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Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
WHY DIRECT SPACE CHARGE WAS / IS BELIEVED TO HAVE ONLY A SMALL EFFECT
ON THE TMCI INTENSITY THRESHOLD OF THE CERN SPS?
E. Métral (following, as usual, very interesting discussions with AlexeyB during HB2016!)
◆ Not only because a very good agreement has already been reached between simulations and measurements without space charge
◆ Not only because some past simulations with space charge revealed only a small effect on the intensity threshold
◆ … but due to a simple model (which as usual needed / needs to be confirmed)…
◆ Appendix: Case of a constant inductive impedance
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
Not only because a very good agreement has already been reached between simulations and
measurements without space charge
0 1 2 3 4 5
x 1011
0
0.1
0.2
0.3
0.4
0.5
N (p/b)
! l (
eV
s)
Measurement − stable
Measurement − unstable (slow losses)
Measurement − unstable (fast losses)
Q20
N (p/b)! l
(e
Vs)
1 2 3 4 5
x 1011
0
0.1
0.2
0.3
0.4
0.5
Ve
rtic
al g
row
th r
ate
(1
/tu
rns)
0
0.005
0.01
0.015
0.02Q20
measurements HEADTAIL simulations
4.5x1011 p/b @ 0.35 eVs
nominal Island of slow instability
Courtesy of Hannes Bartosik et al.
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
Not only because some past simulations with space charge revealed only a small effect on the intensity
threshold
Broad-band WITHOUT SC Broad-band WITH SC
€
ΔQSC /QS ≈ 50
Courtesy of D. Quatraro
Courtesy of B. Salvant (Q26, BB impedance deduced from beam-based
measurements, without SC)
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
Not only because some past simulations with space charge revealed only a small effect on the intensity
threshold
◆ Last pictures from “Effects of direct space charge on transverse mode coupling instabi l i ty” by Quatraro&Rumolo_2010 (https://accelconf.web.cern.ch/accelconf/IPAC10/papers/tupd046.pdf)
=> My simple (too simple ;-)… just to guide...) consideration was / is the following
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
… but due to a simple model (which as usual needed / needs to be confirmed)…
SC ONLY (square-well air-bag, Blaskiewicz1998)
IMPEDANCE ONLY (BB impedance & (very) long-bunch regime)
= 0 here( )
Qc± =12× 2Qy0 + 2m+1( ) Qs +ΔQmS, y +ΔQm+1S, y⎡⎣ ⎤⎦
±12
Qs +ΔQm+1S, y −ΔQm
S, y( )2− 2ΔQm,m+1
S, y( )2
ΔQm≥0y = −
ΔQSC2
+ΔQSC2
⎛
⎝⎜
⎞
⎠⎟2
+ mQs( )2
0 2 4 6 8 10-3
-2
-1
0
1
2
3
ΔQSC /Qs
ΔQy/Q
s
ΔQm
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
◆ Beam stability condition WITHOUT SC
=>
=> This is the equation which leads to
… but due to a simple model (which as usual needed / needs to be confirmed)…
Qs + ΔQm+1S, y − ΔQm
S, y = 2 ΔQm,m+1S, y
~ 0 (see Fig. left of previous page)
Qs ≈ 2 ΔQm,m+1S, y
~ 0 (see Fig. left of previous page)
Nb, thy ∝ η Qy εL
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
… but due to a simple model (which as usual needed / needs to be confirmed)…
◆ Beam stability condition WITH SC
=> For a (very) long bunch, (very) high-order modes are excited and the Equation
becomes
=>
=>
Qs + ΔQm+1S, y − ΔQm
S, y = 2 ΔQm,m+1S, y
ΔQm≥0y = −
ΔQSC2
+ΔQSC2
⎛
⎝⎜
⎞
⎠⎟2
+ mQs( )2
ΔQm>>ΔQSC
2Qs
y ≈ −ΔQSC2
+mQs
ΔQm+1S, y − ΔQm
S, y = ΔQm+1y − ΔQm
y −Qs ≈ 0
Qs ≈ 2 ΔQm,m+1S, y
As for the case without SC
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
◆ Reminder of the picture of beneficial effect of SC for TMCI between modes -1 and 0: See “Stability Issues of Low-Energy Intense Beams” by Ng&Burov_1999 (http://inspirehep.net/record/508683/files/fermilab-fn-0685.pdf)
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
Appendix: Case of a constant inductive impedance
Elias Métral, 11/07/2016, Following discussions @ HB2016, Malmö, Sweden
=> In this case the usual Sacherer’s formula cannot be applied and the full eigenvalue system needs to be solved => Seems at least qualitatively close (to be checked quantitatively) to the simple formula of VladimirK and OliverBF
See also CAS course from Laclare