Diffusion measurement from observed transverse beam echoes

We study the measurement of transverse diffusion through beam echoes. We revisit earlier observations of echoes in the Relativistic Heavy Ion Collider and apply an updated theoretical model to these measurements. We consider three possible models for the diffusion coefficient and show that only one is consistent with measured echo amplitudes and pulse widths. This model allows us to parameterize the diffusion coefficients as functions of the bunch charge. We demonstrate that echoes can be used to measure diffusion much quicker than present methods and could be useful to a variety of hadron synchrotrons.

Figure 1

Left: The echo amplitude as a function of the coefficients $D_0$, $D_1$, and $D_2$ scaled by the value $D_{\text{scale}}=2.4\times {10}^{-15}{\mathrm{m}}^2/\mathrm{s}$. Right: Form of the echo pulse with the $D_1$, $D_2$ model shown in blue. The red curve outlines the upper envelope of the echo. Beam parameters in both plots were taken from Table 1, except for $\mu =0.0077$. $D_1$ and $D_2$ in the right plot were set to the values $D_{1,sc}$ and $D_{2,sc}$, respectively, which are defined in Sec. 3b.

Figure 2

FWHM as a function of the diffusion coefficients $D_0$, $D_1$, and $D_2$ scaled by the value $D_{\text{scale}}$. Each curve shows the impact of the single coefficient with the others set to zero. The FWHM is calculated analytically from Eq. (24) for $D_0$ and $D_1$ and numerically for $D_2$. Parameters were taken from Table 1, except for $\mu =0.0077$.

Figure 3

Comparison of the echo amplitude vs the tune shift strength scan. The data are shown in red with error bars, while the fits shown are with the three models for the diffusion coefficients discussed in the text.

Figure 4

Comparison of the echo amplitude vs the time delay between the dipole and quadrupole kicks. The data are shown with error bars, while the fits are shown with the three models discussed in the text.

Figure 5

The entire centroid position turn by turn (left) and the echo pulse isolated (right) for the data with the shortest FWHM. Here the centroid decoheres cleanly after the dipole kick. The quadrupole kick was applied at turn 450.

Figure 6

The entire centroid position turn by turn (left) and the echo pulse isolated (right) for the data with the largest FWHM. Notice the much larger and longer ringing of the centroid after the dipole kick. The quadrupole kick was applied again at turn 450.

Figure 7

FWHM vs the number of particles per bunch. Except for the points labeled 1 and 2, all the other data points show an increasing FWHM with bunch charge. The blue curve shows a quadratic fit through these points.

Figure 8

Left: Relative echo amplitude (in brown) as a function of the scaled diffusion coefficients $d_1/d_{1,sc}$ and $d_2/d_{2,sc}$ intersected by a plane (in blue) of a particular relative amplitude value, here chosen to be 0.2. The intersection defines the family of solutions for $(d_1,d_2)$ at this amplitude. Right: The FWHM (in brown) as a function of the same scaled variables and the plane (in blue) at a constant FWHM, here chosen to be 60 turns. Again, the intersection defines the family of solutions for the FWHM equation.

Figure 9

Left: Calculated $d_1$ values as a function of the number of particles per bunch and the linear fit to the values. Right: Measured relative echo amplitude (red) at different intensities and compared with the best fit curve (blue) with $(D_1,D_2)$, with D1 from the linear fit in the left plot and D2 independent of the bunch charge.

Figure 10

Dependence of $A_F$, defined in Eq. (38), on the ratio of diffusion coefficients $D_1/D_2$ for three values of the ratio of the action at the absorbing aperture $J_A$ to the initial emittance $J_0$.