PTCVariables and (some) Maps

Piotr Skowroński

25 November 2013MADX Meeting1

MADX Meeting

VariablesThere are 2 sets of variables depending on TIME flag

TIME=false TIME=true

To switch the time flag use the following in MADX scriptptc_setswitch, time=true; 25 November 20132

General Hamiltonian

25 November 20133 MADX Meeting

Integration of straight elements (rectangular bend, quad) uses Cartesian coordinates. It is not done along arc.

MADX Meeting

SYMPLECTIC INTEGRATOR3 Integration algorithms method 2: the naive 2nd order method, 1 kick per

integration step method 4: the Ruth-Neri-Yoshida 4th order method, 3 kicks /

step method 6: the Yoshida 6th order method, 7 kicks / step Controlled with: ptc_create_layout, method=XX;

PTC has a hook so user defined algorithm can be hooked in Each element can be integrated with different algorithm

Magnets are split into smaller slices Number of slices (integration steps) is controlled withptc_twiss, nst=NN; ptc_normal, nst=NN;

Automatic “resplit” sets method (and adjusts also nst), method 2 for drifts (1 step) method 6 for quads and strong bends Method 4 for weak bends

25 November 20134

MADX Meeting

Integration models

In practice, this integration is applying special maps at different positions

Transport between the kicks can be done As in drift: model 1 = DRIFT-KICK-DRIFT Via matrix: model 2 = MATRIX-KICK-MATRIX Like in SixTrack: model 3 = KICK-SIXTRACK-KICK

Controlled with: ptc_create_layout, model=NN;

25 November 20135

MADX Meeting

Hamiltonians

25 November 20136

Different Hamiltonians Expanded Exact Each has different flavors

different “splittings” different reference frames

Depending on the user requirements Precision

Price of computation time, of course Which flavor is preferred

Controlled with: ptc_create_layout, exact=T/F;

MADX Meeting

PTC Settings

Want to be safe, always use

ptc_create_layout, exact=true, model=2, method=6,resplit, xbend;

It willTake exact hamiltonians, Use matrix in between kicksUse 6th order Yoshida methodAutomatically adjusts methods and number of steps for each element

Default isptc_create_layout, exact=false, model=1, method=2;

25 November 20137

MADX Meeting

Quadrupole

25 November 20138

EXACT EXPANDEDModel=1Drift-Kick-Drift STREX_caseDKD DKD2

Model=2Matrix-Kick-Matrix TKTF TKTF

Model=3 Kick-SixTrack-Kick KTK KTK

MADX Meeting

SBEND & default RBEND

Curved reference frame

25 November 20139

EXACT EXPANDEDModel=1Drift-Kick-Drift TEAPOT DKD2

Model=2Matrix-Kick-Matrix TEAPOT TKTF

Model=3 Kick-SixTrack-Kick TEAPOT KTK

MADX Meeting

RBEND, ptcrbend=true;

Straight reference frame

25 November 201310

EXACT EXPANDEDModel=1Drift-Kick-Drift STREX_caseDKD DKD2

Model=2Matrix-Kick-Matrix STREX_caseMKM TKTF

Model=3 Kick-SixTrack-Kick STREX_caseMKM KTK

MADX Meeting

STREX

Quad in model=1 & RBEND, ptcrbend=true;

Hamiltonian:

Docu: CERN-SL-2012-044, Section K.4.12

Implemented in Sh_def_kind.h: INTEP_STREX (example method 2)

Case DKD (model1): DRIFT(L/2); KICKEX(L); DRIFT(L/2); Case MKM (model2): SPAR(L/2); KICKEX(L); SPAR(L/2);

25 November 201311

Model=1

Model=2

MADX Meeting

STREX

25 November 201312

Kick: KICKEX

BYW=bNMUL

BXW=aNMUL

DO J=NMUL-1,1,-1 BYWT=x*BYW-y*BXW+bJ

BXW=y*BYW+x*BXW+aJ

BYW=BYWT ENDDO px=px-l*BYWpy=py+l*BXWif(model1) px=px+l*b1

MADX Meeting

STREX

25 November 201313

Drift (model=1): DRIFT TIME=false TIME=true

MADX Meeting

STREX

25 November 201314

Matrix (model=2): SPAR

TIME=false time=true

MADX Meeting

TEAPOT

25 November 201315

RBEND, SBEND in all models

Hamiltonian:

Docu: CERN-SL-2012-044, Section K.4.9

Implemented in Sh_def_kind.h: INTEP_TEAPOTField strengths needs to renormalized to always provide the requested bending angle

Model=1

Model=2

MADX Meeting

TEAPOTKick: SKICK

m=NMULIPOLES-1 do a=m,1,-1 do j=m-a,1,-1 i=i+1 BTX= (BTX+BF_X(i))*Y BTY= (BTY+BF_Y(i))*Y enddo i=i+1 BTX= (BTX+EL%BF_X(i)) BTY= (BTY+EL%BF_Y(i)) BX= (BX+BTX)*X BY= (BY+BTY)*X enddo BTX=0; BTY=0 do j=m,1,-1 i=i+1 BTX= (BTX+EL%BF_X(i))*Y BTY= (BTY+EL%BF_Y(i))*Y enddo i=i+1 BX= BX+BTX+BF_X(i) BY= BY+BTY+BF_Y(i)

25 November 201316

px=px-l*BYpy=py+l*BX

MADX Meeting

TEAPOT Drift (model=1): SPROT

TIME=false

25 November 201317

MADX Meeting

TEAPOT Matrix (model=2): SSEC

TIME=false

25 November 201318

MADX Meeting

TKTF

25 November 201319

Quad in model=2

Hamiltonian:

Docu: CERN-SL-2012-044, Section K.4.7

Implemented in Sh_def_kind.h: INTEP_TKTF

PUSHTKT7(L/2);KICKPATH(L/2);KICKTKT7(L);KICKPATH(L/2);PUSHTKT7(L/2)

MADX Meeting

TKTF

Matrix: PUSHTKT7 (matrix defined in GETMAT7)

25 November 201320

MADX Meeting

TKTF

KICKPATHTIME=false TIME=true

25 November 201321

MADX Meeting

TKTF

Kick: KICKTKT7

BYW=bNMUL

BXW=aNMUL

DO J=NMUL-1,1,-1 BYWT = x*BYW - y*BXW + bJ

BXW = y*BYW + x*BXW + aJ

BYWc = BYWTENDDO px = px - l*(BYW - b1_0 - b2*x)py = py + l*(BXW - b2*y)

25 November 201322

MADX Meeting

Quad and Multipole FringeHard edge fringe fields effects exact in (1 + s) and consistent with maxwell's equations for rectilinear magnets Forest et.al, NIM in Physics Research A269 (1988) 474-482

It uses Lee-Whiting Formula G.E. Lee-Whiting, Nucl. Instr. and Meth. 83 (1970) 232

Implemented in Sh_def_kind.f90 multipole_fringep

25 November 201323

Sx, Cx,…, are the sin-like and cos-like solutions

In hard edge quad model

MADX Meeting

Summary

DKD2 handles only the expanded Hamiltonian elements. The exact integration are handled by types STREX, TEAPOT, KTK, and TKTF. STREX handles straight elements of all types. TEAPOT handles bends of “cyclotronic” symmetry, i.e., mostly invariant along the ideal trajectoryTKTF and KTK handle straight elements without any ideal bending using a different split from STREX, namely the kick-matrix splits.CAV4 is cavity

Hamiltonians can be found in CERN-SL-2012-044 Section K.4The code is in Sh_def_kind.f90, usually called INTEP_X, where X is STREX, KTK, CAV4, etc.

25 November 201324

MADX Meeting 25 November 201325

MADX Meeting

Quad and Multipole FringeLee-Whitting coefficients

25 November 201326

MADX Meeting

Quad and Multipole FringeLee-Whitting coefficients

25 November 201327

MADX Meeting

Quad and Multipole FringeHard edge fringe fields effects exact in (1 + s) and consistent with maxwell's equations for rectilinear magnets Forest et.al, NIM in Physics Research A269 (1988) 474-

482It uses Lee-Whiting Formula G.E. Lee-Whiting, Nucl. Instr. and Meth. 83 (1970) 232

25 November 201328

MADX Meeting

Quad and Multipole FringeHard edge fringe fields effects exact in (1 + s) and consistent with maxwell's equations for rectilinear magnets Forest et.al, NIM in Physics Research A269 (1988) 474-

482It uses Lee-Whiting Formula G.E. Lee-Whiting, Nucl. Instr. and Meth. 83 (1970) 232

25 November 201329

MADX Meeting

Quad and Multipole Fringe

Implemented in Sh_def_kind.f90 multipole_fringep DEL=1/(1+delta) A=1-FX_X*DEL B= -FY_X*DEL D=1-FY_Y*DEL C= -FX_Y*DEL

x=x-FX*DEL X2=(D*px-B*py)/(A*D-B*C) py=(A*py-C*px)/(A*D-B*C) px=X2 y=y-FY*DEL if(k%TIME) then ct=ct-(1/beta0+pt)*(px*FX+py*FY)*DEL**3 else s=s-(px*FX+py*FY)*DEL**2 endif

25 November 201330