1. Laslett self-field tune spread calculation with momentum dependence (Application to the PSB at 160 MeV) M. Martini

2. Contents 06/07/2012M. Martini2 Two-dimensional binomial distributions Projected binomial distributions Laslett space charge self-field tune shift Laslett space charge tune spread with momentum Application to the PSB

3. Two-dimensional binomial distributions 06/07/2012M. Martini3 Binomial transverse beam distributions The general case is characterized by a single parameter m > 0 and includes the waterbag distribution (uniform density inside a given ellipse), the parabolic distribution... (c.f. W. Joho, Representation of beam ellipses for transport calculations, SIN-Report, Tm-11-14, 1980. The Kapchinsky-Vladimirsky distribution (K-V) and the Gaussian distribution are the limiting cases m  0 and m  . For 0 < m <  there are no particle outside a given limiting ellipse characterized by the mean beam cross-sectional radii a x and a y. Unlike a truncated Gaussian the binomial distribution beam profile have continuous derivatives for m  2.

4. Two-dimensional binomial distributions 06/07/2012M. Martini4 Kapchinsky-Vladimirsky beam distributions (m  0) Define the Kapchinsky-Vladimirsky distribution (K-V) as Since the projections of  B 2D (m,a x,a y,x,y) for m  0 and  KV 2D (m,a x,a y,x,y) yield the same Kapchinsky-Vladimirsky beam profile The 2-dimensional distribution  KV 2D (m,a x,a y,x,y) can be identified to a binomial limiting case m  0

5. Two-dimensional binomial distributions 06/07/2012M. Martini5

6. Two-dimensional binomial distributions 06/07/2012M. Martini6

7. Two-dimensional binomial distributions 06/07/2012M. Martini7

8. Two-dimensional binomial distributions 06/07/2012M. Martini8 Gaussian transverse beam distributions (m   ) The 2-dimensional Gaussian distribution  G 2D (  x,  y,x,y) can be identified to a binomial limiting case m   since

9. Projected binomial distributions 06/07/2012M. Martini9

10. Projected binomial distributions 06/07/2012M. Martini10 m01/213/226  √2√2√3√32√5√5√6√6√14  1/20.5770.6080.6260.6370.6640.683 --10.9840.9750.9600.955

11. Laslett space charge self-field tune shift 06/07/2012M. Martini11 Space charge self-field tune shift (without image field) For a uniform beam transverse distribution with elliptical cross section (i.e. binomial waterbag m=1) the Laslett space charge tune shift is (c.f. K.Y. Ng, Physics of intensity dependent beam instabilities, World Scientific Publishing, 2006; M. Reiser, Theory and design of charged particle beams,Wiley-VCH, 2008). For bunched beam a bunching factor B f is introduced as the ratio of the averaged beam current to the peak current the tune shift becomes Considering binomial transverse beam distributions and using the rms beam sizes  x,y instead of the beam radii a x,y yields

12. Laslett space charge self-field tune shift 06/07/2012M. Martini12 Space charge self-field tune shift (without image field) The self-field tune shift can also be expressed in terms of the normalized rms beam emittances defined as Nonetheless this expression is not really useful due to contributions of the dispersion D x,y and relative momentum spread  to the rms beam sizes

13. Laslett space charge self-field tune shift 06/07/2012M. Martini13 For bunched beam with binomial or Gaussian longitudinal distribution the bunching factor B f can be analytically expressed as (assuming the buckets are filled) m  

14. Laslett space charge tune spread with momentum 06/07/2012M. Martini14 Space charge self-field tune spread (without image field) Tune spread is computed based on the Keil formula (E. Keil, Non-linear space charge effects I, CERN ISR-TH/72-7), extended to a tri-Gaussian beam in the transverse and longitudinal planes to consider the synchrotron motion (M. Martini, An Exact Expression for the Momentum Dependence of the Space Charge Tune Shift in a Gaussian Bunch, PAC, Washington, DC, 1993).

15. Laslett space charge tune spread with momentum 06/07/2012M. Martini15 Tune spread formula In the above formula j 1 +j 2 +j 3 =n where n is the order of the series expansion. The function J(j 1 +j 2 +j 3 ) is computed recursively as It holds for bunched beams of ellipsoidal shape with radii defined as a x,y,z =  2  x,y,z with Gaussian charge density in the 3-dimensional ellipsoid. It remains valid for non Gaussian beams like Binomial distributions with a x,y,z =  (2m+2)  x,y,z (0  m <  ).  x,y are the rms transverse beam sizes and  z the rms longitudinal one, x, y, z are the synchro-betatron amplitudes. Q x,y,z are the nominal betatron and synchrotron tunes. R is the machine radius, the other parameters D x,y, , e, h, E 0... are the usual ones.

16. Application to the PSB 06/07/2012M. Martini16 Tune diagram on a PSB 160 MeV plateau for the CNGS-type long bunch PSB MD: 22 May 2012 Total particle number = 950 10 10 Full bunch length = 627 ns Q x0 = 4.10 (  tr =4) Q y0 = 4.21 E k = 160 MeV  x n (rms) = 15  m  y n (rms) = 7.5  m  p /p = 1.44 10 -3 Bunching factor (meas) = 0.473 RF voltage= 8 kV h = 1 RF voltage= 8 kV h = 2 in anti-phase PSB radius = 25 m  Q x0 = -0.247  Q y0 = -0.365 12 th order run-time  11 h The smaller (blue points) tune spread footprint is computed using the Keil formula using a bi-Gaussian in the transverse planes while the larger footprint (orange points) considers a tri-Gaussian in the transverse and longitudinal planes. All the space-charge tune spread have been computed to the 12 th order but higher the expansion order better is the tune footprint (15 th order is really fine but time consuming)

17. 06/07/2012M. Martini17 PSB MD: 4 June 2012 Total particle number = 160 10 10 Full bunch length = 380 ns Q x0 = 4.10 (  tr =4) Q y0 = 4.21 E k = 160 MeV  x n (rms) = 3.3  m  y n (rms) = 1.8  m  p /p = 2 10 -3 Bunching factor (meas) = 0.241 RF voltage= 8 kV h = 1 RF voltage= 8 kV h = 2 in phase  Q x0 = -0.221  Q y0 = -0.425 Tune diagram on a PSB 160 MeV plateau for the LHC-type short bunch Application to the PSB

18. 06/07/2012M. Martini18 PSB MD: 6 June 2012 Total particle number = 160 10 10 Full bunch length = 540 ns Q x0 = 4.10 (  tr =4) Q y0 = 4.21 E k = 160 MeV  x n (rms) = 3.4  m  y n (rms) = 1.8  m  p /p = 1.33 10 -3 Bunching factor (meas) = 0.394 RF voltage= 8 kV h = 1 RF voltage= 4 kV h = 2 in anti-phase  Q x0 = -0.176  Q y0 = -0.288 Tune diagram on a PSB 160 MeV plateau for the LHC-type long bunch Application to the PSB