Orbit#

Closed orbit search

Functions

`findorbit()`	find the closed orbit
`findorbit4()`	finds closed orbit in the 4-d transverse phase
`findorbit6()`	finds closed orbit in the full 6-d phase space
`findsyncorbit()`	finds closed orbit, synchronous with the RF cavity
`plotcod()`	Plots the closed orbit distortion

findorbit(ring, refpts)#: find the closed orbit

Depending on the lattice, findorbit will:

- use findorbit6 if radiation is ON,

- use findsyncorbit if radiation is OFF and ct is specified,

- use findorbit4 otherwise

[orbit,fixedpoint]=findorbit(ring,refpts)

ORBIT: 6xNrefs, closed orbit at the selected reference points

FIXEDPOINT: 6x1 closed orbit at the entrance of the lattice

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

radiation is OFF (default: 0)

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

radiation is OFF (default: 0)

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

radiation is OFF (default: 0)

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

of the ring, if known. findorbit will then transfer it to the reference points

[…]=findorbit(ring,…,’guess’,guess) The search will start with GUESS as

initial conditions

For other keywords, see the underlying functions.

See also findorbit4(), findsyncorbit(), findorbit6()

findorbit4(ring)#: finds closed orbit in the 4-d transverse phase

space by numerically solving for a fixed point of the one turn

map M calculated with LINEPASS

(X, PX, Y, PY, dP2, CT2 ) = M (X, PX, Y, PY, dP1, CT1)

under the CONSTANT MOMENTUM constraint, dP2 = dP1 = dP and

there is NO constraint on the 6-th coordinate CT

IMPORTANT !!!

findorbit4 gives a wrong result with 6-d rings. If you still want to

get the result, set ‘strict’ to -1 or 0.

findorbit4 imposes a constraint on dP and relaxes

the constraint on the revolution frequency. A physical storage

ring does exactly the opposite: the momentum deviation of a

particle on the closed orbit settles at the value

such that the revolution is synchronous with the RF cavity

HarmNumber*Frev = Frf

To impose this artificial constraint in findorbit4

PassMethod used for any elemen SHOULD NOT

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

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

findorbit4(ring) is 4x1 vector - fixed point at the

entrance of the 1-st element of the RING (x,px,y,py)

findorbit4(ring,refpts) is 4xlength(refpts)

array of column vectors - fixed points (x,px,y,py)

at the entrance of each element indexed by the REFPTS 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

findorbit4(ring,dp,refpts,…) Obsolete syntax

findorbit4(ring, …,’strict’,strict) Default: 0.

If STRICT is -1, check_6d is skipped

If STRICT is 0, check_6d emits a warning for non-6d rings.

If STRICT is 1, check_6d emits an error for non-6d rings.

findorbit4(ring,…,’dp’,dp) Specifies the off-momentum

findorbit4(ring,…,’dct’,dct) Specifies the path lengthening

findorbit4(ring,…,’df’,df) Specifies RF frequency deviation

findorbit4(ring,dp,refpts,guess)

findorbit4(…,’guess’,guess) The search for the fixed point

starts from initial condition GUESS. Otherwise the search starts from

[0; 0; 0; 0; 0; 0]. GUESS must be a 6x1 vector.

findorbit4(…,’orbit’,orbit) Specify the orbit at the entrance

of the ring, if known. findorbit4 will then transfer it to the

reference points. ORBIT must be a 6x1 vector.

[orbit, fixedpoint] = findorbit4( … )

The optional second return parameter is a 6x1 vector:

closed orbit at the entrance of the RING.

See also findsyncorbit(), findorbit6(), atdisable_6d(), check_6d()

findorbit6(ring, refpts, guess)#: finds closed orbit in the full 6-d phase space

by numerically solving for a fixed point of the one turn

map M calculated with RINGPASS

(X, PX, Y, PY, DP, CT2 ) = M (X, PX, Y, PY, DP, CT1)

with constraint % CT2 - CT1 = C*HarmNumber(1/Frf - 1/Frf0)

IMPORTANT!!! findorbit6 is a realistic simulation

1. The Frf frequency in the RF cavities (may be different from Frf0)

imposes the synchronous condition

CT2 - CT1 = C*HarmNumber(1/Frf - 1/Frf0)

2. The algorithm numerically calculates

6-by-6 Jacobian matrix J6. In order for (J-E) matrix

to be non-singular it is NECESSARY to use a realistic

PassMethod for cavities with non-zero momentum kick

(such as RFCavityPass).

3. findorbit6 can find orbits with radiation.

In order for the solution to exist the cavity must supply

adequate energy compensation.

In the simplest case of a single cavity, it must have

‘Voltage’ field set so that Voltage > Erad - energy loss per turn

4. findorbit6 starts the search from [ 0 0 0 0 0 0 ]’, unless

the third argument is specified: findorbit6(ring,refpts,guess)

There exist a family of solutions that correspond to different RF buckets

They differ in the 6-th coordinate by C*Nb/Frf. Nb = 1 .. HarmNum-1

5. The value of the 6-th coordinate found at the cavity gives

the equilibrium RF phase. If there is no radiation the phase is 0;

findorbit6(ring) is 6x1 vector - fixed point at the

entrance of the 1-st element of the RING (x,px,y,py,dp,ct)

findorbit6(ring,refpts) is 6xlength(refpts)

array of column vectors - fixed points (x,px,y,py,dp,ct)

at the entrance of each element indexed by the REFPTS 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

findorbit6(…,’dp’,dp)

Specify the off-momentum. The RF frequency will be adjusted to get the

desired value

findorbit6(…,’dct’,dct)

Specify the path lengthening. The RF frequency will be adjusted to get

the desired value

findorbit6(…,’df’,df)

Specify the RF frequency deviation

findorbit6(ring,refpts,guess)

findorbit6(…,’guess’,guess) The search for the fixed point

starts from initial condition GUESS. Otherwise the search starts from

the synchronous phase. GUESS must be a 6x1 vector.

findorbit6(…,’orbit’,orbit) Specify the orbit at the entrance

of the ring, if known. findorbit6 will then transfer it to the

reference points. ORBIT must be a 6x1 vector.

[orbit, fixedpoint] = findorbit6( … )

The optional second return parameter is a 6x1 vector:

closed orbit at the entrance of the RING.

See also findorbit4(), findsyncorbit()

findsyncorbit(ring)#: finds closed orbit, synchronous with the RF cavity

and momentum deviation dP (first 5 components of the phase space vector)

by numerically solving for a fixed point

of the one turn map M calculated with LINEPASS

(X, PX, Y, PY, dP2, CT2 ) = M (X, PX, Y, PY, dP1, CT1)

under constraints CT2 - CT1 = dCT = C(1/Frev - 1/Frev0) and dP2 = dP1 , where

Frev0 = Frf0/HarmNumber is the design revolution frequency

Frev = (Frf0 + dFrf)/HarmNumber is the imposed revolution frequency

IMPORTANT!!!

findsyncorbit gives a wrong result with 6-d rings. If you still want to

get the result, set ‘strict’ to -1 or 0.

findsyncorbit imposes a constraint (CT2 - CT1) and

dP2 = dP1 but no constraint on the value of dP1, dP2

The algorithm assumes time-independent fixed-momentum ring

to reduce the dimensionality of the problem.

To impose this artificial constraint in findsyncorbit

PassMethod used for any element SHOULD NOT

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

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

findsyncorbit(ring) is 5x1 vector - fixed point at the

entrance of the 1-st element of the RING (x,px,y,py,dP)

findsyncorbit(ring,refpts) is 5xlength(refpts)

array of column vectors - fixed points (x,px,y,py,dP)

at the entrance of each element indexed by the REFPTS 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

findsyncorbit(ring, …,’strict’,strict) Default: 0.

If STRICT is -1, check_6d is skipped

If STRICT is 0, check_6d emits a warning for non-6d rings.

If STRICT is 1, check_6d emits an error for non-6d rings.

See also findsyncorbit(), findorbit6(), atdisable_6d(), check_6d()

plotcod(ring, dp)#: Plots the closed orbit distortion

plotcod(ring,dp) finds and plots closed orbit for a given momentum

deviation DP. It calls FINDORBIT4 which assumes a lattice

with NO accelerating cavities and NO radiation