are expressed (non linearly)
as a function of the instantaneous values of the position and
momentum of the particle, so we say that spin dynamics rides
on top of orbit dynamics.
Properties of the matrix, in particular the use of its eigenvectors
and eigenvalues, as described in the following, are important to
calculate spin coherence.
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EUCard Workshop, Mainz, Sept./10-11 2015
Spin Dynamics: for EDM search
Spin kicks, applied at each Bend and Quad, follow the leapfrog
pattern of the orbit.
At the magic energy we have f = 0 and the spin remains frozen
in its longitudinal direction from injection on. If the proton
has an EDM, the spin is NOT completely frozen: by Relativistic
transformation in the rest frame of the particle, the electric field
appears as a magnetic field B’ ⊥ to E
B0 = γ
~
β
B− ×E
c
−
γ2 ~
~
(β · B)β.
γ+1
(16)
Another small term is added to f in Eq.(15)
~ × E. f := f + η B0 × v.
B0 = −γ β
(17)
The spin will make a precession around this magnetic field and a
spin vertical component will appear, proportional to the electric
dipole moment, that we want to measure. For a magic proton
this is the only non vanishing additional spin component.
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EUCard Workshop, Mainz, Sept./10-11 2015
Spin dynamics of a longitudinally frozen spin
Fig.8 - Longit. component of the frozen spin: red line in accelerator (Frénet-Serret) coordinates, green line, in laboratory
coordinates. The red line shows little wiggles because the responsible proton is in this example on purpose not perfectly
magic and there are also betatron oscillations.
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EUCard Workshop, Mainz, Sept./10-11 2015
Spin coherence
For EDM measurements the spin of the beam should remain
coherent (the beam remains polarized as an ensemble) for a
measurement time of the order of milliseconds.
Spin coherence is conveniently measured as the reciprocal of
the width of a spin tune line, built up from the spin tune of a
representative large ensemble of the particles in the beam (say,
256), tracked at the same time on a computer cluster. For this
we wrote a Linux csh script to:
1. generate a distribution for a polarized particle beam of
given emittance, energy spread and polarization under different distributions, depending on the beam injector;
2. run the tracking code in parallel for all the particles using
the Message Passing Interface (MPI) Library;
3. calculate as a running average of spin components the spin
tune line.
Spin coherence time is the reciprocal of the spin tune linewidth.
It is efficiently controlled using sextupoles in the ring lattice.
We did a very successful simulation of spin coherence with the
code UAL-Spink on polarized runs on COSY with a RF solenoid
[10].
32
Spin tune
Spin tune, number of spin precession per machine turn, is calculated as an eigenvalue of a one-turn spin 3 × 3 matrix M,
calculated in turn when the orbital 6 dimension phase space
vector regains its starting value. Instead of waiting very many
turns for this to happen, spin tune is equivalently calculated by
running average of the spin tune in each orbit turn, over typically
15,000 turns.
Calling T = M11 + M22 + M33 the trace of the spin matrix, the
fractional part of spin tune Qspin or its complement to one is
calculated as
Q̃spin
µ
=
,
2π
µ = arccos
1−T
2
,
(18)
depending on the sign of µ. From other matrix elements, this
sign, and the two Euler angles of the orientation of the spin
precession axis, θ, latitude, and φ, longitude, are





φ = arctan(2) ((M12 − M21 ), (M23 − M32 ))
θ = arctan(2) ((M23 + M32 ) sin φ, (M13 + M31 ))
M12 − M21
sin µ =
2 cos µ
sign(µ) = − arctan(2)(sin µ, cosµ)
if(sign(µ) <= 1){Q̃spin = 1 − Q̃spin }
(19)
(20)
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EUCard Workshop, Mainz, Sept./10-11 2015
References
1. M.Conte Electrostatic Storage Ring
University of Genova, PACS 29.20.db Oct.24, 2012
2. Nicholas D’Imperio (BNL) internal communication
3. L.Verlet Computer experiments in classical fluids
Phys.Rev. 159 98-103
4. J.B.Delambre Tables du soleil 1792
5. R.D.Ruth A Canonical Integration Technique
IEEE Trans. on Nuclear Sciences NS-3, No.4, August 1983
6. L.Schachinger and R.Talman
Teapot: A Thin-element Accelerator Program
Particle Accelerators, 22 1987
7. J.M.Sanz-Serna
Symplectic integrators for Hamiltonian problems
Acta Numerica (1991), 243-286 QA 297,A21 1992
8. W.H.Press et Al. Numerical Recipes Cambridge Uni.Press,1992
9. A.U.Luccio Brookhaven National Laboratory, July 30, 2007
Spink User’s Manual. Version v.2-21-beta
10. A.U.Luccio, E.Stephenson et Al
Proceedings of the the Bad Honnef Meeting on EDM,
July 2012
11. Y.Orlov and Y.Semertzidis
private communication, internal notes
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EUCard Workshop, Mainz, Sept./10-11 2015