Analytical estimations of the chromaticity and corrections to the spin precession frequency in weak focusing magnetic storage rings

Understanding beam and spin dynamics are fundamental in accelerator-based experiments concerning polarized beams. And there has been growing interest in having more precise estimations of the beam and spin dynamics variables, as more high precision particle physics experiments in the intensity frontier appear. This paper provides analytical estimations of some of the important variables such as beam transverse chromaticities and corrections to the spin precession frequency in the simplest type of particle accelerator: a circular magnetic storage ring with weak vertical focusing. We attempt to precisely obtain the next order approximations from the small betatron oscillations or the momentum dispersion, verified by high precision spin tracking simulation. We also discuss a potential way to suppress the corrections to the spin precession frequency, which relevant experiments may find beneficial.

Figure 1

Analytic approximations of the transverse chromaticities as a function of field index in weak magnetic or electric focusing storage rings. In contrast to the magnetic focusing ring, the chromaticities depend on the momentum in the electric focusing ring. (a) Magnetic focusing case. (b) Electric focusing case. The solid lines indicate the low momentum limit $(\gamma =1)$ and the dashed lines indicate the high momentum limit $(\gamma \to \infty )$.

Figure 2

The transverse betatron tunes from the high precision tracking simulation and the analytic estimation. Each point is a betatron tune obtained by fitting the betatron oscillations from simulation, which agree well with the analytical estimations, indicated as dashed lines. The reference tunes $({\nu }_{x0}=\sqrt{1-n},{\nu }_{y0}=\sqrt{n})$ are shown as green horizontal solid lines in each figure. Details of the simulation setup and parameters are described in the text. (a) Horizontal tune $(v_x)$ as a function of the fractional momentum offset. (b) Vertical tune $(v_y)$ as a function of the fractional momentum offset.

Figure 3

The relative differences of the transverse betatron tunes between numerical and analytic estimations. The analytic estimations were obtained to the first order of the normalized momentum offset $\delta$. Details of the simulation setup and parameters are described in the text. (a) Relative difference of the horizontal tune $(\mathrm{\Delta }{v}_x/v_{x0})$ as a function of the fractional momentum offset. (b) Relative difference of the vertical tune $(\mathrm{\Delta }{v}_y/v_{y0})$ as a function of the fractional momentum offset.

Figure 4

The pitch correction per square pitch angle ($C_p/{\psi }_0^2$) in parts-per-$\mathrm{million}/{\mathrm{mrad}}^2$ as a function of the particle momentum and the weak focusing field index. Two particles, the proton with $G=1.79$ and the deuteron with $G=-0.14$ are plotted for both magnetic and electric focusing storage rings. Specific contour lines with the label 0 indicate when the pitch correction vanishes. For deuterons, the hatched regions are near the spin resonance ${\omega }_a^2={\omega }_y^2$ and therefore are excluded in the plot. (a) For protons with magnetic focusing. (b) For protons with electric focusing. (c) For deuterons with magnetic focusing. (d) For deuterons with electric focusing.

Figure 5

The dispersion correction ($C_d$) and pitch correction ($C_p$) for the proton with a momentum $1\mathrm{GeV}/\mathrm{c}$, obtained by the high precision spin tracking simulation and the analytic estimation in the magnetic and electric focusing storage rings. Each point was obtained by fitting the simulated spin precession curve for each particle, and agreed well with the analytical estimations, indicated by the dashed lines labeled “Analytic.” Another theoretical prediction for the pitch correction labeled “Farley” from Ref. [39] was plotted as reference. (a) The dispersion correction. (b) The pitch correction.

Figure 6

The relation between the magnetic and electric field indices ($n_B$ and $n_E$) to cancel out the dispersion correction for given momenta of the muon and the proton. The field indices must lie within a region enclosed by the red solid lines, to obtain the net vertical focusing and positive momentum. The magic momentum case for each particle is shown for validation, where it needs only the electric focusing to be present ($n_B=0$). (a) Muon. (b) Proton.

Figure 7

The spin precession frequency relative to the reference, $\mathrm{\Delta }{\omega }_a/{\omega }_{a0}$, in the combined magnetic and electric focusing storage ring with $5\mathrm{GeV}/\mathrm{c}$ muons from numerical simulations. The magnetic and electric field indices were given such that the dispersion correction $C_d$ is suppressed. Different colors represent different lattice configurations for the focusing elements: (blue) continuous focusing elements around the ring, (red) discrete 8-fold and (green) 40-fold alternating elements.