Connection between zero chromaticity and long in-plane polarization lifetime in a magnetic storage ring

In this paper, we demonstrate the connection between a magnetic storage ring with additional sextupole fields set so that the $x$ and $y$ chromaticities vanish and the maximizing of the lifetime of in-plane polarization (IPP) for a $0.97\text{−}\mathrm{GeV}/c$ deuteron beam. The IPP magnitude was measured by continuously monitoring the down-up scattering asymmetry (sensitive to sideways polarization) in an in-beam, carbon-target polarimeter and unfolding the precession of the IPP due to the magnetic anomaly of the deuteron. The optimum operating conditions for a long IPP lifetime were made by scanning the field of the storage ring sextupole magnet families while observing the rate of IPP loss during storage of the beam. The beam was bunched and electron cooled. The IPP losses appear to arise from the change of the orbit circumference, and consequently the particle speed and spin tune, due to the transverse betatron oscillations of individual particles in the beam. The effects of these changes are canceled by an appropriate sextupole field setting.

Figure 1

Variation of the beta functions ${\beta }_x$ and ${\beta }_y$ along with the dispersion $D$ as a function of the distance $s$ along the reference orbit through the first arc of the COSY storage ring. The opposite arc is similar. The locations of the MXS, MXL, and MXG sextupole magnets are indicated with different colors. All of the magnets in each of these three families (in both arcs) are wired in series and controlled with a single setting of the current.

Figure 2

(from Ref. [1]) Measurements of the IPP as a function of time after the polarization was rotated into the horizontal plane. The scale is set so that the normalized polarization is unity at $t=0\mathrm{s}$. The measurements were made with two beam polarization states, separately normalized and then averaged. (Both the spin tune and the phase were constrained in order to improve the analysis accuracy for small IPP values [1].) The two panels represent two different sextupole magnet settings, resulting in polarization lifetimes, defined as the time for the normalized IPP to reach 0.606, or $64.7\pm 5.4\mathrm{s}$ and $18.6\pm 2.6\mathrm{s}$ in (a) and (b), respectively.

Figure 3

(from Ref. [1]) This figure represents the MXS $\times \mathrm{MXG}$ plane where $B_{\mathrm{MXL}}=-0.14{\mathrm{m}}^{-3}$. The gray bands represent the error about the lines of zero $x$ and $y$ chromaticity. Scans of the MXS and MXG sextupole family currents found several local points of maximal IPP lifetime. The circles represent a beam setup with horizontal heating; the plus signs represent runs with enhanced longitudinal width. The error bars are smaller than the points.

Figure 4

Measurements of the $x$ chromaticity as a function of the sectupole setting for either the MXG or MXS magnet families while the other was held at zero. The lines are the slices through the plane corresponding to the scan, which is a fit to all 24 chromaticity points. The field for MXL is $-0.145{\mathrm{m}}^{-3}$.

Figure 5

Measurements of the $y$ chromaticity in a manner similar to Fig. 1.

Figure 6

Values of the $x$ (green) and $y$ (blue) chromaticities as a function of the fields in the $S$ and $G$ sextupole magnet families. The planes are fits to a set of individual chromaticity measurements. The place where each plane crosses zero chromaticity is indicated by a dashed line.

Figure 7

In this picture of the $S\times G$ plane, four bars show the paths of four planned scans of the IPP lifetime. They are labeled “1” through “4” for later reference.

Figure 8

The normalized value of the measured IPP in time bin 5 as a function of the EDDA detector rate. The curves are a polynomial fit to the measurements. The black curve fits the black points for spin down; the red curve fits the red points for spin up; and the blue curve fits both. The IPP values have been scaled upward by a factor for each polarization state that makes the typical low-rate bin-5 polarization equal to one. In the analysis for all time bins, the measured IPP was adjusted upward by the appropriate factor to compensate for this loss.

Figure 9

Measurement of the IPP for two runs, normalized to one at $t=5\mathrm{s}$ based on the typical asymmetry values described in the text. In the left panel, the detector rate is low and not much change is apparent between red (original measurement) and blue (rate corrected measurement). The right panel contains another example for a higher rate run with a considerable initial slope change.

Figure 10

Panels for representative sextupole values involved in Scan 3 of Fig. 7 for a wide beam and negative polarization. Zero time begins 80 s into the store when the data acquisition was started. The value of $I_G$ in terms of a percentage of full sextupole current are included in each panel. A region (indicated here between vertical dashed lines) near the beginning of each data set was reproduced by either a linear or a quadratic polynomial (heavy red line). The red region shows the extension of this polynomial, including statistical errors from the fitting process. The green line represents the tangent to this polynomial at 5 s where it differs from a nonlinear polynomial fit.

Figure 11

(from Ref. [1]) Measurements of the normalized polarization. Least squares fits were made to the data acquired after rotation into the horizontal plane during four periods when the beam was extracted onto the polarimeter target. Data points taken with target extraction off are not shown. (The fitting template is based on an unfolded Gaussian beam distribution.) The residuals of the fit are given in the lower panel. The IPP lifetime is $782\pm 117\mathrm{s}$.

Figure 12

Representations of the IPP lifetime $\tau$ on a projected graph as a function of $S$ and $G$ field values for the wide beam. The various scans are shown with different colors. Scan 1 is lavender, 2 is black, 3 is blue, and 4 is red. The two polarization states are plotted separately; their values overlap closely and show essentially the same information.

Figure 13

Same as Fig. 12 except for the long beam. Scan 4 did not have enough data to be considered.

Figure 14

Measurements of $1/\tau$ of the IPP for scan 2 and a wide beam. The value of MXG was zero. Errors shown arise only from the fit of the polynomial to the early time measurements as shown in Fig. 10. Only the points for the spin down polarization state are shown. In general, the two polarization states agree well.

Figure 15

$1/\tau$ points for the wide beam scan 1 as a function of the $G$ sextupole field ($S$ is zero). The black triangles are for the down polarization state and the red triangles are for the up polarization state. The filled symbols indicate those points included in the straight line fit. Points shown with open symbols were not included. Weights based on the statistical errors were used. This emphasizes points in the fit that are near the zero crossing.

Figure 16

Same as Fig. 15 but for wide beam scan 2, shown as a function of $S$ ($G$ is zero). This is a subset of the data shown in Fig. 14.

Figure 17

Same as Fig. 15 but for wide beam scan 3, shown as a function of $G$ ($S=1.95{\mathrm{m}}^{-3}$).

Figure 18

Same as Fig. 15 but for wide beam scan 4, shown as a function of $G$ ($S=3.90{\mathrm{m}}^{-3}$).

Figure 19

Measurements made with the long beam of $1/\tau$ as a function of the $G$ sextupole field (Scan 1). The $S$ field is zero. The optimum field value is indicated by a dashed line.

Figure 20

Measurements made with the long beam of $1/\tau$ as a function of the $S$ sextupole field (Scan 2). The $G$ field is zero. The optimum field value is indicated by a dashed line.

Figure 21

Measurements made with the long beam of $1/\tau$ as a function of the $G$ sextupole field (Scan 3). The $S$ field is $1.95{\mathrm{m}}^{-3}$. The optimum field value is indicated by a dashed line.

Figure 22

Two lines with error bands show the places where the $x$ and $y$ chromaticities were consistent with zero. The locations of the points of largest IPP lifetime are shown by the circles for the wide beam and the plus signs for the long beam. For the long beam, the point from Scan 4 is missing.

Figure 23

Measurements of $1/\tau$ as a function of changing MXK field for $S=6.04{\mathrm{m}}^{-3}$ and $G=0{\mathrm{m}}^{-3}$. The black circle points are for polarization negative and the red X points for polarization positive. The dashed line marks the point chosen to bring the $x$ and $y$ chromaticities close together. The corresponding parabolic curves are a guide to the eye for each polarization state.