ESS#Linac#Simulator#vs.#TraceWineval.esss.lu.se/DocDB/0002/000254/001/Seminarino-ELSvsTW.pdf ·...

ESS Linac Simulator vs. TraceWin

Emanuele LafacePhysicist

Accelerator Department

AD Seminarino

February 19th 2013

/[email protected]

TDR 2012Power 5 MW

Peak Power 125 MWPeak Current 50 mAEnergy From 3 MeV to 2.5 GeV

Pulse Length 2.86 msDuty Cycle 4.00%Cavities 208Gradient 40 MV/mLength ~600 m

Simulated parameters for the ESS Proton Linac

1

mailto:[email protected]


/[email protected] 2

DRIFT 156.522 11 0

Available elements:Drift

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

Tracewin input

Matrix




Available elements:Quadrupole

Tracewin input

Matrix

QUAD 22.5 -12.3918 12 0 0 0 0 0

2

6666664

cos(KL) sin(KL)K 0 0 0 0

�K sin(KL) cos(KL) 0 0 0 0

0 0 cosh(KL) sinh(KL)K 0 0

0 0 K sinh(KL) cosh(KL) 0 0

0 0 0 0 1

L�2�2

0 0 0 0 0 1

3

7777775




Available elements:Bending magnet

Tracewin input

Matrix

2

666666664

cos(Kx

L) sin(K

x

L)

K

x

0 0 0

1�cos(K

x

L)

⇢K

x

2

�Kx

sin(Kx

L) cos(Kx

L) 0 0 0

sin(K

x

L)

⇢K

x

0 0 cos(Ky

L) sin(K

y

L)

K

y

0 0

0 0 �Ky

sin(Ky

L) cos(Ky

L) 0 0

� sin(K

x

L)

⇢K

x

� 1�cos(K

x

L)

⇢K

x

L

2 0 0 1 �K

x

L�

2�sin(K

x

L)

⇢

2K

x

3 +

L

�

2

⇣1� 1

⇢

2K

x

2

⌘

0 0 0 0 0 1

3

777777775

BEND -11 9375.67 0 50 1




Available elements:Edge effect

Tracewin input

Matrix

EDGE -5.5 9375.67 50 0.45 2.8 50 1

2

66666664

1 0 0 0 0 0tan(�)|⇢| 1 0 0 0 0

0 0 1 0 0 0

0 0 � tan(�� )|⇢| 1 0 0

0 0 0 0 1 00 0 0 0 0 1

3

77777775




Available elements:DTL Cell

Tracewin input

Matrix

DTL_CEL 71.2854 22.5 22.5 0.00308396 0 -70.2647 156981 -34.6474 10 0 0.0838043 0.78447 -0.36933 -0.151042

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

2

66666664

k1C 0 0 0 0 0kxy

(��)o

k2C 0 0 0 0

0 0 k1C 0 0 0

0 0 kxy

(��)o

k2C 0 0

0 0 0 0 1 0

0 0 0 0 kz

(��)o

(��)i

(��)o

3

77777775

Drift Thin Gap Drift2

6666664





0 0 0 0 1

L�2�2

0 0 0 0 0 1

3

7777775

2

6666664





0 0 0 0 1

L�2�2

0 0 0 0 0 1

3

7777775

Quadrupole Quadrupole




Available elements:Cavity Multi gap

Tracewin input

Matrix

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

2

66666664

k1C 0 0 0 0 0kxy

(��)o

k2C 0 0 0 0

0 0 k1C 0 0 0

0 0 kxy

(��)o

k2C 0 0

0 0 0 0 1 0

0 0 0 0 kz

(��)o

(��)i

(��)o

3

77777775

Drift Thin Gap Drift

NCELLS 1 3 0.5 5.41203e+06 -50.6001 31 0 0.493611 0.488812 12.9855 -14.5316 0.400918 0.679564 0.395851 0.334978 0.657124 0.596556 0.26422 0.661268 0.590482 0.266148

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

2

6666664

1 L 0 0 0 00 1 0 0 0 00 0 1 L 0 00 0 0 1 0 00 0 0 0 1 L

�2�2

0 0 0 0 0 1

3

7777775

2

66666664

k1C 0 0 0 0 0kxy

(��)o

k2C 0 0 0 0

0 0 k1C 0 0 0

0 0 kxy

(��)o

k2C 0 0

0 0 0 0 1 0

0 0 0 0 kz

(��)o

(��)i

(��)o

3

77777775

Drift Thin Gap Drift

...

Taking into account acceleration, time transit factor, etc.




Available elements:Space Charge

U

sc

(x, y, z) =eN

4p⇡

3✏0�

2

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t � 1p(2�

x

2 + t)(2�y

2 + t)(2�z

2 + t)dt

F

x

=e

2N

2p⇡

3✏0�

2x

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)3(2�y

2 + t)(2�z

2 + t)dt

F

y

=e

2N

2p⇡

3✏0�

2y

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)(2�y

2 + t)3(2�z

2 + t)dt

F

z

=e

2N

2p⇡

3✏0�

2z

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)(2�y

2 + t)(2�z

2 + t)3dt





U

sc

(x, y, z) =eN

4p⇡

3✏0�

2

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t � 1p(2�

x

2 + t)(2�y

2 + t)(2�z

2 + t)dt

F

x

=e

2N

2p⇡

3✏0�

2x

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)3(2�y

2 + t)(2�z

2 + t)dt

F

y

=e

2N

2p⇡

3✏0�

2y

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)(2�y

2 + t)3(2�z

2 + t)dt

F

z

=e

2N

2p⇡

3✏0�

2z

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)(2�y

2 + t)(2�z

2 + t)3dt

The three forces must be applied in the framework of the bunch, so they are not the same because z -‐> γz





U

sc

(x, y, z) =eN

4p⇡

3✏0�

2

Z 1

0

e

� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t � 1p(2�

x

2 + t)(2�y

2 + t)(2�z

2 + t)dt

This integral cannot be expressed in terms of elementary functions.




Z 1

0

e� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)3(2�y

2 + t)(2�z

2 + t)dt

Common strategies adopted to solve it:



/[email protected]

Z 1

0

e� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)3(2�y

2 + t)(2�z

2 + t)dt

-‐ Linear approximation and long bunch: x ! 0, y ! 0,�z

� �

x

,�

y

K.Y. Ng, “The transverse Space-Charge force in tri-gaussian distribution”, Fermilab-TM-2331-AD, 2007.

10




http://inspirehep.net/record/700137/files/fermilab-tm-2331-AD.PDF?version=1


/[email protected]

Z 1

0

e� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)3(2�y

2 + t)(2�z

2 + t)dt


� �

x

,�

y


-‐ Elliptical uniform bunch:

P.M. Lapostolle, “Effets de la charge d’espace dans un accelerateur lineaire a protons”, CERN AR/Int. SG/65-15, 15 Juillet 1965.

U

sc

(x, y, z) = V0 �⌧

2✏0

"x

2+ y

2

2

+ a

2 z �x

2+y

2

2

b

2 � a

2

1�

acosh(

ba )bp

b

2 � a

2

!#.

10








/[email protected]

Z 1

0

e� x

2

2�x

2+t

� y

2

2�y

2+t

� z

2

2�z

2+t

p(2�

x

2 + t)3(2�y

2 + t)(2�z

2 + t)dt


� �

x

,�

y


P.M. Lapostolle, “Effets de la charge d’espace dans un accelerateur lineaire a protons”, CERN AR/Int. SG/65-15, 15 Juillet 1965.

U

sc

(x, y, z) = V0 �⌧

2✏0

"x

2+ y

2

2

+ a

2 z �x

2+y

2

2

b

2 � a

2

1�

acosh(

ba )bp

b

2 � a

2

!#.

-‐ Numerical integration: I adopted this solution using an adaptive algorithm for the Gaussian quadrature.R. Pissens et al., “QUADPACK, A Subroutine Package for Automatic Integration”, Berlin : Springer, 1983.

P. Gonnet, “Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants”, ACM Trans. on Math. Soft. Vol. 33, Issue 3, Article 26 (2010).

11


-‐ Elliptical uniform bunch:







http://www.amazon.com/Quadpack-Subroutine-Integration-Computational-Mathematics/dp/3540125531

http://www.amazon.com/Quadpack-Subroutine-Integration-Computational-Mathematics/dp/3540125531

http://dl.acm.org/ft_gateway.cfm?id=1824804&ftid=850093&dwn=1&CFID=154008535&CFTOKEN=92415017

http://dl.acm.org/ft_gateway.cfm?id=1824804&ftid=850093&dwn=1&CFID=154008535&CFTOKEN=92415017

/[email protected]

Some results

DTL to Target October 2012 Lattice

I=0, No Space Charge, Horizontal plane12



/[email protected]

Some results


I=0, No Space Charge, Vertical plane13



/[email protected]

Some results


I=0, No Space Charge, Longitudinal plane14



/[email protected]

Some results


I=50 mA, Horizontal plane15



/[email protected]

Some results


I=50 mA, Vertical plane16



/[email protected]

Some results


I=50 mA, Longitudinal plane17



/[email protected]

Some results


I=50 mA, 10 σ, Horizontal plane with machine aperture18



/[email protected]

Some results


I=50 mA, 10 σ, Vertical plane with machine aperture19



/[email protected]

Future developments

XML input and output functionality.

Matching module for the initial conditions.

Multi-‐particles version.

Better model for RF cavities: analytical `ield solver or `ield map.

User Interface.

20



ESS#Linac#Simulator#vs.#TraceWineval.esss.lu.se/DocDB/0002/000254/001/Seminarino-ELSvsTW.pdf ·...

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