TouschekPiwinski#

Touschek liftime and apertures

Functions

`MomAperture_Project2Start()`	calculates the local momentum aperture.
`MomAperture_allRing()`	returns positive and negative momentum aperture
`TLT_IntPiw()`	integral in Piwinski Formula for the Lifetime
`TLT_IntPiw_k()`	integral in Piwinski Formula for the Lifetime with u=tan^2(k)
`TouschekPiwinskiLifeTime()`	function [Tl,contributionsTL]=(ring,dpp,Ib,…)
`momentum_aperture_at()`	recursively offsets the particle energy and checks

MomAperture_Project2Start(thering)#: calculates the local momentum aperture.

MomAperture_Project2Start is a Bipartition search of the negative and

positive stability thesholds in the 5th dimension (relative momentum).

-The 6D closed orbit is taken into account.

-Particles launched at different REFPTS along the ring are first projected

to the ring last element so that all particles can be tracked together.

[etn, etp] = MomAperture_Project2Start(thering)

Inputs:

THERING: ring used for tracking.

Options:

REFPTS: REFPTS where to calculate the momentum acceptance.

Default 1:numel(THERING);

nturns: Number of turns to track. Default 1000

dptol: resolution in momentum acceptance. Default 1e-4

dpuguess: unstable momentum threshold guess. Default [].

If not given it uses the linear momentum acceptance delta_max

from ringpara.

troffset: [x y] starting transverse offset for the tracking.

Default [1e-6 1e-6]

verbose: boolean indicating verbose mode. Default false.

epsilon6D: if not passed, all particles are tracked.

If epsilon6D is given, we track for nturns only

particles having 6D coordinates different by epsilon6D

after being projected to the end of the ring.

Output:

ETN: stability threshold for positive off momentum particles

ETP: stability threshold for negative off momentum particles

Other functions in the file:

Loste = Multiorigin_ringpass_islost

Returns a boolean array: tells whether the particle launched at

the reference point refpts with positive and negative momentum offset is

lost or not.

MomAperture_allRing(..., nturns)#: returns positive and negative momentum aperture

boundaries where the particle is still alive.

The boundary width (i.e. the uncertainty) is equal to

energystep / (splitdivisor^ntimessplit),

meaning that one more step of this size makes the

particle unstable.

[map_l,map_h] …

= **MomAperture_allRing**(

THERING, …

POINTS …

)

[…] = MomAperture_allRing(…, nturns)

Tracks over NTURNS to get the momentum aperture. Default 100

e.g [dppm,dppp]=MomAperture_allRing(thering,positions,nturns)

[…] = MomAperture_allRing(…, ‘reforbit’,orbitin)

The initial particle coordinates are taken from ORBITIN.

Default zeros(6,length(POINTS))

[…] = MomAperture_allRing(…, ‘xyinitoffsets’,[x y])

The transverse offsets to add to the reference orbit.

Default 1e-5*ones(length(POINTS),2)

[…] = MomAperture_allRing(…, ‘deltalimits’,[deltapos deltaneg])

The energy offset limits. Default [0.1 -0.1]

[…] = MomAperture_allRing(…, ‘initialguess’,[posguess negguess])

The starting point of the recursive energy offsets.

Default [0.0 0.0]

[…] = MomAperture_allRing(…, ‘energysteps’,[posstep negstep])

The positive and negative initial energy steps.

Default [0.01 -0.01]

[…] = MomAperture_allRing(…, ‘ntimessplit’,nsplit)

The number of recursive calls reducing the step size. Default 2

[…] = MomAperture_allRing(…, ‘splitdivisor’,splittdivisor)

The step divisor every time we split the step. Default 10.

[…] = MomAperture_allRing(…, ‘verbose’,verbose)

Print info the current position. Default 0.

If set to 1 it will print info at every reference point.

If set to 2 it will print info at each energy step.

ex: [map_l,map_h] = MomAperture_allRing(thering,points,nturns);

ex: [map_l,map_h] = …

MomAperture_allRing(thering,points,nturns,’reforbit’,findorbit6(thering,points));

TLT_IntPiw()#: integral in Piwinski Formula for the Lifetime

TLT_IntPiw_k()#: integral in Piwinski Formula for the Lifetime with u=tan^2(k)

TouschekPiwinskiLifeTime(..., 'abstol', abstol)#: function [Tl,contributionsTL]=(ring,dpp,Ib,…)

evaluates Touschek Lifetime using Piwinski formula

INPUT

**TouschekPiwinskiLifeTime**(

ring,

momentumaperturecolumnvector, column array (size of r or positions)

it can be length(r)x2, for positive and

negative aperture

current per bunch in A, scalar

positions where to evaluate, default: all elements with length>0 column array

emittancex, default: atx modemittance(1) scalar

emittancey, default: emittancex/2 scalar

integration_method, default: ‘integral’, may be: ‘integral’, ‘quad’, ‘trapz’)

energy_spread, scalar

bunch_length, scalar

)

OUTPUT

contributionsTL 1/T contribution at each element

if dpp has positive and negative apertures, then contributionTL is a 2 columns vector

Tl Lifetime in seconds 1/Tl=sum(contributionsTL.*L)/sum(L);

NAME-VALUE PAIRS

TouschekPiwinskiLifeTime(…, ‘abstol’, abstol)

Absolute tolerance for the ‘integral’ function. Default: 1.0e-16

TouschekPiwinskiLifeTime(…, ‘reltol’, abstol)

Relative tolerance for the ‘integral’ function. Default: 1.0e-12

“The Touscheck Effect in strong focusing storage rings”

A.Piwinski, DESY 98-179, November 1998

“Touscheck Effect calculation and its applications to a transport line”

A.Xiao M. Borland, Proceedings of PAC07, Albuquerque, New Mexico, USA

momentum_aperture_at(..., 'reforbit', orbitin)#: recursively offsets the particle energy and checks

for survival over n turns of tracking.

Returns the stable energy boundary.

deltamax …

= **momentum_aperture_at**( …

THERING,…

deltalimit,… [min max]

initcoord,… [x y] initial coordinate

delta,… current energy offset

precdelta,… previous energy offset

deltastepsize,…

splits,… number of times splitting the deltastepsize

split_step_divisor, divides the step size at every split

nturns

)

… = momentum_aperture_at(…, ‘reforbit’,orbitin)

Use ORBITIN to define the reference. Useful when the closed orbit

is not zero.

Adapted routine based on ELEGANT

1. Start with delta = 0, i.e., zero momentum offset.

2. If the limit has been reached stop, otherwise

If the number of step divisions is done, stop. Otherwise …

Track the particle

If it survives, increase the energy by one step, and start 2) again.

Otherwise, go back one step in energy, and divide the step.

Count the number of times the step has been divided.

Start 2) with the new step.

Debugging info prints are commented to avoid speed reduction,

The ELEGANT routine:

https://ops.aps.anl.gov/manuals/elegant_latest/elegantsu53.html

ex: [deltamax]=momentum_aperture_at(thering,0.1,[10^-6 10^-6],0,0,0.01,3,10,100)

ex: [deltamin]=momentum_aperture_at(thering,-0.1,[10^-6 10^-6],0,0,-0.01,3,10,100)