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MotivationHas been examined...
- flat-beam transformation
- trans-long transformation
Using another suggestion of Derbenev, can we ...
- make the flat beam smaller in the bigger dimension?
- ameliorate space charge effects by making a ring at the cathode?
A Possible ScenarioUse photoinjector with masked laser, say, to produce “ring” on the cathode to mitigate space charge
pass through the flat beam transformer
- this produces “no emittance” in one plane, and a “ring” in the phase space of the other d.o.f.
inject into “new device” to reduce the larger dimension
perform x-z exchange if desired
... could give much flexibility to a facility
S.C. Force within a ringassume uniform density
make ring of ave. radius a0, thickness da, with same area as circle of radius a (<a0); i.e., choose:
space charge force at radius a of smaller circle is
s.c. force at radius a0+da/2 of the “ring” is
So, might expect a reduction by factor...
2!a0da = !a2
Fring
Fcircle=
2da
a= 2
da
a0
a0
a=
a2
a20
a0
a=
a
a0a
a0
x
y
F ! e2!
2"2#0
r2 " b2
r, b < r < b + da
F ! e2!
2"2#0· 2da
F ! e2!
2"2#0a
Flat Beam x-form of a “ring”
!X0 =
!
""#
x0
0y0
0
$
%%& =
!
""#
r0 cos "0
0r0 sin "0
0
$
%%& !Xs =
!
""#
1 0 0 00 1 !k 00 0 1 0k 0 0 1
$
%%& !X0 =
!
""#
x0
!k y0
y0
k x0
$
%%& .
!Xf = R!1McR !Xs R =!
22
·
!
""#
1 0 "1 00 1 0 "11 0 1 00 1 0 1
$
%%& , Mc =
!
""#
1 0 0 00 1 0 00 0 0 !0
0 0 "1/!0 0
$
%%&
!Xf =
!
""#
x0 ! y0
k(x0 + y0)00
$
%%& =
!
""#
r0(cos "0 ! sin "0)kr0(cos "0 + sin "0)
00
$
%%& =
!
""#
rc cos "c
krc sin "c
00
$
%%& .
Initial Conditions: Exiting solenoid field:
Through a skew quad channel:
Final result:
Flat Beam x-form of a “ring”
!X0 =
!
""#
x0
0y0
0
$
%%& =
!
""#
r0 cos "0
0r0 sin "0
0
$
%%& !Xs =
!
""#
1 0 0 00 1 !k 00 0 1 0k 0 0 1
$
%%& !X0 =
!
""#
x0
!k y0
y0
k x0
$
%%& .
!Xf = R!1McR !Xs R =!
22
·
!
""#
1 0 "1 00 1 0 "11 0 1 00 1 0 1
$
%%& , Mc =
!
""#
1 0 0 00 1 0 00 0 0 !0
0 0 "1/!0 0
$
%%&
!Xf =
!
""#
x0 ! y0
k(x0 + y0)00
$
%%& =
!
""#
r0(cos "0 ! sin "0)kr0(cos "0 + sin "0)
00
$
%%& =
!
""#
rc cos "c
krc sin "c
00
$
%%& .
Initial Conditions: Exiting solenoid field:
Through a skew quad channel:
Final result:
Turn “ring” into “bunch”Derbenev noted the process in NIM-A 441 (2000) 221-233.
Akin to adiabatic capture in synchrotron RF systems (h=1)
Assume a focusing channel or ring, with a repeat period of length
L that has phase advance near, but just below, 2π;
- Use octupole fields within the channel to give an amplitude dependent tune shift; adjust octupole to produce phase advance
of exactly 2π at an amplitude of a0 (which will be the radius of the initial ring distribution in phase space).
- Then, introduce a gradually increasing dipole field at the end of each repeat period (could be distributed, w/ proper phasing)
Parameterizationsmall amplitude phase advance of channel: 2π - ∆φ
phase shift with amplitude due to octupole field...
- do not want octupole field to contribute to ∆a; only ∆ψ; then, yields ∆a = 0
- define , then
dipole kick:
total kick (depicting ramped field):
resulting Hamiltonian:
!8 ! (B!!!/6)0 a30L/(B!)
(See details in Edwards/Syphers note)
!!8 =34
"#0
L
#0"8
a30
(a2 ! a20)
!2 ! B0L/(B!)
!B8 =!
B!!!(s)6
"x3 =
!!0
!
"3 !B!!!
6
"
0
x3
!2 · b, where 0 < b < 1
H = k!3!816 (r2 ! 1)2 !!2br cos !
"
k ! !"0L
"0a0
r ! a/a0
Resulting Hamiltonianfixed Points at ψ = 0 (stable), ψ = π (unstable)
for full “b”, island at r ~ 1 (i.e., a~a0) of width:
island “tune”:
- must be sufficiently small, compared with unity, for adiabatic process
w = 2!
83
!2!8
"1/2
!s !!
332!2 k2!2!8b
"1/2
Expected Phase Spaceuse as input parameters:
- given L, determine β0
- scale dimensions w/ a0
- choose ∆φ, w
- from which:
•
•
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
-2
-1.6
-1.2
-0.8
-0.4
0.4
0.8
1.2
1.6
2
!8 = 4La03!"2
0"!
!2 = 332!8w2
x/a0
!0x!/a0
Expected Phase Spaceuse as input parameters:
- given L, determine β0
- scale dimensions w/ a0
- choose ∆φ, w
- from which:
•
•
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
-2
-1.6
-1.2
-0.8
-0.4
0.4
0.8
1.2
1.6
2
!8 = 4La03!"2
0"!
!2 = 332!8w2
x/a0
!0x!/a0
w
Expected Phase Spaceuse as input parameters:
- given L, determine β0
- scale dimensions w/ a0
- choose ∆φ, w
- from which:
•
•
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
-2
-1.6
-1.2
-0.8
-0.4
0.4
0.8
1.2
1.6
2
!8 = 4La03!"2
0"!
!2 = 332!8w2
x/a0
!0x!/a0
w
Expected Phase Spaceuse as input parameters:
- given L, determine β0
- scale dimensions w/ a0
- choose ∆φ, w
- from which:
•
•
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
-2
-1.6
-1.2
-0.8
-0.4
0.4
0.8
1.2
1.6
2
!8 = 4La03!"2
0"!
!2 = 332!8w2
x/a0
!0x!/a0
w
A Device?How would one make a device?
- FODO channel -- how long?
• communications from others who have looked at this -- "process takes long time"
- use FODO ring? -- how big?
- try weak focus ring?
• easiest to quickly examine...
Numerical Simulationassume weak focusing, circumference L
- tune = √(1-n); as an example, choose n such that tune
= 15/16, ∆φ = 2π/16
take u = x/a0, v = β0 x’/a0 ; choose w = 0.5
divide ring into m sections, at the center of which an octupole kick is received:
- ∆v = -3/8 ∆φ/m u3
after m sections, s = L and dipole kick is provided:
- ∆v = 1/4 ∆φ w2 b, 0<b<1 (ramped dipole)
Result
adiabatically ramp the dipole
-2 -1 0 1 2
-2-1
01
2
u
vrev= 0
-2 -1 0 1 2
-2-1
01
2
u
v
rev= 10000
0 2000 4000 6000 8000 10000
0.0
0.2
0.4
0.6
0.8
1.0
revolution number
dip
ole
pro
file
0 2000 4000 6000 8000 10000
0.0
0.2
0.4
0.6
0.8
1.0
revolution number
u_
rms (
red
), v
_rm
s (
bla
ck)
initialdistribution
“final”distribution
rms of distribution
position (red)angle (black)
Animation
Parameters of presumed ring:
L = 2 mB0 = 0.314 T
ν = Δφ/2π = 15/16
β0 = 0.34 mpc = 30 MeVa0 = 4 mm
w = 0.5
Θ8 = 12.3 mr
Θ2 = 0.289 mrB2L = 0.03 T-mm
u = x/a0
v = β0 x’/a0
Animation
Parameters of presumed ring:
L = 2 mB0 = 0.314 T
ν = Δφ/2π = 15/16
β0 = 0.34 mpc = 30 MeVa0 = 4 mm
w = 0.5
Θ8 = 12.3 mr
Θ2 = 0.289 mrB2L = 0.03 T-mm
u = x/a0
v = β0 x’/a0
Animation
Parameters of presumed ring:
L = 2 mB0 = 0.314 T
ν = Δφ/2π = 15/16
β0 = 0.34 mpc = 30 MeVa0 = 4 mm
w = 0.5
Θ8 = 12.3 mr
Θ2 = 0.289 mrB2L = 0.03 T-mm
u = x/a0
v = β0 x’/a0
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4
-2
-1.6
-1.2
-0.8
-0.4
0.4
0.8
1.2
1.6
2
RemarksType of device
- if channel, very very long
- if ring, worry about bunch lengthening (isochronous ring?), chromatic effects, etc.
Possible Further Studies
- distribute dipole field along channel/ring
- how far from adiabaticity can be tolerated?
With Liouville in mind, only makes sense if a “ring beam” off the cathode helps with space charge effects
RemarksCould make nice student project; “too large” (?) for one student, if large-scale ring or channel is involved; could be enough work for several students
- needs further thought -- can be scaled down?
References
[1] Ya. S. Derbenev, “Advanced optical concepts for electron cooling”, Nucl.Instr. and Meth. A 441 (2000) 221-233. See the discussion beginning at thebottom of page 226.
[2] D. A. Edwards, M. J. Syphers, “Introduction to the Physics of High EnergyAccelerators”, J. Wiley and Sons, 1993. The material used here begins onpage 116 of this textbook.
[3] Private communications from R. Brinkmann, K. Flottmann, S. Nagaitsev
[4] D.A. Edwards, M.J. Syphers, “Comments on an Experiment on TransverseCoalescing,” unpublished.