Magnetic-field measurement and analysis for the Muon $g-2$ Experiment at Fermilab

The Fermi National Accelerator Laboratory (FNAL) Muon $g-2$ Experiment has measured the anomalous precession frequency $a_{\mu }^{}\equiv (g_{\mu }^{}-2)/2$ of the muon to a combined precision of 0.46 parts per million with data collected during its first physics run in 2018. This paper documents the measurement of the magnetic field in the muon storage ring. The magnetic field is monitored by systems and calibrated in terms of the equivalent proton spin precession frequency in a spherical water sample at $34.7{}^{\circ }\mathrm{C}$. The measured field is weighted by the muon distribution resulting in ${\tilde{\omega }}_p^{\prime }$, the denominator in the ratio ${\omega }_a^{}$/${\tilde{\omega }}_p^{\prime }$ that together with known fundamental constants yields $a_{\mu }^{}$. The reported uncertainty on ${\tilde{\omega }}_p^{\prime }$ for the Run-1 data set is 114 ppb consisting of uncertainty contributions from frequency extraction, calibration, mapping, tracking, and averaging of 56 ppb, and contributions from fast transient fields of 99 ppb.

Figure 1

An image of the storage ring as prepared for Run-1. Credit: Fermilab.

Figure 2

A cross section of the storage ring magnet featuring the components used to generate the highly uniform 1.45 $\mathrm{T}$ magnetic field in the Run-1 configuration.

Figure 3

The coordinate systems used in this paper. The muon beam nominal orbital radius is at $r=7.112\mathrm{m}$ in the $ry\varphi$ basis, equivalent to $x=0\mathrm{c}\mathrm{m}$ in the $xyz$ basis.

Figure 4

Schematic drawing of the calibration probe used to calibrate the trolley probe measurements.

Figure 5

Schematic drawing of the NMR probe for field mapping and monitoring.

Figure 6

The layout of the 17 probes in the trolley. Positive $x$ is towards higher radius. The fixed probe locations on the top (T) and bottom (B) of the storage region are shown as well. For the fixed probes, the six-probe stations have probes in the inner (I), middle (M), and outer (O) positions. In the four-probe stations, only the middle and outer probes are present.

Figure 7

A flow chart of the field analysis showing the calibration chain through the data processing. The muon distribution is an input that is external to the field analysis, and is required to calculate the muon-weighted field average. Bold items show input measurements to the analysis. Not shown is the NMR frequency extraction step required for each of the field measurements.

Figure 8

A typical free induction decay (FID) from a trolley probe. The zoomed inset shows the periodic behavior that is used to measure $\omega \left(t\right)$.

Figure 9

The upper panel shows $\mathrm{\Phi }\left(t\right)$ as extracted from the Hilbert transform of the signal in Fig. 8. The blue points in the lower panel are the difference between $\mathrm{\Phi }\left(t\right)$ and a linear fit $\mathrm{\Phi }{\left(t\right)}_{\mathrm{lin}}^{}$ to $\mathrm{\Phi }\left(t\right)$. The black line is a polynominal fit to these residuals, the dotted part shows the extrapolation outside the fit range to $t=0$.

Figure 10

Oscillatory signal as measured by the fixed probes (black dots), where linear drift corrections have been applied. The calibration probe data before (blue diamonds) and after (orange crosses) the correction is overlaid to show the typical size of corrections.

Figure 11

$\mathrm{\Delta }{\omega }_{j,x}^{\mathrm{tr}}$ (left) and $\mathrm{\Delta }{\omega }_{j,y}^{\mathrm{tr}}$ (right) measured by the trolley. The graphs were fitted to a two-dimensional polynomial to extract the large imposed field gradient. These gradients are used to uniquely identify a probe's location in the $xy$ plane.

Figure 12

A typical field map from a trolley run ($25\mathrm{th}$ of April 2018, approximately at 3 a.m.). On top, the raw, relative frequency $[{\omega }_j(\varphi ,0)-〈{\omega }_j〉]/〈{\omega }_j〉$ for the central trolley probe $j=1$. The lower three plots show the corresponding lowest-order multipoles, dipole (black), normal quadrupole (green), and skew quadrupole (red), as a function of azimuth. The dipole distribution has an RMS of 16 ppm with a peak-to-peak variation of 101 ppm.

Figure 13

Variations in the azimuthally averaged, relative frequency $[{\omega }_j^{\mathrm{tr}}(\varphi ,0)-〈{\omega }_j^{\mathrm{tr}}〉]/〈{\omega }_j^{\mathrm{tr}}〉$ for the central probe ($j=1$). The locations of the 17 trolley probes are indicated by (x). Their raw frequencies are averaged and the field variations are interpolated.

Figure 14

The difference between moving and static trolley measurements. (a) Comparison of the frequencies measured in a selected azimuthal region for the normal trolley motion and the stop-and-go operation. (b) The distribution of the dipole differences between motion and static measurements over the full ring.

Figure 15

Surveyed offsets (top: radial with a mean of 0.2 $\mathrm{m}\mathrm{m}$ and RMS of 0.5 $\mathrm{m}\mathrm{m}$, middle: vertical with a mean of −0.6 $\mathrm{m}\mathrm{m}$ and RMS of 0.6 $\mathrm{m}\mathrm{m}$) and rotation (bottom: roll with a mean of −${0.3}^{\circ }$ and a RMS of ${0.6}^{\circ }$) of the trolley rails.

Figure 16

The trolley footprint as seen by the fixed probe station, and the replacement data after vetoing the perturbed region.

Figure 17

Before the backward correction, the uncorrected tracking curve (light) can disagree with the measurement from the second trolley run. After the correction (dark), ${\mathbf{m}}_s^{\mathrm{tr}}\left(t\right)$ is equal to the corresponding trolley measurements ${\mathbf{m}}_s^{\mathrm{tr}}\left(0\right)$ (left diamond) and ${\mathbf{m}}_s^{\mathrm{tr}}(T=74\mathrm{h})$ (right diamond) at both bookending trolley runs. Note that this plot shows only a single station.

Figure 18

The first two moments (dipole and normal quadrupole), each azimuthally averaged, tracked over the four major data subsets in Run-1. The average field $m_1^{}$ is reported as ppm away from a reference value of 61.79 $\mathrm{M}\mathrm{Hz}$, so the value is $(f-61.79\mathrm{M}\mathrm{Hz})/61.79\mathrm{M}\mathrm{Hz}$. The moment $m_2^{}$ is also reported as ppm of 61.79 $\mathrm{M}\mathrm{Hz}$, but with a central value of 0 $\mathrm{Hz}$. The major discontinuities coincide with magnet ramps and major configuration changes.

Figure 19

A typical example of the muon distribution measured by the trackers after integrating for several hours. This distribution is used to weight the field map.

Figure 20

The number of muons integrated per bin over a typical trolley run pair and the azimuthally averaged dipole field over the same time. The number of integrated muons, represented by the number of observed decay positrons, is used to weight the field when evaluating the integral over time in Eq. (26).

Figure 21

The amplitudes of the muon beam moments $k_i^{}$ decrease as $i$ increases. The moments with positive (negative) amplitudes are shown in dark (light). This decrease implies that the effect of higher-order moments on the average field is suppressed by the muon distribution. The muon distribution can be thought of as a low-pass filter on the moments of the field. The vertical line shows the truncation order. All moments (and beam parameters) to the right of the vertical line are truncated.

Figure 22

(a) Schematic of the fiber magnetometer. The device is about 6 $\mathrm{c}\mathrm{m}$ tall. (b) The signal measured by the fiber magnetometer after subtracting the vibration background. The measurements and a fit to the transient are shown. The gray shaded band represents the associated uncertainty of $\pm 0.6\mu \mathrm{T}$. Muon data are fit from 30 to 700 $\mu \mathrm{s}$ after the kick.

Figure 23

(a) The time structure of the ESQ transient is determined by scanning the delay time between the pulse trigger and the NMR measurement. The gray region corresponds to the time intervals in which the ESQ are charged and muons can be used for the muon precession fits. (b) The same time structure zoomed in to a single beam pulse. The black dashed line indicates the time of the muon injection, the dotted line the earliest start of the precession fits.

Figure 24

The relative field shift caused by the transient as a function of the azimuthal position with respect to the center ($\varphi =0^{\circ }$) of a long ESQ. The transient is strongest in the center of the ESQ section, falling off toward the edges.

Figure 25

The distribution of the observed ESQ transient effect over all stations and sections. The full width of the distribution is used as the uncertainty ($\pm 178$ ppb) in the ESQ region and scaled down by the geometrical coverage factor of the ESQ in the storage ring (0.433).

Figure 26

FID with two RF pulses overlayed to a nominal FID (gray) with a single RF pulse for reference.

Figure 27

The field step size as a function of azimuth for all field jumps during the Run-1a data subsets. The vertical gray lines indicate the positions of the radial stops (solid: top, dashed: bottom).

Figure 28

This shows a pure normal sextupole ($m_5^{}$) field. The fixed probes (light circles) are all located in a low region (dark). If we simply relate the dipole component in the muon storage region (depicted by the large circle) to the average of all six fixed probes, then the presence of a nonzero normal sextupole would bias the calculation.

Figure 29

The geometry of an offset six-probe station and the corresponding Jacobian matrix. The change-of-basis matrix for offset stations is not corrected; instead, the correction for the offset is done with the Jacobian.

Figure 30

The Jacobians for all four different fixed probe layouts present in the experiment. Recall that $m_5^{}$ at four-probe stations is estimated using the average of their nearest neighbors. These estimates are used to make corrections to the measured values, as seen in the $5\times 5$ Jacobians for the four-probe stations.