NonLinearDynamics#

Non-linear beam dynamics

Functions

`RDTfluctuationIndicator()`	quantitative representation of RDT fluctuations
`atnuampl()`	computes tune shift with amplitude
`computeRDT()`	Computes Hamiltonian resonance driving terms (RDTs)
`computeRDTfluctuation()`	Computes Hamiltonian resonance driving terms (RDTs)
`tunespaceplot()`	draws a tune diagram

RDTfluctuationIndicator(ring, varargin)#: quantitative representation of RDT fluctuations

This function calls computeRDTfluctuation(ring, varargin) to compute

RDT fluctuations, and provides one example to quantitatively represents

the RDT fluctuations.

[h3ave,h4ave,h3chroave,adts]=RDTfluctuationIndicator(ring,varargin)

ring is the AT lattice

The additional argument:

nslices: number of slices of each sextupole, which affects the

crossing terms. default: 4.

h3ave and h4ave quantitatively represents 3rd- and 4th-order geometric

RDT fluctuations, respectively.

h3ave + w * h4ave can be used to represents geometric RDT fluctuations.

The coefficient w can be estimated by action of particle at the

anticipated DA, w ~ (2J_x)^0.5 = x / betax^0.5, usually 0.01 is OK.

h3chroave is the fluctuation of 3rd-order chromatic RDTs,

defined similarly to h3ave.

adts is the sum of three ADTS terms, which are also used in onlinear optimization.

The fluctuations of ADTS terms are not considered.

It is calculated here to avoid duplicate computation.

Noting:

1.This function provides one example to quantitatively represents the

RDT fluctuations similar to that in Ref.[1]. But in Ref.[1], only

the RDTs at the locations of nonlinear magnets are considered.

We think the differences are small and it’s more important to

keep the function simple.

2.People can call computeRDTfluctuation(ring, varargin) directly,

and try other quantitative representations.

3.The build-up RDT fluctuation can also be used, see Ref.[1].

If the build-up RDT fluctuations are used, it is better to calculate

the RDT build-up fluctuations for multiple periods to have

better convergence of calculation.

REFERENCE:

[1] B. Wei, Z. Bai, J. Tan, L. Wang, and G. Feng, Phys. Rev. Accel. Beams 26, 084001 (2023)

atnuampl(ring, amplitude)#: computes tune shift with amplitude

[nux,nuz]=atnuampl(ring,amplitude)

[nux,nuz]=atnuampl(ring,amplitude,1)

Computes tunes for the specified horizontal amplitudes

[nux,nuz]=atnuampl(ring,amplitude,3)

Computes tunes for the specified vertical amplitudes

atnuampl(…)

Plots the computed tunes in the current axes

atnuampl(…,name,value)

Uses additional options specified by one or more Name,Value pairs.

Possible values are:

orbit: initial closed orbit

nturns: specify the number of turns for tracking (default 256)

method: specify the method for tune determination

1: Highest peak in fft

2: Interpolation on fft results

3: Windowing + interpolation (default)

4: NAFF

Other options are transmitted to the plot function

computeRDT(ring, index, varargin)#: Computes Hamiltonian resonance driving terms (RDTs)

This function calls RDTElegantAT mex function and returns the

hamiltonian resonance driving terms, using the elegant c++

function computeDrivingTerms().

rdt=computeRDT(ring, index, varargin)

ring is the AT lattice

index is the vector of indexes where one wants to compute RDTs

The additional arguments can be up to five strings:

chromatic, coupling, geometric1, geometric2 and tuneshifts

example:

rdt=computeRDT(ring, indexbpm, ‘geometric1’, ‘tuneshifts’);

creates an array of structs (the length of the array is the number of

indexes where you want to compute driving terms) with first order

geometric driving terms and tune shifts with amplitude.

The driving terms are complex numbers, the tune shifts are real.

computeRDTfluctuation(ring, varargin)#: Computes Hamiltonian resonance driving terms (RDTs)

This function is based on simplestoragering and returns the RDTs

and their longitudinal fluctuations.

[rdt,buildup_fluctuation,natural_fluctuation]=computeRDTfluctuation(ring, varargin)

ring is the AT lattice

The additional arguments:

nslices: number of slices of each sextupole, which affects the crossing

terms. default: 4.

nperiods: number of periods. RDTs and RDT build-up fluctuations will be

computed for n periods. default: 1.

natural RDT fluctuation of different periods are the same.

So the results contain only one period.

RDT: struct, RDTs (complex numbers) and

amplitude-dependent tune shifts (real)

(ADTS are calculated using h22000, h11110 and h00220)

buildup_fluctuation: a struct of complex arrays,

accumulated RDTs from s=0,

showing the build-up and cancellation of RDTs along the

longitudinal position.

natural_fluctuation: a struct of double arrays,

absolute values of one-period RDTs observed at different

longitudinal starting position.

same as s_dependent_driving_terms in ELEGANT.

REFERENCES

[1] Johan Bengtsson, SLS Note 09/97, (1997)

[2] S. C. Leemann, A. Streun, Phys. Rev. ST Accel. Beams 14, 030701 (2011)

[3] A. Franchi, L. Farvacque, F. Ewald, G. Le Bec, and K. B. Scheidt, Phys. Rev. ST Accel. Beams 17, 074001 (2014)

[4] B. Wei, Z. Bai, J. Tan, L. Wang, and G. Feng, Phys. Rev. Accel. Beams 26, 084001 (2023)

tunespaceplot()#: draws a tune diagram

resonance lines: m*nu_x + n*nu_y = p