LinearOptics#

Linear beam dynamics

Functions

`amat()`	find A matrix from one turn map matrix T such that:
`atdampingrates()`	find tunes and damping rates from one map matrix with radiation
`atlinopt()`	Performs 4D linear analysis of the COUPLED lattices
`atlinopt2()`	Performs the linear analysis of UNCOUPLED lattices
`atlinopt4()`	Performs the 4D linear analysis of COUPLED lattices
`atlinopt6()`	Performs linear analysis of the lattice
`beam22()`	computes the beam matrix from the 1-turn transfer matrix
`beam44()`	computes the coupled beam matrices
`find_betaoids()`	[H1 H2 H3]=(A)
`find_etaoids()`	Given the normalizing matrix A, we compute the etaoids
`find_inv_G()`	This function computes the invariants of a one turn map matrix.
`find_inv_G_fromA()`	This function computes the invariant matrices G1,G2,G3
`findelemm44()`	numerically finds the 4x4 transfer matrix of an element
`findelemm66()`	numerically finds the 6x6 transfer matrix of an element
`findm44()`	numerically finds the 4x4 transfer matrix of an accelerator lattice
`findm66()`	numerically finds the 6x6 transfer matrix of an accelerator lattice
`get_dispersion_from_etaoids()`	computes dispersion functions (x,px,y,py) at refpts
`jmat()`	Compute antisymmetric Matrix [O 1; -1 0]
`linopt()`	performs linear analysis of the COUPLED lattices
`mkSRotationMatrix()`	(PSI) coordinate transformation matrix
`plotbeta()`	plots UNCOUPLED! beta-functions
`r_analysis()`	bk=cellfun(@(slc) ai(:,slc)*ai(:,slc)’,slices,’UniformOutput’,false); % Only if symplectic

amat()#: find A matrix from one turn map matrix T such that:

[Rotx 0 0 ]

inv(A)*T*A=[ 0 Rotz 0 ]

[ 0 0 Rots]

Order it so that it is close to the order of x,y,z

also ensure that positive modes are before negative so that

one has proper symplecticity

B. Nash July 18, 2013

we order and normalize the vectors via

v_j’ jmat(3) v_k = i sgn(j) delta(j,k)

atdampingrates(m66)#: find tunes and damping rates from one map matrix with radiation

[nu,chi]=atdampingrates(m66)

note that in order to find the damping times, one needs the revolution

time T0, then

tau1 = T0/chi1, tau2 = T0/chi2, tau3 = T0/chi3

atlinopt(ring, dp, refpts)#: Performs 4D linear analysis of the COUPLED lattices

atlinopt only works for CONSTANT energy. So no PassMethod should:

1. change the longitudinal momentum dP (cavities, magnets with radiation)

2. have any time dependence (localized impedance, fast kickers etc)

For a more general computation, see ATLINOPT6

lindata = atlinopt(ring,dp,refpts) is a MATLAB structure array with fields

ElemIndex - ordinal position in the RING

SPos - longitudinal position [m]

ClosedOrbit - 4x1 closed orbit vector with components

x, px, y, py (momentums, NOT angles)

Dispersion - 4x1 dispersion with components

eta_x, eta_prime_x, eta_y, eta_prime_y

M44 - 4x4 transfer matrix M from the beginning of RING

to the entrance of the element [2]

A - 2x2 matrix A in [3]

B - 2x2 matrix B in [3]

C - 2x2 matrix C in [3]

gamma - gamma parameter of the transformation to eigenmodes

mu - [mux, muy] horizontal and vertical betatron phase advances

beta - [betax, betay] vector

alpha - [alphax, alphay] vector

All values are specified at the entrance of each element specified in REFPTS.

REFPTS is an array of increasing indexes that select elements

from the range 1 to length(LINE)+1. Defaults to 1 (initial point)

See further explanation of REFPTS in the ‘help’ for FINDSPOS

[lindata,nu] = atlinopt() returns a vector of linear tunes

[nu_u , nu_v] for two normal modes of linear motion [1]

[lindata,nu, ksi] = atlinopt() returns a vector of chromaticities ksi = d(nu)/(dp/p)

[ksi_u , ksi_v] - derivatives of [nu_u , nu_v]

[…] = atlinopt(…,’orbit’,orbitin)

[…] = atlinopt(ring,dp,refpts,orbitin) (deprecated syntax)

Do not search for closed orbit. Instead ORBITIN,a 6x1 vector

of initial conditions is used: [x0; px0; y0; py0; DP; 0].

The sixth component is ignored.

This syntax is useful to specify the entrance orbit if RING is not a

ring or to avoid recomputing the closed orbit if is already known.

[…] = atlinopt(…,’coupled’,flag)

If flag is false, a faster computation is performed

assuming no coupling in the lattice. Default: true

[…] = atlinopt(…,’ct’,ct)

Instead of computing the linear optics for the specified DP/P,

computes for the path lenghening specified by CT.

The DP argument is ignored.

[…] = atlinopt(…,’twiss_in’,twissin)

Computes the optics for a transfer line.

TWISSIN is a scalar structure with fields:

ClosedOrbit - 4x1 initial closed orbit. Default: zeros(4,1)

Dispersion - 4x1 initial dispersion. Default: zeros(4,1)

mu - [ mux, muy] horizontal and vertical betatron phase

beta - [betax0, betay0] vector

alpha - [alphax0, alphay0] vector

Difference with linopt: Fractional tunes 0<=tune<1

Dispersion output

Alpha output

Phase advance output

Option to skip closed orbit search

REFERENCES

[1] D.Edwars,L.Teng IEEE Trans.Nucl.Sci. NS-20, No.3, p.885-888, 1973

[2] E.Courant, H.Snyder

[3] D.Sagan, D.Rubin Phys.Rev.Spec.Top.-Accelerators and beams, vol.2 (1999)

See also atlinopt2(), atlinopt4(), atlinopt6(), atx(), tunechrom()

atlinopt2(ring, refpts)#: Performs the linear analysis of UNCOUPLED lattices

[ringdata,elemdata] = atlinopt2(ring,refpts)

IMPORTANT!!! atlinopt2 assumes a constant momentum deviation.

PassMethods used for any element in the RING SHOULD NOT

1.change the longitudinal momentum dP

(cavities , magnets with radiation, …)

2.have any time dependence (localized impedance, fast kickers, …)

RINGDATA is a structure array with fields:

tune 1x2 tunes

chromaticity 1x2 chromaticities (only with get_chrom or get_w flags)

ELEMDATA is a structure array with fields:

SPos - longitudinal position [m]

ClosedOrbit - 4x1 closed orbit vector with components

x, px, y, py (momentums, NOT angles)

Dispersion - [eta_x; eta’_x; eta_y; eta’_y] 4x1 dispersion vector

M - 4x4 transfer matrix M from the beginning of RING

to the entrance of the element [2]

beta - [betax, betay] vector

alpha - [alphax, alphay] vector

mu - [mux, muy] horizontal and vertical betatron phase advances

W - [Wx, Wy] Chromatic amplitude function [3] (only with the

get_w flag)

All values are specified at the entrance of each element specified in REFPTS.

REFPTS is an array of increasing indexes that select elements

from the range 1 to length(LINE)+1. Defaults to 1 (initial point)

See further explanation of REFPTS in the ‘help’ for FINDSPOS

Use the Matlab function “cat” to get the data from fields of ELEMDATA as MATLAB arrays.

Example:

>> [ringdata, elemdata] = atlinopt2(ring,1:length(ring));

>> beta = cat(1,elemdata.beta);

>> s = cat(1,elemdata.SPos);

>> plot(S,beta)

[…] = atlinopt2(…,’get_chrom’)

Trigger the computation of chromaticities

[…] = atlinopt2(…,’get_w’)

Trigger the computation of chromatic amplitude functions (time consuming)

[…] = atlinopt2(…,’dp’,dpp)

Analyses the off-momentum lattice by specifying the central

off-momentum value

[…] = atlinopt2(…,’ct’,dct)

Analyses the off-momentum lattice by specifying the path lengthening

corresponding to a frequency shift. The resulting deltap/p is returned

in the 5th component of the ClosedOrbit field.

[…] = atlinopt2(…,’orbit’,orbitin)

Do not search for closed orbit. Instead ORBITIN,a 6x1 vector

of initial conditions is used: [x0; px0; y0; py0; DP; 0].

The sixth component is ignored.

This syntax is useful to specify the entrance orbit if RING is not a

ring or to avoid recomputing the closed orbit if is already known.

[…] = atlinopt2(…,’twiss_in’,twissin)

Computes the optics for a transfer line.

TWISSIN is a scalar structure with fields:

ClosedOrbit - 4x1 initial closed orbit. Default: zeros(4,1)

Dispersion - 4x1 initial dispersion. Default: zeros(4,1)

mu - [ mux, muy] horizontal and vertical betatron phase

beta - [betax0, betay0] vector

alpha - [alphax0, alphay0] vector

REFERENCES

[1] D.Edwards,L.Teng IEEE Trans.Nucl.Sci. NS-20, No.3, p.885-888, 1973

[2] E.Courant, H.Snyder

[3] Brian W. Montague Report LEP Note 165, CERN, 1979

See also atlinopt(), atlinopt4(), atlinopt6(), tunechrom()

atlinopt4(ring, refpts)#: Performs the 4D linear analysis of COUPLED lattices

[ringdata,elemdata] = atlinopt4(ring,refpts)

IMPORTANT!!! atlinopt4 assumes a constant momentum deviation.

PassMethods used for any element in the RING SHOULD NOT

1.change the longitudinal momentum dP

(cavities , magnets with radiation, …)

2.have any time dependence (localized impedance, fast kickers, …)

RINGDATA is a structure array with fields:

tune 1x2 tunes

chromaticity 1x2 chromaticities (only with get_chrom or get_w flags)

ELEMDATA is a structure array with fields:

SPos - longitudinal position [m]

ClosedOrbit - 4x1 closed orbit vector with components

x, px, y, py (momentums, NOT angles)

Dispersion - [eta_x; eta’_x; eta_y; eta’_y] 4x1 dispersion vector

M - 4x4 transfer matrix M from the beginning of RING

to the entrance of the element [2]

A - 2x2 matrix A in [4]

B - 2x2 matrix B in [4]

C - 2x2 matrix C in [4]

gamma - gamma parameter of the transformation to eigenmodes [4]

beta - [betax, betay] vector

alpha - [alphax, alphay] vector

mu - [mux, muy] Betatron phase advances

W - [Wx, Wy] Chromatic amplitude function [3] (only with the

get_w flag)

Use the Matlab function “cat” to get the data from fields of ELEMDATA as MATLAB arrays.

Example:

>> [ringdata, elemdata] = atlinopt4(ring,1:length(ring));

>> beta = cat(1,elemdata.beta);

>> s = cat(1,elemdata.SPos);

>> plot(S,beta)

All values are specified at the entrance of each element specified in REFPTS.

REFPTS is an array of increasing indexes that select elements

from the range 1 to length(LINE)+1. Defaults to 1 (initial point)

See further explanation of REFPTS in the ‘help’ for FINDSPOS

[…] = atlinopt4(…,’get_chrom’)

Trigger the computation of chromaticities

[…] = atlinopt4(…,’get_w’)

Trigger the computation of chromatic amplitude functions (time consuming)

[…] = atlinopt4(…,’dp’,dpp)

Analyses the off-momentum lattice by specifying the central

off-momentum value

[…] = atlinopt4(…,’ct’,dct)

Analyses the off-momentum lattice by specifying the path lengthening

corresponding to a frequency shift. The resulting deltap/p is returned

in the 5th component of the ClosedOrbit field.

[…] = atlinopt4(…,’df’,df)

Analyses the off-momentum lattice by specifying the RF frequency shift.

The resulting deltap/p is returned in the 5th component of the ClosedOrbit field.

[…] = atlinopt4(…,’orbit’,orbitin)

Do not search for closed orbit. Instead ORBITIN,a 6x1 vector

of initial conditions is used: [x0; px0; y0; py0; DP; 0].

The sixth component is ignored.

This syntax is useful to specify the entrance orbit if RING is not a

ring or to avoid recomputing the closed orbit if is already known.

[…] = atlinopt4(…,’twiss_in’,twissin)

Computes the optics for a transfer line.

TWISSIN is a scalar structure with fields:

ClosedOrbit - 4x1 initial closed orbit. Default: zeros(4,1)

Dispersion - 4x1 initial dispersion. Default: zeros(4,1)

mu - [ mux, muy] horizontal and vertical betatron phase

beta - [betax0, betay0] vector

alpha - [alphax0, alphay0] vector

REFERENCES

[1] D.Edwards,L.Teng IEEE Trans.Nucl.Sci. NS-20, No.3, p.885-888, 1973

[2] E.Courant, H.Snyder

[3] Brian W. Montague Report LEP Note 165, CERN, 1979

[4] D.Sagan, D.Rubin Phys.Rev.Spec.Top.-Accelerators and beams, vol.2 (1999)

See also atlinopt2(), atlinopt6(), tunechrom()

atlinopt6(ring, refpts)#: Performs linear analysis of the lattice

[ringdata,elemdata] = atlinopt6(ring,refpts)

For circular machines, atlinopt6 analyses

the 4x4 1-turn transfer matrix if radiation is OFF, or

the 6x6 1-turn transfer matrix if radiation is ON.

For a transfer line, The “twiss_in” intput must contain either:

- a field ‘R’, as provided by atlinopt6, or

- the fields ‘beta’ and ‘alpha’, as provided by ATLINOPT and atlinopt6

RINGDATA is a structure array with fields:

tune Fractional tunes

damping_time Damping times [s]

chromaticity Chromaticities

ELEMDATA is a structure array with fields:

SPos - longitudinal position [m]

ClosedOrbit - 6x1 closed orbit vector with components

x, px, y, py, dp/d, ct (momentums, NOT angles)

Dispersion - [eta_x; eta’_x; eta_y; eta’_y] 4x1 dispersion vector

R - DxDx(D/2) R-matrices

A - DxD A-matrix

M - DxD transfer matrix M from the beginning of RING

beta - [betax, betay] 1x2 beta vector

alpha - [alphax, alphay] 1x2 alpha vector

mu - [mux, muy] Betatron phase advances

W - [Wx, Wy] Chromatic amplitude function [3] (only with the

get_w flag)

Use the Matlab function “cat” to get the data from fields of ELEMDATA as MATLAB arrays.

Example:

>> [ringdata, elemdata] = atlinopt6(ring,1:length(ring));

>> beta = cat(1,elemdata.beta);

>> s = cat(1,elemdata.SPos);

>> plot(S,beta)

All values are specified at the entrance of each element specified in REFPTS.

REFPTS is an array of increasing indexes that select elements

from the range 1 to length(LINE)+1. Defaults to 1 (initial point)

See further explanation of REFPTS in the ‘help’ for FINDSPOS

[…] = atlinopt6(…,’get_chrom’)

Trigger the computation of chromaticities

[…] = atlinopt6(…,’get_w’)

Trigger the computation of chromatic amplitude functions (time consuming)

[…] = atlinopt6(…,’orbit’,orbitin)

Do not search for closed orbit. Instead ORBITIN,a 6x1 vector

of initial conditions is used: [x0; px0; y0; py0; DP; 0].

The sixth component is ignored.

This syntax is useful to specify the entrance orbit if RING is not a

ring or to avoid recomputing the closed orbit if is already known.

[…] = atlinopt6(…,’twiss_in’,twissin)

Computes the optics for a transfer line.

TWISSIN is a scalar structure with fields:

R 4x4x2 R-matrices

or

beta [betax0, betay0] vector

alpha [alphax0, alphay0] vector

[…] = atlinopt6(…,’dp’,dp)

Specify off-momentum

[…] = atlinopt6(…,’dct’,dct)

Specify the path lengthening

[…] = atlinopt6(…,’df’,df)

Specify the RF frequency deviation

REFERENCES

[1] Etienne Forest, Phys. Rev. E 58, 2481 – Published 1 August 1998

[2] Andrzej Wolski, Phys. Rev. ST Accel. Beams 9, 024001 –

Published 3 February 2006

[3] Brian W. Montague Report LEP Note 165, CERN, 1979

See also atlinopt2(), atlinopt4(), tunechrom()

beam22(t)#: computes the beam matrix from the 1-turn transfer matrix

%

beam=beam22(t)

T: 1-turn transfer matrix

BEAM: envelope matrix

[beam,tune]=beam22(t)

also returns the tune

beam44(a, b, c, gamma)#: computes the coupled beam matrices

[beam1,beam2]=beam44(a,b,c,gamma)

A,B,C,gamma: Coupling parameters, see [1]

BEAM1,BEAM2: Eigen modes

[beam1,beam2]=beam44(lindata)

LINDATA: structure with fields A,B,C,gamma

[beam1,beam2,tune1,tune1]=beam44(…)

also returns the tunes

[1] Sagan, Rubin, “Linear Analysis of Coupled Lattices”

Phys.Rev.Spec.Top. - Accelerators and Beams, vol2, 1999

find_betaoids()#: [H1 H2 H3]=(A)

Given the normalizing matrix A, we compute the betaoids

(in Forest’s terminology)

these can be related to the invariants

find_etaoids()#: Given the normalizing matrix A, we compute the etaoids

(in terminology of E. Forest)

these can be related to the dispersion

find_inv_G()#: This function computes the invariants of a one turn map matrix.

The resulting invariant matrices G1,G2,G3 satisfy

M^T G_a M = G_a for a=1,2,3

Algorithm from PhD thesis of B. Nash

find_inv_G_fromA()#: This function computes the invariant matrices G1,G2,G3

starting from the A matrix computed via the function amat().

The resulting invariant matrices G1,G2,G3 satisfy

M^T G_a M = G_a for a=1,2,3

findelemm44(elem, methodname)#: numerically finds the 4x4 transfer matrix of an element

findelemm44(elem, methodname)

ELEM - the element data structure

METHODNAME - name of the pass-method function

(default: ELEM.PassMethod)

m66=findelemm44(…,’orbit’,orbitin) (deprecated syntax)

m66=findelemm44(elem, methodname, orbitin)

ORBITIN - 6x1 phase space coordinates at the entrance

(default: zeros(6,1))

The transverse matrix is momentum-dependent,

the 5-th component of ORBITIN is used as the DP value

m66=findelemm44(…,’energy’,energy)

Use ENERGY and ignore the ‘Energy’ field of elements

m66=findelemm44(…,’particle’,particle)

Use PARTICLE (default is relativistic)

See also findelemm66()

findelemm66(elem, methodname)#: numerically finds the 6x6 transfer matrix of an element

m66=findelemm66(elem, methodname)

ELEM - the element data structure

METHODNAME - name of the pass-method function

(default: ELEM.PassMethod)

m66=findelemm66(elem, methodname, orbitin) (deprecated syntax)

m66=findelemm66(…,’orbit’,orbitin)

ORBITIN - 6-by-1 phase space coordinates at the entrance

(default: zeros(6,1))

m66=findelemm66(…,’energy’,energy)

Use ENERGY and ignore the ‘Energy’ field of elements

m66=findelemm66(…,’particle’,particle)

Use PARTICLE (default is relativistic)

See also findelemm44()

findm44(lattice)#: numerically finds the 4x4 transfer matrix of an accelerator lattice

for a particle with relative momentum deviation DP

IMPORTANT!!!

findm44 gives a wrong result with 6-d rings.

findm44 assumes constant momentum deviation.

PassMethod used for any element in the LATTICE SHOULD NOT

1.change the longitudinal momentum dP

(cavities , magnets with radiation, …)

2.have any time dependence (localized impedance, fast kickers, …)

m44 = findm44(lattice) finds the full one-turn

matrix at the entrance of the first element

!!! With this syntax findm44 assumes that the LATTICE

is a ring and first finds the closed orbit

[m44,t] = findm44(lattice,refpts) also returns

4x4 transfer matrixes between entrance of

the first element and each element indexed by REFPTS.

T is 4x4xlength(REFPTS) 3 dimensional array

so that the set of indexes (:,:,i) selects the 4-by-4

matrix at the i-th reference point.

Note: REFPTS is an array of increasing indexes that

select elements from range 1 to length(LATTICE)+1.

See further explanation of REFPTS in the ‘help’ for FINDSPOS

When REFPTS= [ 1 2 .. ] the fist point is the entrance of the

first element and T(:,:,1) - identity matrix

Note: REFPTS is allowed to go 1 point beyond the

number of elements. In this case the last point is

the EXIT of the last element. If LATTICE is a RING

it is also the entrance of the first element

after 1 turn: T(:,:,end) = M

[…] = findm44(ring,…,’dp’,dp)

[…] = findm44(ring,dp,refpts,…) (deprecated syntax)

Computes for the off-momentum DP

[…] = findm44(ring,…,’dct’,dct)

Computes for the path lenghening specified by CT.

[…] = findm44(ring,…,’df’,df)

Computes for a deviation of RF frequency DF

[…] = findm44(ring,…,’orbit’,orbitin)

[…] = findm44(ring,dp,refpts,orbitin) (deprecated syntax)

Do not search for closed orbit. Instead ORBITIN,a 6x1 vector

of initial conditions is used: [x0; px0; y0; py0; DP; 0].

The sixth component is ignored.

This syntax is useful to specify the entrance orbit if RING is not a

ring or to avoid recomputing the closed orbit if is already known.

[…] = findm44(…,’full’)

Same as above except that matrices returned in T are full 1-turn

matrices at the entrance of each element indexed by REFPTS.

[m44,t,orbit] = findm44(…)

In addition returns the closed orbit at the entrance of each element

indexed by REFPTS.

See also findm66(), findorbit4(), atenable_6d(), atdisable_6d(), check_6d()

findm66(ring)#: numerically finds the 6x6 transfer matrix of an accelerator lattice

by differentiation of LINEPASS near the closed orbit

findm66 uses FINDORBIT6 to search for the closed orbit in 6-d

In order for this to work the ring MUST have a CAVITY element

m66 = findm66(ring) finds the full one-turn 6-by-6

matrix at the entrance of the first element

[…]=findm66(ring,…,’dp’,dp) Specify the momentum deviation when

radiation is OFF (default: 0)

[…]=findm66(ring,…,’dct’,dct) Specify the path lengthening when

radiation is OFF (default: 0)

[…]=findm66(ring,…,’df’,df) Specify the RF frequency deviation

radiation is OFF (default: 0)

[…]=findm66(ring,…,’orbit’,orbit) Specify the orbit at the entrance

of the ring, if known.

[m66,t] = findm66(ring,refpts) in addition to M finds

6-by-6 transfer matrixes between entrances of

the first element and each element indexed by REFPTS.

T is 6-by-6-by-length(REFPTS) 3 dimentional array.

REFPTS is an array of increasing indexes that select elements

from the range 1 to length(RING)+1.

See further explanation of REFPTS in the ‘help’ for FINDSPOS

Note:

When REFPTS= [ 1 2 .. ] the first point is the entrance of the first element

and T(:,:,1) - identity matrix

When REFPTS= [ .. length(RING)+1] the last point is the exit of the last element

and the entrance of the first element after 1 turn: T(:,:, ) = M

[…] = findm66(ring,refpts,orbitin) (deprecated syntax)

[…] = findm66(…,’orbit’,orbitin)

Do not search for closed orbit. This syntax is useful to avoid

recomputing the closed orbit if it is already known.

[m66,t,orbit] = findm66(…)

In addition returns the closed orbit at the entrance of each element

indexed by REFPTS.

See also findm44(), findorbit6()

get_dispersion_from_etaoids()#: computes dispersion functions (x,px,y,py) at refpts

using the etaoids (E. Forest terminology) which are computed from the one turn map

jmat()#: Compute antisymmetric Matrix [O 1; -1 0]

INPUTS

1. dim - 1,2,3 Dimension of the sqare matrix

OUPUTS

2. mat - Antisymmetric block Matrix [O 1; -1 0]

See also symplectify()

linopt(ring, dp, refpts)#: performs linear analysis of the COUPLED lattices

Notation is the same as in reference [3]

lindata = linopt(ring,dp,refpts) is a MATLAB structure array with fields

ElemIndex - ordinal position in the RING

SPos - longitudinal position [m]

ClosedOrbit - closed orbit column vector with

components x, px, y, py (momentums, NOT angles)

Dispersion - dispersion orbit position vector with

components eta_x, eta_prime_x, eta_y, eta_prime_y

calculated with respect to the closed orbit with

momentum deviation DP

M44 - 4x4 transfer matrix M from the beginning of RING

to the entrance of the element for specified DP [2]

A - 2x2 matrix A in [3]

B - 2x2 matrix B in [3]

C - 2x2 matrix C in [3]

gamma - gamma parameter of the transformation to eigenmodes

mu - [ mux, muy] horizontal and vertical betatron phase

beta - [betax, betay] vector

All values are specified at the entrance of each element specified in REFPTS.

REFPTS is an array of increasing indexes that select elements

from the range 1 to length(LINE)+1.

See further explanation of REFPTS in the ‘help’ for FINDSPOS

[lindata,nu] = linopt() returns a vector of linear tunes

[nu_u , nu_v] for two normal modes of linear motion [1]

[lindata,nu, ksi] = linopt() returns a vector of chromaticities ksi = d(nu)/(dp/p)

[ksi_u , ksi_v] - derivatives of [nu_u , nu_v]

[1] D.Edwars,L.Teng IEEE Trans.Nucl.Sci. NS-20, No.3, p.885-888, 1973

[2] E.Courant, H.Snyder

[3] D.Sagan, D.Rubin Phys.Rev.Spec.Top.-Accelerators and beams, vol.2 (1999)

See also findspos(), twissring(), tunechrom()

mkSRotationMatrix()#: (PSI) coordinate transformation matrix

that describes s-rotation of the ELEMENT

| cos(psi) 0 sin(psi) 0 0 0 |

| 0 cos(psi) 0 sin(psi) 0 0 |

| -sin(psi) 0 cos(psi) 0 0 0 |

| 0 -sin(psi) 0 cos(psi) 0 0 |

| 0 0 0 0 1 0 |

| 0 0 0 0 0 1 |

Note: AT defines ‘positive’ s-rotation as clockwise,

looking in the dirction of the beamm

plotbeta(ring)#: plots UNCOUPLED! beta-functions

plotbeta(ring) calculates beta functions of the lattice RING

plotbeta with no argumnts uses THERING as the default lattice

Note: plotbeta uses FINDORBIT4 and LINOPT which assume a lattice

with NO accelerating cavities and NO radiation

See also plotcod()

r_analysis()#: bk=cellfun(@(slc) ai(:,slc)*ai(:,slc)’,slices,’UniformOutput’,false); % Only if symplectic