Recent developments of the HEADTAIL code

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ABP
-

LHC Injector Synchtrons Section

GSI, Darmstadt, 18.02.2009

Giovanni Rumolo

1

Recent

developments

of
the

HEADTAIL
code

G.
Rumolo
,

G.
Arduini
, E. Benedetto, E.
Métral
, D.
Quatraro
, B.
Salvant
,
D.
Schulte,
R. Tomás, F.
Zimmermann

CERN/GSI Meeting, GSI, Darmstadt, 18
-
19/02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

2

Overview


Description

of
the

HEADTAIL
code



history

and
model



features

and
motivations

for

upgrades


Upgrades and
applications


Transverse

plane


Transport
based

on
maps



Selection

of
interaction/observation

points


Application


Longitudinal plane:


Bunch

flattening

with

double
rf

system

or

rf

dipole

kick


Bunch

lengthening

and

microwave

instability


Accelerating

bucket

and
transition

crossing


Outlook

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

3

Localized impedance source

"
Electron

cloud

simulations
:
beam

instabilities

and
wakefields
"

G.
Rumolo

and F. Zimmermann,
PRST
-
AB 5, 121002 (2002)

The

collective

interaction

is

lumped

in
one

or

more

points

along

the

ring (
kick
points
),
where

the

subsequent

slices

of a
bunch

(
macroparticles
)
interact

with

an
electron

cloud

(
macroelectrons
)
or

an
impedance

(
wake
)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

4

Slice 1

W
1
N
1
+W
0
N
2

Σ

W
k
N
i
-
k

Slice 2

K
=0

i
-
1

Slice i

Σ

W
k
N
i
-
k

K=1

N
s
-
1

Slice N
s

1.
Bunch macroparticles are
transported across different
interaction points through
the sector matrices

2.
At each interaction point
macroparticles in each slice
receive the kick from the
wakes of the preceding slices

3.
Slicing is refreshed at each
turn taking into account the
longitudinal motion

W
0
N
1

Longitudinal


W
i

=
W
L
(i

D
z
)

Energy loss

GSI, Darmstadt, 18.02.2009

1

2

i

N
s

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

5

Slice 1

N
1
(W
1d
x
1
+W
1q
x)

Σ

N
k
(W
kd
x
k
+W
kq
x
)

Slice 2

K=1

i
-
1

Slice i

Σ

N
k
(W
kd
x
k
+W
kq
x
)

K=1

N
s
-
1

Slice N
s

1.
Bunch macroparticles are
transported across different
interaction points through
the sector matrices

2.
At each interaction point
macroparticles in each slice
receive the kick from the
wakes of the preceding slices

3.
Slicing is refreshed at each
turn taking into account the
longitudinal motion

Transverse (
x
)

dipolar:

W
id

=
W
dx
(i

D
z
)

quadrupolar
:

W
iq

=
W
qx
(i

D
z
)

x
i

centroid

of slice
i

x

position of particle


GSI, Darmstadt, 18.02.2009

1

2

i

N
s

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

6

Slice 1

Slice 2

Slice i

Slice N
s

1.
Bunch
macroparticles

are
transported across different
interaction points through
the sector matrices

2.
At each interaction point
macroparticles

in each slice

interact with the electron
cloud, as it was modified by
the interaction with
the
preceding slices

3.
Slicing

is updated

Electrons step 1

Electrons step 0

Electrons step i
-
1

Electrons step N
s
-
1





GSI, Darmstadt, 18.02.2009

1

2

i

N
s

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

7

What the HEADTAIL model includes (I)


Synchrotron
motion

included


Single
bunch

with

longitdinal

distribution

that

can

be

Gaussian

or

uniform

(
barrier

bucket
, 2002). Longitudinal
dynamics

is

solved

in a
linear,
sinusoidal

(2004)
voltage

or

no
bucket

(


debunching
)
.


Chromaticity

and
detuning

with

amplitude


Dispersion

at
the

kick
section(s
).


Electron

cloud

kick(s
):


Soft
Gaussian

approach

with

finite
size

electrons

(
used

till

2001, obsolete)


PIC
module

on a
grid

inside

the

beam

pipe

(2001)


PIC
solver

with

optional
conducting

boundary

conditions

(GR, D. Schulte, E.
Benedetto, 2003)


Uniform
or

1
-
2
stripes

initial

e
-
distributions

(GR, E. Benedetto, 2005)


Kicks
can

be

given

at
locations

with

different
beta

functions

(2004)


Electrons

can

move

in
field

free

space

or

in
certain

magnetic

field

configurations
,
like

dipole
,
solenoid
,
combined

function

magnet

(2002)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

8


Short range wake field due to a

broad band impedance


or

to
classical

thick

resistive

wall.


x,y

components

(
driving

and
detuning
) of
the

wakes

can

be

weighted

by

the

Yokoya

coefficients

to
include

the

effect

of

flat

chamber
.


Space

charge
:
each

bunch

particle

can

receive

a
transverse

kick
proportional to
the

local

bunch

density

around

the

local

centroid
.


Linear
coupling

between

transverse

planes

What the HEADTAIL model includes (II)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

9

Outputs of HEADTAIL (I)


The main direct
output files

of HEADTAIL give:



Bunch
centroid

positions,
rms
-
sizes

and

emittances

(horizontal, vertical
and longitudinal) as a function of time


Slice by slice centroid positions and rms
-
sizes
. Coherent intra
-
bunch
patterns can be resolved using this information.


Transverse and longitudinal
phase space

of the bunch with a sub
-
sample
of macroparticles and
bunch longitudinal distributions



Off line analysis of the HEADTAIL output allows evaluating
tune
shifts, growth rates, mode spectra

(B. Salvant)


Instability thresholds

can be determined through massive simulation
campaigns with different bunch intensities

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

10



Instability

thresholds

are inferred by HEADTAIL tracking when unstable coherent
motion of the bunch centroid with exponential growth suddenly appears for a tiny
change of bunch current.

Advantages of

HEADTAIL
wrt analytical
formulae that can be used to determine the

instability thresholds
:



It allows for simulations with
several types of
impedance

and with
dipole and quadrupole

components of the wake



It allows for simulations in
non
-
ideal conditions

(correct longitudinal motion, chromaticity,
amplitude detuning, linear coupling, space charge)



It gives as an output the
full bunch dynamics

in
the unstable regime.

Outputs of HEADTAIL (II)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

11

HEADTAIL

Predictions of tune
shifts and instability
thresholds (design
of new machines)

Comparison with
beam
-
based
measurements of
collective effects
(PSB, PS, SPS, LHC)

MAD
-
X

Z
-
Base

Other EM codes…

HFSS

Particle Studio

Gdfidl

RESWALL

Bench measurements

Lattice

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

12

Recent upgrades of HEADTAIL (I)


Since 2006 a number of modifications have been introduced into the
HEADTAIL

code, mainly in order to:



Broaden the range of problems that can be studied and understood using
the code (see following slides)


Improve computation speed and accuracy of the results


Revisit some parts of the code to optimize calculations over some loops or
minimize conditional statements


Introduce frozen models for electron cloud and wake fields (only applicable
in some specific cases)


Make it more user
-
friendly and thus increase the number of potential
users of the code inside and outside of CERN.



Practical User Guide for HEADTAIL

G. Rumolo and F. Zimmermann,
CERN
-
SL
-
Note
-
2002
-
036
-
AP


HEADTAIL
upgrade

D.
Quatraro
, G
.
Rumolo

and

B.
Salvant

(
work

in
progress
)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

13

Recent upgrades of HEADTAIL (II)


Features
which

have

been

added

to
the

HEADTAIL

code


Transverse

plane:

.


The

initial

distribution

of
electrons

can

be

self
-
consistently

loaded

from

a
build
-
up

code

(
ECLOUD)
run


More

wake

field

options

have

been

included
:


The

interaction

with

the

resistive

wall
impedance

has
been

extended

to
include

the

inductive

by
-
pass

effect

and
near
-
wall

effects



The

wake

field

can

be

loaded

from

an
external

table
,
calculated

as
Fourier

transform

of a
known

impedance

(
e.g
.
kicker

or

resistive

wall in
low

energy
).
It

accepts

an
input

having

the

Z
-
BASE
output

format
.


Interaction of
the

bunch

with

several

different
resonators

placed

at
locations

with

different
beta

functions

(
the

list
needs

to
be

input

on a
separated

file
)


GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

14

Recent upgrades of HEADTAIL (III)


Features
which

have

been

added

to
the

HEADTAIL

code


Transverse

plane (
cont‘d
):



The

beam

transport

with

a simple rotation
one
-
turn

matrix

can

be

optionally

replaced

by

transport

using

the

correct

lattice

of
the

machine


Sector

maps

generated

by

MAD
-
X
between

selected

points

of
the

ring
are

loaded

and
used

for

the

transport

between

these

points

(R. Tomás
2006)


The

beta

functions
, as
read

from

a
MAD
-
X
Twiss

file
,
are

used

for

building

the

linear
transport

matrices

between

kick
points

(D.
Quatraro

2007,
see

next

slides
)


This

has
the

advantage

of
easier

implementation

of
chromaticity

through

adjustment

of
the

phase

advance

between

kick
points

by

the

fractional

part

of
the

chromatic

shift


The

signal

from

several

BPMs

can

be

saved

and
used

for

further

analysis

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

15

Recent upgrades of HEADTAIL (IV)


Features which have been added to the
HEADTAIL

code


Longitudinal plane:



A
higher harmonic rf system

has been introduced with adjustable
relative phase to the main rf system (e.g., Bunch Shortening and Bunch
Lengthening modes) and a voltage ramp.


Full motion inside an
accelerating bucket

has been implemented
(GR & B. Salvant 2008)


Phenomena on the energy ramp can be simulated without approximations


Transition crossing can be modeled in detail


So far without
g
tr
-
jump scheme


With and without higher order terms of
h

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

16

Transverse

plane (I)

Transport
matrices

built

from

a MAD
-
X
Twiss

file

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

17

Transverse

plane (
II)

Transport
matrices

built

from

a MAD
-
X
Twiss

file


HEADTAIL

reads

tune and

chromaticity

values

from

the

standard

input

file

.
cfg


MAD
-
X

is

run

internally

and
the

lattice

is

matched

to
the

given

tune and
chromaticity

values


Transport
matrices

are

then

built

from

the

Twiss

file

output

by

MAD
-
X


The

local

chromaticities

x
j+1,j

are

also
contained

in
the

Twiss

file
, and
they

are

used

to
give

particles

their

correct

phase

advances

at
each

turn
according

to
their

momenta

(
evolving

according

the

synchrotron

motion
)




GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

18

Transverse

plane
(III)

Interaction/observation

points

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

19

Transverse

plane
(IV)

Space

charge

kicks

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

20

Transverse

plane
(V)

E
-
cloud/wake

field/observation

points

GSI, Darmstadt, 18.02.2009

MBB

200 interaction points with space charge randomly chosen

Interaction with electron cloud in all the MBB dipoles

Interaction with wake fields at all the kickers

Observation points at all the
BPMs

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

22

Transverse

plane
(VI)

TMCI in
the

SPS
from

the

kicker

impedance

(mode
shifts
)

GSI, Darmstadt, 18.02.2009

Mode shifting and coupling
has been studied for an SPS bunch under the action of the wake fields from

all the kickers. Kicks (20 per turn) were applied to the bunch particles exactly at the kickers’ locations.

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

23

Transverse

plane
(VII)

TMCI in
the

SPS
from

the

kicker

impedance

(mode
shifts
)

GSI, Darmstadt, 18.02.2009

The red lines correspond to the
one
-
kick approximation
. The wake fields from the different kickers have

been weighted by the beta’s in the kicker locations, added up and applied to the bunch once per turn.

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

24

Transverse

plane
(VIII)

TMCI in
the

SPS
from

the

kicker

impedance

(growth
rates
)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

24

Longitudinal plane (I)

Production

of
flat

bunches
: double
rf
-
system

in
the

SPS

0

0.7 MV

Idea from “Studies of beam behavior in a double RF system“, E. Shaposhnikova in
APC Meeting

06.07.2007

GSI, Darmstadt, 18.02.2009

ABP
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LHC Injector Synchtrons Section

Giovanni Rumolo

26

Longitudinal plane (II)

Production of flat bunches: double rf
-
system in the SPS

Importance of this option:



SPS: The
800 MHz cavity is used in BS mode in normal operation

to keep the beam stable



LHC upgrade: Stability studies for a
beam in a double rf
-
system in BL mode

(flat bunch)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

26

Longitudinal plane (III)

Production of flat bunches: longitudinal dipole kick

Importance of this option:



LHC upgrade: Simulation studies of stability of flat hollow bunches

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

27

Longitudinal plane (IV)

Bunch lengthening and microwave instability in the SPS

Potential Well

Bunch Lengthening

regime

Microwave

Instability

regime

Broad
-
band,
Z

/n
=10
W,
f
r
=700 MHz

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

28

Longitudinal plane (V)

Bunch lengthening and microwave instability in the SPS

Bunch shape evolution in the regime of
bunch lengthening

(10
11

ppb, left movie)
and just above the threshold for
microwave instability

(1.5 x 10
11

ppb, right movie)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

29

Longitudinal plane (VI)

Accelerating bucket and transition crossing



Phenomena on the energy ramp
can be simulated without
approximations



Transition crossing can be
modeled in detail


So far without
g
tr
-
jump scheme


With and without higher order
terms of
h


GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

30

Longitudinal plane (VII)

Transition crossing in the PS

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

31

Longitudinal plane (VII)

Transition crossing in the PS

To have a better picture of
the longitudinal phase space,
only few particles at defined
synchrotron amplitudes are
plotted (10 subsequent turns
for each particle)

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

32

Longitudinal plane (VIII)

Transition crossing in the PS

Analytical solution by Elias anticipated exactly the same type of evolution of
the phase space ellipse when crossing transition

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

33

Longitudinal plane (IX)

Transition crossing in the PS

The agreement between the analytically calculated evolution and the one simulated
with HEADTAIL is very good.

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

34

Longitudinal + Transverse....

Transition crossing in the PS with a BB impedance

Relativistic
gamma

Vertical
position of
centroid [m]

Linear scale <y>

Log scale <y>

g

Unstable at

g

= 5.25

Growth rate


~ 60 µs

GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

35

Conclusions & outlook


HEADTAIL
is

a

multi
-
purpose

tool

that

can

be

used

to do
particle

tracking

with

a
variety

of
collective

interactions

(
electron

cloud
,
resonator

impedances
,
resistive

wall,
space

charge
)


HEADTAIL

has
been

improved


is

interfaced

with

MAD
-
X and Z
-
BASE to
track

a
single

bunch

in a real
lattice

with

localized

impedance

sources


can

track

a
single

bunch

in a double
harmonic

rf

system

and in an
accelerating

bucket

(also
across

transition
)


HEADTAIL

is

constantly

under

development


To
make

it

more

performant

and
user
-
friendly


To
add

features

that

can

enlarge

its

range

of
applicability


Near

future

upgrade

plans
:


A robust
model

for

longitudinal
space

charge


Correctly

include

wake

fields

in
the

low

energy

range


Extension to
multi
-
bunch

simulations


GSI, Darmstadt, 18.02.2009

ABP
-

LHC Injector Synchtrons Section

Giovanni Rumolo

21

Transverse

plane
(VI)

E
-
cloud/wake

field/observation

points

GSI, Darmstadt, 18.02.2009