why direct space charge was / is believed to have …...charge revealed only a small effect on the...

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


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.


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)


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


… 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


◆  Beam stability condition WITHOUT SC

=>

=> This is the equation which leads to


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



◆  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


◆  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)


Appendix: Case of a constant inductive impedance


=> 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

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