1

Tracking and Analysis Code for Beam Dynamics in SPring-8:CETRA

J. SCHIMIZU : JASRI/SPring-8

Co-workers: M. Takao and H. Tanaka

SAD2006 : Sep. 05

2

Objectives

・ Understanding of Beam Dynamics in SPring-8

・ Upgraid of Beam Quality in SPring-8・ Treatment as exact as possible

J. Schimizu, K. Soutome, M. Takao, H. Tanaka, Proc, 13th Symp. on Accelerator Science and Technology, Suita, Osaka, Japan, October 29-31, 2001, p.80.

3

Code Construction

(Fortran77 : Xeon 3.6GHz x 2, Memory 4 GB, Redhat Enterprise Linux 3)

*：４ × ４、 **：unfinished

Base : RACETRACK, A. Wrulich, DESY 84-026 (1984).

Differential Algebra is used for map extraction [M. Berz, Particle Accelerators, 1989, Vol.24,

pp.109-124.]

4

Outline of the Code

Model Calibration and Analysis Packages• Model Calibration package H. Tanaka, et. al., Proc, 13th Symp. on Acclerator Science and Technology, Suita, Osaka, October 2001, p.83. • Analysis packages ・ Nonlinear dispersion H. Tanaka, M. Takao, K. Soutome, H. Hama, M. Hosaka, NIM A431 (1999), pp.396-404.

・ Nonlinear chromaticity M. Takao, H. Tanaka, K. Soutome, J. Schimizu, Phys. Rev. E70 (2004) 0616501.

5

Hamiltonian and DevicesNormalized Hamiltonian

€

ˆ H = Pσ − h 1+ δ( )2

− ˆ p x −q

p0

Ax

⎛

⎝ ⎜

⎞

⎠ ⎟

2

− ˆ p y −q

p0

Ay

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

1/ 2

−q

p0

As

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

€

δ =p − p0

p0

, ˆ p x,y =px,y

p0

, σ = s − v0 ⋅ t , Pσ =E − E0

p0v0

€

h =1+ Kx x + Ky y

Devices・ Kicks BH, Q, Sx, ... , 20-pole ; BV, skew Q, skew Sx, ... , skew 20-pole

・ Symplectic Integrators BH†, Q †, Sx †, BV †, skew Q † ＊ , ID ... Include radiation energy losses+

Drift Space, Solenoid † ＊

・ Others Cavity, Wakefield , BPM, Aperture Limit, Marker † possible to include fringe effect、 ＊ impossible to use for map extracion + after each symplectic integration steps, calls radation loss routine if nessesary

6

Radiation loss

After closed orbit search, function f(x) is calculated and srored as f(x) table 、 X = 0.001、maximum of x is 6.Photon energy u=uc*x is determined by linear interpolation of function f(x), where f(x) isdetermined by random number r .

M. Sands, Proc. Int. School of Physics <<Enrico FERMI>> Course XLVI, edited by B. Touschek, June 1969, pp.257-411.

€

ωc =3

2

cγ 3

ρ, uc = hωc , Pγ =

cCγ

2π

EG4

ρ 2

€

S ξ( ) =9 3

8πξ K5 / 3 η( )

ξ

∞

∫ dη

spectum function, uc = critical energy

€

n(u) =Pγ

uc2 F

u

uc

⎛

⎝ ⎜

⎞

⎠ ⎟ photon distribution

€

F ξ( ) =1

ξS ξ( ), f (X) = F ξ( )

0

x

∫ dξ / F ξ( )0

6

∫ dξ

F() = photon number spectrummax of X(=u/uc ) is assumed as 6.0

€

N =15 3

8

Pγ

uc

(photons/sec)

7

Symplecti Integrators

・ Bending magnet （ BM ） and Fringe field Sector and Rectangular Bends: Analytic solutions E. Forest, M. F. Reusch, D. L. Bruhwiler, A. Amiry, Particle Accelerators, 1994, Vol. 45, pp.65-94.

Fringes of BH, BV, Q, skew Q and Sx: All the same treatment

・ Q, skew Q and Sx 4th order leap-frog method: H = T(p) + V(q) H. Yoshida, Celestial Mechanics and Dynamical Astronomy 56, 27-43, 1993.

・ Insertion devise （ ID ） and Solenoid Linearized Hamiltonian in x' and y' Explicit solution by generating function E. Forest and K. Ohmi, KEK Report 92-14 September 1992 A.

2 models for solenoid: Sharp-edge model and Soft-edge Bump function model A. J. Dragt, Nuclear Instruments and Methods in Physics Research A298 (1990) 441-459.

8

・ Drift space

€

ˆ H = pσ − 1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2

, Ax = Ay = As = 0

€

x'=ˆ p x

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2 ; ˆ p x '= 0, y'=ˆ p y

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2 ; ˆ p y '= 0

€

σ '=1−1+ δ( )β 0 /β

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2 ; pσ '= 0

€

x f = x i + x'⋅L; ˆ p xf = ˆ p x

i , y f = y i + y'⋅L; ˆ p yf = ˆ p y

i

€

σ f = σ i + σ '⋅L; pσf = pσ

i

・ Bendig magnet and fringe Sector Bend

€

ˆ H = pσ − 1+ Kx ⋅ x + Ky ⋅ y( ) 1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2

+1

21+ Kx ⋅ x + Ky ⋅ y( )

2, Ax = Ay = 0

€

ˆ p xf = ˆ p x

i cos Kx ⋅s( ) + 1+ δ( )2

− ˆ p xi2 − ˆ p y

i2 − Kx

1

Kx

+ x i ⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎩

⎫ ⎬ ⎭sin Kx ⋅s( )

€

x f =1

Kx2 Kx 1+ δ( )

2− ˆ p x

f 2

− ˆ p yi2 −

∂ˆ p xf

∂s− Kx

⎧ ⎨ ⎩

⎫ ⎬ ⎭

€

ˆ p yf = ˆ p y

i , δ f = δ i = δ( )

9

Rectangular Bend: Cartesian coordinate

€

y f = y + ˆ p yi ⋅s +

ˆ p yi

Kx

arcsinˆ p x

i

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟− arcsin

ˆ p xf

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

€

σ f = σ i + 1+ δ( )s +1+ δ( )Kx

arcsinˆ p x

i

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟− arcsin

ˆ p xf

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪− s

€

H = pσ − 1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2

+ Kx ⋅ x

€

M = Yrot

θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟Fringe(1)M||Fringe(2)Yrot

θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

Body M||

€

ˆ p xf = ˆ p x

i − Kx ⋅z, ˆ p yf = ˆ p y

i

€

x f = x i +1

Kx

1+ δ( )2

− ˆ p xf 2

− ˆ p yi2 − 1+ δ( )

2− ˆ p x

i2 − ˆ p yi2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

€

y f = y i +ˆ p y

i

Kx

arcsinˆ p x

i

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟− arcsin

ˆ p xf

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

€

σ f = σ i +1+ δ( )Kx

arcsinˆ p x

i

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟− arcsin

ˆ p xf

1+ δ( )2

− ˆ p yi2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪− s

€

z = 2sin θ /2( ) /Kx : trajectory length

10

Coordinate rotation

€

x f =x i

cos θ( ) 1−ˆ p x

itan θ( )

1+ δ( )2

− ˆ p xi2 − ˆ p y

i2

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

, ˆ p xf = ˆ p x

i cos θ( ) + sin θ( ) 1+ δ( )2

− ˆ p xi2 − ˆ p y

i2

€

y f = y i +ˆ p y

ix i ⋅ tan θ( )

1+ δ( )2

− ˆ p xi2 − ˆ p y

i2 1−ˆ p x

itan θ( )

1+ δ( )2

− ˆ p xi2 − ˆ p y

i2

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

, ˆ p yf = ˆ p y

i

€

σ f = σ i +1+ δ( )x i ⋅ tan θ( )

1+ δ( )2

− ˆ p xi2 − ˆ p y

i2 1−ˆ p x

itan θ( )

1+ δ( )2

− ˆ p xi2 − ˆ p y

i2

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪

Fringes(B, Q, skew Q and Sx) BH case

€

H = − 1+ δ( )2

− ˆ p x2 − ˆ p y

2 ˆ p xax

1+ δ( )2

− ˆ p x2 − ˆ p y

2+ B0(z)x

€

b0(z) =q

p0

B(z), ax =q

p0

Ax

€

g±(q, p)=Δ lim0 H±−Δ

Δ

∫ (q, p)ds = ±Kx

y 2 ˆ p x /2

1+ δ( )2

− ˆ p x2 − ˆ p y

2→ ±

1

2Kx

y 2 ˆ p x1+ δ

€

ˆ p x1+ δ

⎛

⎝ ⎜

⎞

⎠ ⎟2

<<1,ˆ p y

1+ δ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

<<1 : assumption

11

・ Q, skew Q and Sx

€

ˆ H = pσ − 1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2

+1

2g0 x 2 − y 2

( ) − n0x ⋅ y +1

6λ 0 x 3 − 3xy 2

( ), Ax = Ay = 0

€

x'=∂ ˆ H

∂ˆ p x=

ˆ p x

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2 , ˆ p x '= −∂ ˆ H

∂x= −g0x + n0y −

1

2λ 0 x 2 − y 2

( )

€

y'=∂ ˆ H

∂ˆ p y=

ˆ p y

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2 , ˆ p y '= −∂ ˆ H

∂y= +g0y + n0x +

1

2λ 0xy

€

σ '=∂ ˆ H

∂pσ

=1−1+ δ( )β 0 /β

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2 , pσ '= −∂ ˆ H

∂σ= 0

€

qi = qi−1 + Δs ⋅c i

∂T

∂ˆ p

⎛

⎝ ⎜

⎞

⎠ ⎟ˆ p = ˆ p

i−1

, ˆ p i = ˆ p i−1 −Δs ⋅di

∂V

∂q

⎛

⎝ ⎜

⎞

⎠ ⎟q= q

i−1

€

σ i = σ i−1 + Δs ⋅c i 1−1+ δ( )β 0 /β

1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ˆ p = ˆ p

i−1

€

c1 = c4 =1

2 2 − 21/ 3( )

, c2 = c3 =1− 21/ 3

2 2 − 21/ 3( )

, d1 = d3 =1

2 − 21/ 3, d2 =

−21/ 3

2 − 21/ 3, d4 = 0

4th order leap-frog method

12

・ Insertion Device（ ID) and Solenoid

€

ˆ H = Pσ − 1+ δ( )2

− ˆ p x −q

p0

Ax

⎛

⎝ ⎜

⎞

⎠ ⎟

2

− ˆ p y −q

p0

Ay

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

1/ 2

, As = 0

Firsrt order in x' and y'

€

ˆ H ⇒ Pσ −δ +ˆ p x − f( )

2

2 1+ δ( )+

ˆ p y − g( )2

2 1+ δ( ), f =

q

p0

Ax , g =q

p0

Ay

€

ˆ p xf =

ˆ p xi − f ⋅ fx + g ⋅gx( )Δz{ } 1− gyΔz( ) + ˆ p y

i − f ⋅ fy + g ⋅gy( )Δz{ }gxΔz

1− fxΔz( ) 1− gyΔz( ) − fx ⋅gyΔz2

€

ˆ p yf =

ˆ p yi − f ⋅ fy + g ⋅gy( )Δz{ } 1− fxΔz( ) + ˆ p x

i − f ⋅ fx + g ⋅gx( )Δz{ } fyΔz

1− fxΔz( ) 1− gyΔz( ) − fx ⋅gyΔz2

€

x f = x i + Δsˆ p x

f − f

1+ δ, y f = y i + Δs

ˆ p yf − g

1+ δ, Δz =

Δs

1+ δ

€

σ f = σ i + Δs 1−β 0

β

⎛

⎝ ⎜

⎞

⎠ ⎟−

Δs

2

ˆ p xf − f

1+ δ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

+ˆ p y

f − g

1+ δ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

β 0

β

13

Insertion Device（ ID）

€

Ax =1

ρ 0ks

cos kx x( )cosh ky y( )sin kss( ) −1

ρ1kskx1

sin kx1x( )sinh ky1y( )sin kss − φ( )

€

Ay =1

ρ 0ksky

sin kx x( )sinh ky y( )sin kss( ) −1

ρ1ks

cos kx1x( )cosh ky1y( )sin kss − φ( )

€

Ax =1

ρ 0ks

cos kx x( )cosh ky y( )sin kss( ) −1

ρ1kskx1

sinh kx1x( )sin ky1y( )sin kss + φ( )

or

€

Ay =1

ρ 0ksky

sin kx x( )sinh ky y( )sin kss( ) −1

ρ1ks

cosh kx1x( )cos ky1y( )sin kss + φ( )

Solenoid・ Sharp-edge

€

Ax = −1

2b0 s( )y , Ay = +

1

2b0 s( )x b0(s)=0 for outside, b0(s)=B0 for inside of solenoid

・ Soft-edge Bump function

€

Bs 0,0,s( ) = B0 ⋅bump s,λ ,L( ), bump s,λ ,L( ) =1

2tanh

s

λ

⎛

⎝ ⎜

⎞

⎠ ⎟− tanh

s − L

λ

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

L = length of solenoid body, λ= depth of fringe region

€

Ax = −yU s,ρ 2( ), Ay = +xU s,ρ 2

( ), Az = 0

€

U s,ρ 2( ) =

−1( )n

ρ 2( )

nb2n

22n +1n! n +1( )!{ }n= 0

∞

∑ , b2n =∂ 2nBs 0,0,s( )

∂s2n, ρ 2 = x 2 + y 2

14

Cavity・ Normal Operation

€

ˆ H = Pσ − 1+ δ( )2

− ˆ p x2 − ˆ p y

2[ ]

1/ 2

−L

2πh

q

p0cV s( )cos

2πh

Lσ + ϕ 0

⎛

⎝ ⎜

⎞

⎠ ⎟, Ax = Ay = 0

€

V s( ) = V0δ s( )

・ Power-off (under construction： temporary treatment) ・ Intact cavity（ power is constant during m sec period）

€

PT =V0

2

R+

U0 ⋅ IeNc

= const

€

tan φ( ) =sin α( ) − I ⋅R ⋅sin θ( ) / 1+ β( ) /V0

cos α( )= −2 ⋅QL

ωRF −ω0

ω0

€

θ =ϕ 0 −α , ω0 n( ) = ω0 n = 0( ) + Δω 1− exp −n

nτ

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

n = revolution after power-off, Vb = reverse voltage, = coupl. const, = tuner offset,ω0 = resonance angular frequency(determined at steady state), nτ = time const.

Step and sawtooth corrections are under consideration for 0 ： riseτ1 、 downτ2

PT = total power / cavity, U0 = radiation loss / revolution, Nc = number of cavity, I = stored current, R = shunt impedance

€

PL =I

N pn=1

Np

∑ V0 ⋅sin2πh

Lσ n + ϕ 0

⎛

⎝ ⎜

⎞

⎠ ⎟ : Load, after power-off, V0 => VRF

€

VRF = PT − PL( ) ⋅R

・ Power-off cavity

€

Vb =1

2

I ⋅R ⋅cos2 φ n( )( )

1+ β

A-, B-, C-, D-RF Stations8 cavities/RF StationTotal 32 cavities

Co-workers ：T. Ohshima, T. Fujita, T. Takashima,

M. Hara, H. Yonehara

L = circumference, h = harmonic number

15

Wakefield・ Longitudinal (Energy loss)

€

U j =I ⋅Cl

M σ s

N j

I = current/bunch, M = number of macro particles, Nj = number of particles going before j-th particle, σs = bunch length, C = coefficient

・ Transverse (kick)

€

θ j =I ⋅Ct 0 ⋅ σ s

Mx i

i

∑ or θ j =I ⋅Ct ⋅σ s

Mx i

i

∑

xi = transverse amplitude of i-th macro particle Summation of amplitude is performed for macro-particles going before j-th particle.

Wakefield is usually neglected（ option off）

K. Bane and M. Sands, SLAC-PUB-4441, November 1987 (A).K. Ohmi and Y. Kobayashi, Phys. Rev. E59 (1999),p.1167.

16

Comparisons with experiments・ Chromaticity: Effect of thick S

xGreen dotted line is the results of CETRA

・ Tune shift: Effect of ID ・ Ping: Effect of wakefield (transverse only)

red ： experimentblue ： CETRA middle ： with wakefield bottom ： w/o wakefield

17

Examples of Tracking simulation

・ Injection efficiency

18

Initial distribution at injection point： w/o window effect (achromat)

Initial distribution at injection point： with window effect (Low Emittance)

19

Multiple scattering effect in septum chamberE(GeV) condition θ (mrad)

8

w/o window 0.000

window, He 0.075

window, Air 0.125

4

w/o window 0.00

window, He 0.15

window, Air 0.25

window ： Be 0.5mm Al 0.1mm Kapton 0.1mm

20

xy profiles and σ-δ phase plots of injected beam (1,000 particles injection). Gaps of ID are full open.

8 GeV 1st trun 4 GeV 1st turn

8 GeV 201th trun 4 GeV 2,001th turn

21

8 GeV 601th trun 4 GeV 6,001th turn

8 GeV 1,000th trun 4 GeV 10,000th turn

22

・Resonances

Beam size and dy/dx of beam profiles in the center of cell-14 BM near the differential resonance (x=y)

The chromaticity dependence of beam size in the center of cell-14 upstream BM near the third-order resonance (2x=y)

23

・ RF power off

All the RF station power off; trajectories of 1st, 10th, ・・・ , 70th revolution（ Cavity model off 、 classical radiation loss assumed ）

In the Low Emittance operation, most particles bombards the injection chamber for all rf power off condition. （ red lines ） .Inner radius of injection chamber = 18.6 mm. upstream : Al alloy (r = 20.0 mm) downstream: SUS (r = 18.6 mm)Horizontal radius of other chambers = 35.0 mm.

Energy loss by BM ： about 9.23MeV/revEnergy loss by IDs ： usually about 1MeV/rev (max is over 2MeV/rev ）

A-, B-, C-, D-RF Stations8 cavities/RF StationTotal 32 cavities

0.5MV× 32=16MV

24

4 cavities in downstream of RF-A station powered off（ 16 MV→14 MV）blue dots : synchronous phase is mannually +8.8° corrected, with assumption of time const. = 2 revs. (~10 sec) [test simulation by temporary model]red dots : without correction 、 large σ(phase) and δ oscillation

Topup operation can be maintained for 2MV off case.

Sawtooth phase correction ： stored current is 100mA　 τ1=3rev, τ2=10,000rev 　（ 4.79μs/rev)

　 Phase in figure shows △.

　 [Results of test simulation with temporary model]　　

The case of 4MV off needs +15° correction, when I is small.

　 +22° correction, beam current decreases

　 +28° correction, beam is not lost

All (eight) cavities of RF-B station powered off（ 16 MV→12 MV ）

25

・ without phase correction （ red ）：　 Momentum and phase oscillations are large, most particles are lost　　　　　　　　　→ topup operation stopped, beam should be re-injected

・ with phase correction ：　 Phase correction corresponds to 4MV off （ +15° ） is not sufficient for 100 mA.　 About twice phase correction is required for 100mA.　 with +28°(green) correction, small phase oscillation → possibility to maintain topup operation When IDs under operation increase, what happens?

26

Summary

・ Use exact equations of motion derived from exact Hamiltonian・ Use symplectic integrators for real devices in SPring-8, if possib

le・ Equations of motion for ID and solenoid are derived from

expanded Hamiltonian in first order of x’ and y’ ・ Nonlinear chromaticity can be reproduced with thick Sx・ Useful tool for investigation and improvement of injection

efficiency for topup operation Upgraid injection efficiency with slits in SSBT Multiple scattering at injection chamber contributes degradation

of injection

efficiency, especially in 4 GeV operation ・ Possible tool for investigation of beam dynamics in RF power

off 2MV off; simulation can reproduce the beam behavior, fairly well 4MV off; possibility to avoid beam abort

・ Home made code to respond to the demands of the times

27

AppendixSymplecticityWhen MTJM=J, matrix M is symplectic, where M and J are 6 by 6 matrices for us.

M is one turn matrix. A sample of check write is as follows: 1 turn matrix am 1 5.232489E-01 1.822018E+01 0.000000E+00 0.000000E+00 0.000000E+00 5.012416E-02 2 -3.585530E-02 6.626100E-01 0.000000E+00 0.000000E+00 0.000000E+00 3.742622E-03 3 0.000000E+00 0.000000E+00 -9.237297E-01 4.860075E+00 0.000000E+00 0.000000E+00 4 0.000000E+00 0.000000E+00 -1.366636E-01 -3.635311E-01 0.000000E+00 0.000000E+00 5 -3.755540E-03 -3.497847E-02 0.000000E+00 0.000000E+00 1.000000E+00 -1.324282E-01 6 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 1.000000E+00 det = 0.999999999999944 amt x J x am 1 0.000000E+00 1.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -1.456734E-15 2 -1.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 -7.476658E-14 3 0.000000E+00 0.000000E+00 0.000000E+00 1.000000E+00 0.000000E+00 0.000000E+00 4 0.000000E+00 0.000000E+00 -1.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 5 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 1.000000E+00 6 1.456734E-15 7.476658E-14 0.000000E+00 0.000000E+00 -1.000000E+00 0.000000E+00

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M =

mx,x mx,px mx,y mx,py mx,σ mx,δ

mpx,x mpx,px mpx,y mpx,py mpx,σ mpx,δ

my,x my,px my,y my,py my,σ my,δ

mpy,x mpy,x mpy,y mpy,py mpy,σ mpy,δ

mσ ,x mσ ,px mσ ,y mσ ,py mσ ,σ mσ ,δ

mδ ,x mδ ,px mδ ,y mδ ,py mδ ,σ mδ ,δ

€

J =

0 1 0 0 0 0

−1 0 0 0 0 0

0 0 0 1 0 0

0 0 −1 0 0 0

0 0 0 0 0 1

0 0 0 0 −1 0