lattice design ii: insertions - cern · 1 lattice design ii: insertions bernhard holzer, desy...
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Lattice Design II: InsertionsBernhard Holzer, DESY
���
�
�
���
�
�
−=�
�
���
�=
ψψ
ψψ
cossinK
sinK
cos'S'C
SCM
1
���
����
�=�
�
���
�
0
0
'xx
*M)s('x)s(x
K*s=ψ
... Particle trajectories in a storage ring are defined by the external fields of the magnets that are placed in the lattice.The coordinates of this trajectories (with respect to the closed orbit) can be calculated using a matrix formalism that describes the effect of the lattice elements.
0=+ x*)s(K''x
Reminder:
equation of motion
particle coordinates
e.g. matrix for a quadrupole lens:
L
x
s
21
ρ+−= kK
2
Dispersion:Dispersion:
describes the motion of particles with momentum deviation ∆p/p
ppx*)s(K''x ∆
ρ1=+
�special solution of the inhomogeneous differential equation:
p/p*)s(D)s(xi ∆=
� extend the matrices to include D, D´
��
���
�+��
���
���
���
�=��
���
�
'DD
pp
'xx
*'S'C
SC'x
x
S
∆
0
0
100 �����
�
�
�����
�
�
���
�
�
���
�
�
=
�����
�
�
�����
�
�
pp'x
x*'D'S'C
DSC
pp'x
x
S
∆∆
particles with different momentum are running on a different closed orbit
Dispersion:the dispersion function D(s) is (...obviously) defined by the focusing properties of the lattice and from position s0 to s in the lattice it is given by:
! weak dipoles � large bending radius � small dispersion
Example: Drift
��
���
�=10
1 �
DM �� −= s~d)s~(S)s~(
*)s(Cs~d)s~(C)s~(
*)s(S)s(D������
ρρ11
= 0 = 0
���
�
�
���
�
�
=→10001001 �
DM ...in similar way for quadrupole matrices,...in similar way for quadrupole matrices,!!! in a !!! in a quite differentquite different way for way for dipole matrixdipole matrix (see appendix)(see appendix)
�−�=S
S
S
Ss~d)s~(S
)s~(*)s(Cs~d)s~(C
)s~(*)s(S)s(D
00
11ρρ
3
Dispersion in a FoDo Cell:Dispersion in a FoDo Cell:
lD
QF QD
in analogy to the derivations of∨
ββ ,ˆ
* thin lens approximation: QQK
f ��
>>= 1
L, DQ 210 =→≈ ��
ff~ 211 =
* length of quad negligible
* start at half quadrupole
1.) calculate the matrix of the FoDo half cell in thin lens approximation:
L
llDD
!! we have now introduced dipole magnets in the FoDo:!! we have now introduced dipole magnets in the FoDo:�� we still neglect the weak focusing contribution 1/we still neglect the weak focusing contribution 1/ρρ22
�� but take into account but take into account 1/1/ρρ for the dispersion effectfor the dispersion effect
matrix of the half cell��
�
�
��
�
�−�
�
���
���
�
�
��
�
�= 11
01
101
1101
f~**
f~M D
Cellhalf
�
... neglecting as usual the weak focusing term of the dipole
����
�
�
����
�
�
+−
−=�
�
���
�=
f~f~
f~
'S'CSC
MDD
DD
Cellhalf��
��
1
1
2
���
���
� −� −��
���
� −=DD
dss**f~
dsf~s**)(D D
DD
�� ���
00
1111ρρ
ρρρρρρ f~f~**
f~f~)(D DDDDDDD
DD
D 222211
2
323222�������
��
� +−−=��
���
� −−���
����
�−=
2.) calculate the dispersion terms D, D´ from the matrix elements
ρ2
2D
D )(D �� =
�−�=S
S
S
Ss~d)s~(S
)s~(*)s(Cs~d)s~(C
)s~(*)s(S)s(D
00
11ρρ
4
in full analogy on derives for D´:
�������
�
�
�������
�
�
++−
−
=���
�
�
���
�
�
=
1002
11
21
1002
2
)f~
(f~f~
f~
'D'S'CDSC
M DDDD
DD
D
Cellhalf����
��
�
ρ
ρ
and we get the complete matrix including the Dispersion terms D, D‘
boundary conditions for the transfer from the center of the foc. to the center of the defoc. quadrupole
�����
�
�
�����
�
�
=����
�
�
����
�
� ∨
10
10 21
D̂*M
D
/
��
���
� +=f~
)('D DDD 2
1 ���
ρ
Dispersion in a FoDo CellDispersion in a FoDo Cellρ2
12
DD )f~
(D̂D �� +−=→∨
)f~
(D̂*f~
DDD
210 2
��� ++−=→ρ
2
2211
2
2
µ
µ
ρ sin
)sin(*D̂ D
+= �
2
2211
2
2
µ
µ
ρ sin
)sin(*D D
−=
∨�
where µ denotes the phase advance of the full celland l/f = sin(µ/2)
0 30 60 90 120 150 1800
2
4
6
8
1010
0.5
D max µ( )
D min µ( )
1801 µ
∨
D
D̂
Nota bene:
! small dispersion needs strong focusing→ large phase advance
!! ↔ there is an optimum phase for small β
!!! ...do you remember the stability criterion?½ trace = cos µ ↔ µ < 180°
!!!! … life is not easy
5
Reminiszenz: Synchrotron light sources
the emittance of an electron beam in a storage ring:
2
32
1
1
33255
RJ
)s(HR
mcx
x γε �=
γ = relativistic gamma,R = bending radius of the dipole magnetsJx = damping partition number ≈ 1
22 2 'D'DDD)s(H βαγ ++=
„low emittance lattices“ � low Dispersion lattices(see lecture of A.Streun in this school)
Chromaticity Correction in lattice cellsChromaticity Correction in lattice cells
Remember Definitionpp*Q ∆ξ∆ =
in FoDo Cells: 21 µπ
ξ tan*Cell −=
Example HERA: µ ≈ 90° , number of cells 104 � ξ ≈ -33including mini beta sections …� ξ ≈ -42…-80∆p/p ≈ 0.5*10-3 �∆Q ≈ 0.04compare to nominal tune: Q = 0.292
Chromaticity Correction in lattice cellsChromaticity Correction in lattice cells
1.) sort the particles as a function of their momentum pp*)s(D)s(x ∆=
2.) create magnetic field with linear increasing gradient const'g,x'*gx
By ==∂
∂
normalised to the momentum: m6pol = g´/(p/e)
6
3.) calculate resulting quadrupol strengthpp*D*mx*mK polpol
∆66 ==
4.) resulting overall chromaticity of the storage ring
{ }ds)s()s(D)s(m)s()s(Ktotal � −−= ββπ
ξ4
1
Hints for the lattice design:Hints for the lattice design:
! avoid large β values
! provide space where D(s) ≠0for installation of 6pol magnets
!! put sextupoles at places where βis large �� close to the quadsclose to the quads
!!! in general ξ is created at locations in the lattice where it cannot be corrected( … as D(s) = 0)
!!!! … life is not easy… life is not easy part of LEP lattice for ξ correction
Lattice Design: InsertionsLattice Design: Insertions
I.) ... the most complicated one: the drift space
Question to the auditorium: what will happen to the beam parameters α, β, γ if westop focusing for a while …?
022
22
2
2
���
�
�
���
�
�
���
�
�
���
�
�
−−+−
−=
���
�
�
���
�
�
γαβ
γαβ
*'S'C'S'C
'SSC'S'SC'CCSSCC
S
transfer matrix for a drift: ��
���
�=��
���
�=10
1 s'S'C
SCM
2000 2 ss)s( γαββ +−=
s)s( 00 γαα −=
0γγ =)s(
„0“ refers to the position of the last lattice element
„s“ refers to the position in the drift
7
location of the waist:location of the waist:
given the initial conditions α0, β0, γ0: where is the point of smallest beam dimension in the drift … or at which location occurs the beam waist ?
beam waist: �� *)( 000 γαα =→=
0
0
γα=�
)()()()l(
)()(
��
�
�
�
ββαγ
αγγ 110
20 =+=→���
==
beam size at that position:
0
1γβ =)( �
s
β beam waist: α = 0
l
ββ--Function in a Drift:Function in a Drift:
let‘s assume we are at a symmetry point in the center of a drift.
2000 2 ss)s( γαββ +−=
as00
20
00110ββ
αγα =+=→= ,
and we get for the β function in the neighborhood of the symmetry point
Nota bene: 1.) this is very bad !!!2.) this is a direct consequence of the
conservation of phase space density(... in our words: ε = const) … and there is no way out.
3.) Thank you, Mr. Liouville !!!
! ! !! ! !
Joseph Liouville,1809-1882
0
2
0 βββ s)s( +=
8
Optimisation of the beam dimension:Optimisation of the beam dimension:
0
2
0 βββ �
� +=)(
Find the β at the center of the drift that leads to the lowest maximum β at the end:
If we cannot fight against Liouvuille theorem ... at least we can optimise
01 20
2
0
=−=ββ
β �
dˆd !!
�=→ 0β
02ββ =→ ˆ
If we choose β0 = ℓ we get the smallest β at the end of the drift and the maximum β is just twice the distance ℓ
**
l l
β0
... clearly there is another problem !!!... clearly there is another problem !!!
Example: Luminosity optics at HERA: β* = 18 cmfor smallest βmax we have to limit the overall length of the drift to L = 2*ℓL = 36 cm
But: ... unfortunately ... in general high energy detectors that are installed in that drift spaces are a little bit bigger ...
9
The MiniThe Mini--ββ Insertion:Insertion:
Event rate of a collider ring for a reaction with cross section σR : L*R Rσ=
Luminosity: given by the total stored beam currents and the beam size at the collision point (IP)
y*
x* *
I*I*bfe
Lσσπ
21
0241=
How to create a mini β insertion:
* symmetric drift space (length adequate for the experiment)* quadrupole doublet on each side (as close as possible)* additional quadrupole lenses to match twiss parameters to
the periodic cell in the arc
MiniMini--ββ Insertions: Insertions: BetafunctionsBetafunctions
A mini-β insertions is always a kind of special symmetric drift space.�greetings from Liouville
at a symmetry point β is just the ratio of beam dimension and beam divergence.… which gives us the size of β at the first quadrupole
** s)s(
βββ
2+=
*
***
'σσβα =→= 0
size of β at the second quadrupole lens (in thin lens approx):… after some transformations and a couple of beer …
2
1
2121
2
1
2 11 ���
����
�+++��
�
����
�+=
fllll*
fl)s( *
*
βββ
IPIP
**l1 l2
10
Now in a mini β insertion:
MiniMini--ββ Insertions: Insertions: Phase advancePhase advance
By definition the phase advance is given by: �= ds)s(
)s(β
Φ 1
)s()s( 20
2
0 1β
ββ +=
4 3.2 2.4 1.6 0.8 0 0.8 1.6 2.4 3.2 49070503010103050709090
90−
f L( )
44− L
summing the drift spaces on both sides of the IP the phase advanceof a mini β insertion is approximately π, in other words:the tune will increase by half an integer.
Φ(s)
s
0020
20 1
11βββ
Φ Larctands/s
)s(L
=� +=→
are there any problems ??are there any problems ??
sure there are...sure there are...
* large * large ββ values at the doublet values at the doublet quadrupolesquadrupoles �� large contribution to large contribution to chromaticity chromaticity ξξ …… and no local correctionand no local correction
* * aperture of mini aperture of mini ββ quadrupolesquadrupolescan limit the luminositycan limit the luminosity
� −−= s~d)Q)s()s~(cos()s~(
)s~(*Qsin)s()s(x πφφ
ρβ
πβ 1
2
•field quality and magnet stability most critical at the high β sectionsorbit distortion due to a kick:
beam envelope at the first beam envelope at the first mini mini ββ quadrupolequadrupole lens in lens in
the HERA proton storage ringthe HERA proton storage ring
� keep distance „s“ to the first mini β quadrupole as small as possible
{ }ds)s()s(K�−= β
πξ
41
11
MiniMini--ββ Insertions: some Guide linesInsertions: some Guide lines
Optics guide line: * calculate the periodic solution in the arc* introduce the drift space needed for the insertion device (detector ...)* put a quadrupole doublet (triplet ?) as close as possible* introduce additional quadrupole lenses to match the beam parametersα, β, γ, D, D’ o the values at hte beginning of the arc structure
parameters to be optimised & matched to the periodic solution:
→
��
�
��
�
�
yx
xx
yy
xx
Q,Q'D,D
,,
βαβα
8 individually powered quadrupole magnets are needed to match the insertion (...at least)
Dispersion SuppressorsDispersion Suppressors
There are two comments of paramount importance about dispersion:! it is nasty!! it is not easy to get rid of it.
remember: oscillation amplitude for a particle with momentum deviation p
p*)s(D)s(x)s(x ∆β +=
Example: beam size at the IP in HERAaverage dispersion in the arc:typical momentum spread:
Dispersion spoils the luminosity and leads to additional stop bands (synchro-betatron resonances) in RF sections and at the IP
optical functions of a FoDo Cell without dipoles: D=0
xβ yβ
0=D
m,m *y
*x µσµσ 32118 ==
mm.x*
pp
m.)s(DD 750
105
51
4 ≈→��
��
�
≈
=
−∆
12
Optical functions in a FoDo Cellincluding the effect of the bendingmagnets
2
2211
2
2
µ
µ
ρ sin
)sin(*D
−=
∨�
2
2211
2
2
µ
µ
ρ sin
)sin(*D̂
+= �
xβ
yβ
Dispersion SuppressorsDispersion Suppressors
There are several ways to suppress the dispersion function in a lattice:
I.) The straight forward one: use additional quadrupole lenses to match the optical parameters ... including the D(s), D´(s) terms
optical functions to match:
→��
��
�
)s(),s()s(),s()s('D),s(D
yy
xx
αβαβ 6 additional quadrupole
lenses required
Optical functions in a FoDo Cell
•Dispersion suppressed by 2 quadrupole lenses,
•β and α restored to the values of the periodic solution by 4 additional quadrupoles
periodic FoDo structure
matching sectionincluding 6 additionalquadrupoles
Dispersion free section, regular FoDo without dipoles
Advantage: ! easy,! flexible: it works for any phase
advance per cell ! does not change the geometry
of the storage ring, ! can be used to match between different lattice
structures (i.e. phase advances)
Dispersion SuppressorsDispersion Suppressors
Disadvantage:
! additional power supplies needed
(→ expensive)
! requires stronger quadrupoles
! due to higher β values: more aperture required
13
Dispersion SuppressorsDispersion Suppressors II.) The clever way: half bend schemes
Desperate statement of the lecturer:the mathematical derivation of the dispersion suppressor is a bit awkward.
�put it into the appendix�state the following conditions:
arcrsup
C
arcC
rsup
)nsin(
)n(sinδδ
Φ
δΦδ21
02
2 2
=��
��
�
=→
=→
...,,k,*kn C 31== πΦ
strength of suppressor dipoles is half as strongas that of arc dipoles
in the n suppressor cells the phase advancehas to accumulate to a odd multiple of π
Example:
phase advance in the arc ΦC = 60°number of suppr. cells n = 3 δsuppr = 1/2 δarc
III.) The clever way: missing bend schemesDispersion SuppressorsDispersion Suppressors
at the end of the arc: add m cells without dipoles followed by n regular arc cells.condition for dispersion suppression:
212
22 πΦ )k(nm
C +=+
...,k,nsin
or...,k,nsin
C
C
3121
2
2021
2
=−=
==
Φ
Φ
Example:
phase advance in the arc ΦC = 60°number of suppr. cells m = 1 number of regular cells n = 1
14
Resume‘Resume‘1.) Dispersion in a FoDo cell:
small dispersion ↔ large bending radiusshort cellsstrong focusing 2
2211
2
2
µ
µ
ρ sin
)sin(*D̂
+= �
2.) Chromaticity of a cell: small ξ ↔ weak focusing
small β{ }ds)s()s(D)s(m)s()s(Ktotal � −−= ββ
πξ
41
3.) Position of a waist at the cell end: α0 β0 = values at the end of the cell
0
0
γα=�
0
1γβ =)( �
5.) Mini β insertionsmall β↔ short drift space requiredphase advance ≈ 180 ° 0
2
0 βββ �
� +=)(
4.) β function in a drift 2000 2 ss)s( γαββ +−=
Dispersion in a FoDo Cell:lD
L
QF QD
in analogy to the derivations of∨
ββ ,ˆ
* thin lens approximation:Q
Q
lKl
f >>= 1
Ll,l DQ 210 =→≈
ff~ 211 =
* length of quad negligible
* start at half quadrupole
22QFBQDCellHalf M*M*MM =
Appendix: some usefull formulae in more detail
15
matrix of the half cell��
�
�
��
�
�−�
�
���
���
�
�
��
�
�= 11
01
101
1101
f~*
l*
f~M
... neglecting as usual the weak focusing term of the dipole
����
�
�
����
�
�
+−
−=�
�
���
�=
f~l
f~l
lf~l
'S'CSC
M1
1
2
�� −= ds)s(S)s(
*)l(Cds)s(C)s(
*)l(S)l(Dρρ
11
���
���
� −� −��
���
� −=ll
dss**f~lds
f~s**l)l(D
00
1111ρρ
ρρρρρρ f~ll
f~lll**
f~l
f~lll
222211
2
323222
+−−=��
���
� −−��
���
� −=
ρ2
2l)l(D =
�� −= s~d)s~(S)s~(
*)l('Cs~d)s~(C)s~(
*)l('S)l('Dρρ
11
�� +��
���
� −+=ll
dss**f~lds
f~s**)
f~l()l('D
02
0
1111ρρ
21
211
2
2
2 l**f~l)
f~ll(**
f~l
ρρ+−�
�
���
� +=
Expression for D´
2
32
221
f~l)
f~ll(*
f~l
ρρρ+−�
�
���
� +=
ρρ f~ll
2
2
+=
)f~
l(l)l('D2
1+=ρ
16
�������
�
�
�������
�
�
++−
−
=���
�
�
���
�
�
=
1002
11
21
1002
2
)f~
l(lf~l
f~l
llf~l
'D'S'CDSC
Mρ
ρ
Complete Matrix including the terms for D, D‘:
����
�
�
����
�
�
=����
�
�
����
�
�∨
10
10 21
D̂*M
D
/
require boudary conditions for half cell solution:
)(l)f~l(D̂D 1
21
2
ρ+−=→
∨
)()f~
l(lD̂*f~l 2
210
2++−=→
ρ
f~*)()( 21 − )f~
l(f~llD̂D2
12
2
+−+=→∨
ρρ
ρρρρf~lD̂lf~llD̂D −=+−+=
∨
22
22
ρρ 2
2lf~lD̂D̂f~lD̂ +−=−→put result in (1)
f~lD̂lf~l =+
ρρ 2
2
)f~
l(f~lf~f~D̂2
12
22
+=+=ρρρ
remember: 2µsin
f~l =
)sin(f~D̂22
112 µ
ρ+=→ )sin(f~D
2211
2 µρ
−=→∨
)sin(llf~D̂
2211
2
22 µρ
+=
����
�
�
����
�
� +=
2
2211
2
2
µ
µ
ρ sin
sinlD̂����
�
�
����
�
� −=
∨
2
2211
2
2
µ
µ
ρ sin
sinlD
17
Dispersion: Example: Dipole sector magnet
�� −= s~d)s~(S)s~(
*)s(Cs~d)s~(C)s~(
*)s(S)s(Dρρ
11
����
�
�
����
�
�
−=���
�
�
���
�
�
−=�
�
���
�=
ρρρ
ρρ
ρ
ψψ
ψψ��
��
cossin
sincos
cossinK
sinK
cos'S'C
SCM 1
1
K*�=ψ 210 ρ== K,k
K describes in this case only the weak focusing term in the dipole
)(cos*cos*sin
*cos***cossin***sin
1
111
2 −��
���
���
���
�+��
���
�=
���
����
���
���
�−��
���
�−��
���
���
���
�=
ρρρ
ρρ
ρρ
ρρρρ
ρρρ
ρ
���
����
���
���
��
�
�−��
�
�+��
�
�=ρρρ
ρ ��� coscossin 22
)cos(*)l(Dρ
ρ �−= 1 )sin()l('Dρ�=
����
�
�
����
�
�−
−
=
100
11
ψψψρ
ψρψρψ
sincossin
)cos(sincos
M ρψ �=,
Dispersion: Example: Dipole sector magnet
Using these expressions for D and D‘ the extended matrix of a Using these expressions for D and D‘ the extended matrix of a dipole sector magnet is given by:dipole sector magnet is given by:
18
The MiniThe Mini--ββ InsertionInsertion::
thin lens approximation for the mini β doublet:
IPIP
**l1 l2
1/f1 -1/f2
��
���
�=10
1 1
1
lM D
marices for the elements in the mini-β region
��
���
�=10
1 2
2
lM, D
��
�
�
��
�
�= 11
01
1
1
fMF �
�
�
�
��
�
�−= 11
01
2
2
fM, F
-1/f2 =hor. fok. lens
��
���
���
�
�
��
�
���
���
���
�
�
��
�
�−=
101
1101
101
1101
1
1
2
2
l*
f*
l*
fMtotal
����
�
�
����
�
�
++−−−−−
+++=
1
1
2
2
21
21
2
1
21
2
21
1
2121
1
2
111
1
fl
fl
ffll
fl
ffl
ff
fllll
fl
Mtotal
value of the β function at the position of the second lens
transformation of twiss parameters:
0
22
22
2
2
���
�
�
���
�
�
���
�
�
���
�
�
−−+−
−=
���
�
�
���
�
�
γαβ
γαβ
*'S'C'S'C
'SSC'S'SC'CCSSCC
S
0
2
00
2 2 γαββ *S*SC*C)s( +−=
now we add the boundary condition for a symmetric problem:α0 = 0
00
2
0
0
11ββ
αγ =+=→
0
2
0
2 βββ /S*C)s( +=
2
1
21
21
0
0
2
1
211 ��
�
����
�++++=
fllll*)
fl()s(
βββ
using the matrix elements calculcated above for the doublet:using the matrix elements calculcated above for the doublet:
19
Dispersion SuppressorsDispersion Suppressors... the calculation in full detail (for purists only)
1.) the lattice is split into 3 parts: (Gallia divisa est in partes tres)
* periodic solution of the arc periodic β, periodic dispersion D* section of the dispersion suppressor periodic β, dispersion vanishes* FoDo cells without dispersion periodic β, D = D´ = 0
2.) calculate the dispersion D in the periodic part of the lattice
transfer matrix of a periodic cell:
����
�
�
����
�
�
−+−−
+=→
)sin(cossin)(cos)(
sin)sin(cosM
SSS
SS
SS
S
φαφββ
ββφααφαα
φββφαφββ
0
0
00
000
0 1
for the transformation from one symmetriy point to the next (i.e. one cell) we have: ΦC = phase advance of the cell, α = 0 at a symmetry point. The index “c” refers to the periodic solution of one cell.
����
�
�
����
�
�−=
���
�
�
���
�
�
=
100
1
100)l('Dcossin
)l(Dsincos
'D'S'CDSC
M CCC
CCC
Cell ΦΦβ
ΦβΦ
�� −=ll
s~d)s~(S)s~(
*)l(Cs~d)s~(C)s~(
*)l(S)l(D00
11ρρ
The matrix elements D and D‘ are given by the C and S elements in the usual way:
�� −=ll
s~d)s~(S)s~(
*)l('Cs~d)s~(C)s~(
*)l('S)l('D00
11ρρ
20
here the values C(l) and S(l) refer to the symmetry point of the cell (middle of the quadrupole) and the integral is to be taken over the dipole magnet where ρ ≠ 0. For ρ = const the integral over C(s) and S(s) is approximated by the values in the middle of the dipole magnet.
φm
ΦC /2
-φm
dipole magnet dipole magnet
Transformation of C(s) from the symmetry point in the foc. quad to the center of the dipole:
)cos(cosC mC
C
m
C
mm ϕΦ
ββ∆Φ
ββ ±==
2)sin(S m
CCmm ϕΦββ ±=
2
where βC is the periodic β function at the beginning and end of the cell, βm its value at the middle of the dipole and φm the phase advance from the quadrupole lens to the dipole center.
Now we can solve the intergal for D and D’:
�� −=ll
s~d)s~(S)s~(
*)l(Cs~d)s~(C)s~(
*)l(S)l(D00
11ρρ
)sin(*L*cos)cos(**L*sin)l(D mC
CmCmC
C
mCC ϕΦββ
ρΦϕΦ
ββ
ρΦβ ±−±=
22
−���
��
���
−++= )cos()cos(sin)l(D mC
mC
CCm ϕΦϕΦΦββδ22
�����
���
−++− )sin()sin(cos mC
mC
C ϕΦϕΦΦ22
remember the relations 22
2 yxcos*yxcosycosxcos −+=+
222 yxcos*yxsinysinxsin −+=+
���
���−= m
CCm
CCCm cos*sin*coscos*cos*sin)l(D ϕΦΦϕΦΦββδ
22
22
���
���−=
222 C
CC
CmCm sin*cos*cos*sincos*)l(D ΦΦΦΦϕββδ
remember: xcos*xsinxsin 22 =xsinxcosxcos 222 −=
���
���−−=
2222222 222 CCCCC
mCm sin*)sin(coscos*sincos*)l(D ΦΦΦΦΦϕββδ
I have put δ = L/ρ for the strength of the dipole
21
���
���+−=
2222
22 222 CCCC
mCm sincoscossin*cos*)l(D ΦΦΦΦϕββδ
22 C
mCm sin*cos*)l(D Φϕββδ=
in full analogy one derives the expression for D‘:
22 C
mCm cos*cos*/)l(D Φϕββδ=
As we refer the expression for D and D‘ to a periodic struture, namly a FoDo cell we require periodicity conditons:
���
�
�
���
�
�
=���
�
�
���
�
�
11C
C
CC
C
'DD
*M'DD
and by symmetry: 0=C'D
With these boundary conditions the Dispersion in the FoDo is determined:
CC
mCmCC Dsin*cos*cos*D =+2
2 ΦϕββδΦ
2C
mCmC sin/cos*D Φϕββδ=
This is the value of the periodic dispersion in the cell evaluated at the position of the dipole magnets.
3.) Calculate the dispersion in the suppressor part:
We will now move to the second part of the dispersion suppressor: The section where ... starting from D=D‘=0 the dispesion is generated ... or turning it around where the Dispersion of the arc is reduced to zero.The goal will be to generate the dispersion in this section in a way that the values of the periodic cell that have been calculated above are obtained.
φm
ΦC /2
-φm
dipole magnet dipole magnet
�� −=ll
s~d)s~(S)s~(
*)l(Cs~d)s~(C)s~(
*)l(S)l(D00
11ρρ
The relation for D, generated in a cell still holds in the same way:
(A1)
22
−±� −== C
mmC
n
iCrsupCCn *)icos(**nsinD
ββϕΦΦδΦβ
1 21
as the dispersion is generated in a number of n cells the matrix for these n cells is
����
�
�
����
�
�−==
100
1nCC
C
nCCC
nCn 'Dncosnsin
Dnsinncos
MM ΦΦβ
ΦβΦ
)isin(***ncos mC
n
iCCmrsupC ϕΦΦββδΦ ±� −−
=1 21
))isin((ncos**))icos((**nsin*D m
n
i
CCrsupCmm
n
i
CrsupCCmn ϕΦΦδββϕΦδΦββ ±� −−±� −=
== 11 212
212
remember:22
2 yxcos*yxsinysinxsin −+=+22
2 yxcos*yxcosycosxcos −+=+
−� −==
m
n
i
CCCmrsupn cos*))icos((*nsin**D ϕΦΦββδ
12
212
m
n
i
CCCmrsup cos*))isin((ncos** ϕΦΦββδ � −−
=12
212
replacing the integral over the n cells of the suppressor by thereplacing the integral over the n cells of the suppressor by the sum over the n cells we obtain for D:sum over the n cells we obtain for D:
����
���
� −−−== =
n
i
n
iC
CC
CmCmrsupn ncos*))isin((nsin*))icos((cos**D
1 1 212
2122 ΦΦΦΦϕββδ
��
��
�
��
��
�
��
��
�
��
��
�
−��
��
�
��
��
�
=
2
22
2
222C
CC
CC
CC
CmCmrsupn
sin
nsin*nsin*ncos
sin
ncos*nsinnsincos**D Φ
ΦΦ
ΦΦ
ΦΦ
Φϕββδ
���
���−=
2222
2 2 CC
CCC
C
mCmrsupn
nsin*ncosncos*nsin*nsinsin
cos**D ΦΦΦΦΦΦ
ϕββδ
set for more convenience x = nΦC/2
}{ xsin*xcosxcos*xsin*xsinsin
cos**D
C
mCmrsupn
222
2
2−= Φ
ϕββδ
}{ xsin)xsinx(cosxsinxcos*xcosxsinsin
cos**D
C
mCmrsupn
2222
2
2−−= Φ
ϕββδ
23
22
2 2 C
C
mCmrsupn
nsin*sin
cos**D Φ
Φϕββδ
=
and in similar calculations:
CC
mCmrsupn nsin*
sin
cos**'D ΦΦ
ϕββδ
2
=
This expression gives the dispersion generated in a certain number of n cells as a function of the dipole kick δ in these cells.At the end of the dispersion generating section, the value obtained for D(s) and D‘(s) has to be equal to the value of the periodic solution:
�equating (A1) and (A2) gives the conditions for the matching of the periodic dispersion in the arc to the values D = D‘= 0 afte the suppressor.
(A2)
22
2
2 2
C
mCmarc
C
C
mCmrsupn
sin
cos*nsin*sin
cos**D Φ
ϕββδΦΦ
ϕββδ==
arcrsup
C
arcC
rsup
)nsin(
)n(sinδδ
Φ
δΦδ21
02
2 2
=��
��
�
=→
=→
and at the same time the phase advance in the arc cell has to obey the relation:
...,,k,*kn C 31== πΦ