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Calculation of Laslett Self-Field Tune Shift in PSB at 160 MeV with Momentum Dependence

This study explores the two-dimensional binomial distributions applicable to space charge effects, focusing on the Laslett self-field tune shift and its momentum dependence at 160 MeV in the PSB. It describes the characteristics of projected distributions, the relevance of the Kapchinsky-Vladimirsky distribution, and compares Gaussian and binomial distributions. Additionally, the research delves into the space charge tune spread and its analytical expression for bunched beams, providing detailed insights into the impacts of beam parameters and corrections for accurate modeling in high-energy physics applications.

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Calculation of Laslett Self-Field Tune Shift in PSB at 160 MeV with Momentum Dependence

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

  2. Contents • 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 M. Martini

  3. Two-dimensional binomial distributions • 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 ax and ay. • Unlike a truncated Gaussian the binomial distribution beam profile have continuous derivatives for m  2. M. Martini

  4. Two-dimensional binomial distributions • Kapchinsky-Vladimirsky beam distributions (m  0) • Define the Kapchinsky-Vladimirsky distribution (K-V) as • Since the projections of B2D(m,ax,ay,x,y) for m  0 and KV2D(m,ax,ay,x,y) yield the same Kapchinsky-Vladimirsky beam profile • The 2-dimensional distribution KV2D(m,ax,ay,x,y) can be identified to a binomial limiting case m  0 M. Martini

  5. Two-dimensional binomial distributions M. Martini

  6. Two-dimensional binomial distributions M. Martini

  7. Two-dimensional binomial distributions M. Martini

  8. Two-dimensional binomial distributions • Gaussian transverse beam distributions (m ) • The 2-dimensional Gaussian distribution G2D(x,y,x,y)can be identified to a binomial limiting case m  since M. Martini

  9. Projected binomial distributions M. Martini

  10. Projected binomial distributions M. Martini

  11. Laslett space charge self-field tune shift • Space charge self-field tune shift (without image field) • For a uniform beam transverse distribution with elliptical cross section (i.e. binomial waterbagm=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 Bf 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,yinstead of the beam radii ax,y yields M. Martini

  12. Laslett space charge self-field tune shift • 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 Dx,y and relative momentum spread to the rms beam sizes M. Martini

  13. Laslett space charge self-field tune shift • For bunched beam with binomial or Gaussian longitudinal distribution the bunching factor Bf can be analytically expressed as (assuming the buckets are filled) m   M. Martini

  14. Laslett space charge tune spread with momentum • 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). M. Martini

  15. Laslett space charge tune spread with momentum • Tune spread formula • In the above formula j1+j2+j3=n where n is the order of the series expansion. The function J(j1+j2+j3) is computed recursively as • It holds for bunched beams of ellipsoidal shape with radii defined as ax,y,z= 2x,y,zwith Gaussian charge density in the 3-dimensional ellipsoid. It remains valid for non Gaussian beams like Binomial distributions with ax,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.Qx,y,z are the nominal betatron and synchrotron tunes. • R is the machine radius, the other parameters Dx,y, , e, h, E0... are the usual ones. M. Martini

  16. Application to the PSB • All the space-charge tune spread have been computed to the 12th order but higher the expansion order better is the tune footprint (15th order is really fine but time consuming) • PSB MD: 22 May 2012 • Total particlenumber = 950 1010 • Full bunchlength = 627 ns • Qx0 = 4.10 (tr=4) • Qy0 = 4.21 • Ek = 160 MeV • xn (rms) = 15 m • yn (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 • Qx0 = -0.247 • Qy0 = -0.365 • 12th order run-time  11 h Tune diagram on a PSB 160 MeV plateau for the CNGS-type long bunch 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. M. Martini

  17. Application to the PSB • PSB MD: 4 June 2012 • Total particlenumber = 160 1010 • Full bunchlength = 380 ns • Qx0 = 4.10 (tr=4) • Qy0 = 4.21 • Ek = 160 MeV • xn (rms) = 3.3 m • yn (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 • Qx0 = -0.221 • Qy0 = -0.425 Tune diagram on a PSB 160 MeV plateau for the LHC-type short bunch M. Martini

  18. Application to the PSB • PSB MD: 6 June 2012 • Total particlenumber = 160 1010 • Full bunchlength = 540 ns • Qx0 = 4.10 (tr=4) • Qy0 = 4.21 • Ek = 160 MeV • xn (rms) = 3.4 m • yn (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 • Qx0 = -0.176 • Qy0 = -0.288 Tune diagram on a PSB 160 MeV plateau for the LHC-type long bunch M. Martini

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