Surface Processing and Discharge-Conditioning of High Voltage Electrodes for the Ra EDM Experiment
Roy A. Ready, Gordon Arrowsmith-Kron, Kevin G. Bailey, Dominic Battaglia, Michael Bishof, Daniel Coulter, Matthew R. Dietrich, Ruoyu Fang, Brian Hanley, Jake Huneau, Sean Kennedy, Peyton Lalain, Benjamin Loseth, Kellen McGee, Peter Mueller, Thomas P. O'Connor, Jordan O'Kronley, Adam Powers, Tenzin Rabga, Andrew Sanchez, Eli Schalk, Dale Waldo, Jacob Wescott, Jaideep T. Singh
SSurface Processing and Discharge-Conditioning of High Voltage Electrodes for theRa EDM Experiment
Roy A. Ready a , Gordon Arrowsmith-Kron a , Kevin G. Bailey b , Dominic Battaglia a , Michael Bishof b , Daniel Coulter a ,Matthew R. Dietrich b , Ruoyu Fang a , Brian Hanley a , Jake Huneau a , Sean Kennedy a , Peyton Lalain a ,Benjamin Loseth a , Kellen McGee a , Peter Mueller b , Thomas P. O’Connor b , Jordan O’Kronley a , Adam Powers a ,Tenzin Rabga a , Andrew Sanchez a , Eli Schalk a , Dale Waldo a , Jacob Wescott a , Jaideep T. Singh a a National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing,Michigan 48824, USA b Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
Abstract
The Ra EDM experiment uses a pair of high voltage electrodes to measure the atomic electric dipole moment of
Ra.We use identical, plane-parallel electrodes with a primary high gradient surface of 200 mm to generate reversible DCelectric fields. Our statistical sensitivity is linearly proportional to the electric field strength in the electrode gap. Weadapted surface decontamination and processing techniques from accelerator physics literature to chemical polish andclean a suite of newly fabricated large-grain niobium and grade-2 titanium electrodes. Three pairs of niobium electrodesand one pair of titanium electrodes were discharge-conditioned with a custom high voltage test station at electric fieldstrengths as high as +52.5 kV/mm and − . ±
20 kV / mm with steady-stateleakage current ≤
25 pA (1 σ ) and a polarity-averaged 98 ±
19 discharges per hour. These electrodes were installed in theRa EDM experimental apparatus, replacing a copper electrode pair, and were revalidated to ±
20 kV / mm. The niobiumelectrodes perform at an electric field strength 3.1 times larger than the legacy copper electrodes and are ultimatelylimited by the maximum output of our 30 kV bipolar power supply. Keywords: high voltage electrode conditioning, leakage current, large-grain niobium, radium-225, atomic electricdipole moment, magnetic Johnson noise
1. Ra EDM Motivation and Requirements
Violation of combined charge conjugation ( C ) and par-ity ( P ) symmetry, or CP , is a necessary ingredient of theobserved dominance of matter over antimatter, or baryonasymmetry of the universe (BAU). CP violation is en-coded in the Standard Model (SM) by a complex phaseterm in the Cabibbo-Kobayashi-Maskawa (CKM) quarkmixing matrix [1]. The SM critically underestimates theBAU, suggesting that new sources of CP violation haveyet to be discovered [2].Permanent electric dipole moments (EDMs) violatetime-reversal ( T ) and P symmetry. Assuming CP T con-servation, EDMs also violate CP . Neutron, electron,molecular, and atomic EDM experiments have been car-ried out over the last seven decades in an effort to measurea nonzero EDM magnitude. While nonzero EDMs remainout of reach for now, the precision of EDM measurementscontinues to improve. Observing a nonzero EDM near sen- Email address: [email protected] (Roy A. Ready) sitivities of today’s leading experiments would provide aclean signature of Beyond the Standard Model physics [1].The atomic EDM of
Ra (spin I = 1 /
2) is enhancedby the octupole deformation (“pear shape”) of its nucleus.Radium-225 has a 55 keV parity doublet ground statestructure, compared to approximately 1 MeV in spheri-cally symmetric nuclei [3]. This enhances the observablecomponent of the EDM, characterized by the nuclear Schiffmoment. The Schiff moment of
Ra is predicted to be upto three orders of magnitude larger than that of diamag-netic atoms with spherically symmetric nuclei [4, 5, 6, 7].The Ra EDM experiment (Argonne National Lab,Michigan State University) measures the precession fre-quency of
Ra in a controlled, uniform magnetic andelectric field between two high voltage electrodes in anoptical dipole trap (ODT). EDM precession measure-ments are performed at Argonne National Lab (ANL),while offline upgrades such as the high voltage de-velopment discussed in this report are carried out atMichigan State University (MSU). In the proof ofprinciple measurement, the EDM 2 σ upper limit wasmeasured to 5 . × − e cm [8]. This was reduced to1 . × − e cm in the subsequent run [9]. Hereafter we a r X i v : . [ phy s i c s . i n s - d e t ] F e b igure 1: Cross-sectional electrode schematic. Surfaces have a flat-ness tolerance of 25.4 µ m and a parallelism of 50.8 µ m. The topsurface is polished to an average roughness of 0.127 µ m. The base ismounted by a 10-32 tapped hole. will refer to these as the ‘first generation’ measurements.The shot noise-limited EDM standard error σ EDM ( e cm) is given by: σ EDM = ¯ h E √ (cid:15)N T τ , (1) where E (V / cm) is the external electric field,¯ h (eV s) is the reduced Planck constant, (cid:15) (unitless) is the atom detection efficiency, N (unitless) is the number of atoms per sample, T (s) is the total measurement time, and τ (s) is the measurement time per cycle.As seen in Equation 1, the statistical sensitivity of theEDM measurement scales linearly with the electric fieldstrength. The Ra EDM experiment will be significantlyimproved with targeted upgrades to the experimental ap-paratus over the next several ‘second generation’ mea-surements. In particular, we will use a new atom detec-tion method to increase (cid:15) and new electrodes to increase E . We will surpass the 10 − e cm sensitivity level dur-ing this phase and the Ra EDM limit will constrainhadronic CP -violating parameters alongside other EDMexperiments [10]. The EDM couples to an external electric field analo-gously to the coupling of the atomic magnetic dipole mo-ment to an external magnetic field. The Hamiltonian ofan atom in the presence of a perfectly uniform electric andmagnetic field is defined as: H = − µ (cid:32) (cid:126)S · (cid:126)BS (cid:33) − d (cid:32) (cid:126)S · (cid:126)ES (cid:33) , (2)where µ = − . × − eV/T is the atomic magnetic dipolemoment of Ra [11], (cid:126)S is the atomic spin, (cid:126)B (T) is the applied magnetic field, d ( e cm) is the atomic EDM, and (cid:126)E (V/cm) is the applied electric field. Table 1: Ra EDM systematic requirements at the10 − e cm sensitivity level. Detailed descriptions of eval-uations of θ EB , | ∆ E | /E, and ¯ I can be found in our previouswork [9]. ∆ B is determined by Equation 5. We describe ourcalculation of the Johnson noise limit in Appendix A. description systematic limit (cid:126)E, (cid:126)B alignment θ EB ≤ | ∆ E | E ≤ B ≤
100 fT a steady-state leakage current ¯ I ≤
100 pA a magnetic Johnson noise (cid:114) dB n dν ≤
15 pT a √ Hz a per measurement cycle The
Ra atoms will precess with frequency ω + ( ω − )when (cid:126)E is parallel (antiparallel) to (cid:126)B : ω ± = 2¯ h ( µB ± dE ) rad / s (3)In the most recent Ra EDM experiment we applied a2 . µ T magnetic field and measured a precession frequencyof 181 . ± . d = ¯ h ∆ φ Eτ , (4)where ∆ φ (rad) is the difference in accumulated phase be-tween the two “field-on” configurations. With a perfectlyuniform and static magnetic field under all configurations,the phase difference between the parallel and antiparallelfield configurations is purely due to the EDM interactionwith the electric field. A higher electric field strength willgenerate a larger accumulated phase and improve EDMsensitivity.During each measurement cycle, one electrode is chargedto ≤ +30 kV (positive polarity) while the other isgrounded. The atom precession lifetime is currently abouttwenty seconds. We expect to increase the precession life-time to one hundred seconds [12] as improvements aremade to the ODT. The charged electrode is then rampedto zero voltage and remains grounded for a period of 60 s2 igure 2: Left: assembly of the niobium pair Nb at 1 mm gap inMacor holder. Right: a slit centered on the gap shields the electrodesurfaces from heating by the atom-trapping and excitation lasers. while a new sample of atoms is prepared. The cyclerestarts and the electrode is charged to the same voltagemagnitude at negative polarity. We repeat this processuntil the atomic oven is depleted after approximately twoweeks.Now we’ll discuss EDM measurement systematics re-lated to the high voltage system. Our requirements foreach systematic are given in Table 1.The electric field between the electrodes must be sym-metric, uniform, and reversible to minimize systematiceffects. The alignment between (cid:126)E and (cid:126)B is fixed aftermounting the electrodes to the Macor holder, as shown inFigure 2. In the experimental apparatus, the holder andelectrodes rest within a borosilicate glass tube. We willuse vector fluxgates with a system of autocollimators tooptically determine the field uniformity and alignment forthe second generation EDM measurements [13]. The fieldreversibility is measured with a calibrated high voltage di-vider (Ross Engineering V30-8.3-A).Magnetic field fluctuations caused by current in the elec-trodes, or magnetic Johnson noise, limits the choice ofelectrode materials and geometries that are suitable for anEDM measurement. The magnetic field scales as ρ − / ,where ρ (Ω m) is the resistivity. For two niobium elec-trodes separated by 1 mm with the geometry shown inFigure 1, we estimate the magnetic Johnson noise per sam-ple to be 2.48 pT/ √ Hz. For an EDM measurement lasting T = 15 days with an atom precession time of τ = 100 sand an electric field of 30 kV/mm (see Equation 1), mag-netic Johnson noise will only become significant at the10 − e cm level. A detailed description of magnetic John-son noise calculations is given in Appendix A.We consider an additional systematic in which the mag-netization of a fraction of the impurities in the electrodesdepends on the polarity of the charging current. A suffi-ciently high concentration of paramagnetic impurities nearan electrode primary surface could perturb the magneticfield in the radium cloud region. This would generate anatomic precession frequency mimicking an EDM signal,which can be expressed as a “false” EDM d ∆ B : d ∆ B = µ ∆ BE , (5)where ∆ B is the local magnetic field change from magnetic Figure 3: A COMSOL meshed model of the electrodes with theregion of interest shaded blue. The origin is 0.5 mm below the topelectrode surface, centered on the top electrode. When the bottomelectrode is aligned with the top electrode, the origin is exactly inthe center of the electrode gap. impurities in the electrodes as the electric field is reversed.For a local magnetic field change ∆ B ≈
100 fT per30 kV/mm field reversal, this systematic will only becomesignificant at the 10 − e cm level. Measuring a magneticfield strength of this magnitude will require more sensi-tive techniques than the low-noise fluxgate magnetometers(Bartington Mag-03MSL70) we currently use.To minimize systematic effects due to magnetic impu-rities, we use high-grade electrode materials and surfaceprocessing techniques that remove contaminants. Table 2and Table 3 list the material properties and processingtechniques that we use. We’ll discuss electrode materialselection and surface processing in detail in Section 2. A radium sample is trapped in the electrode gap byan ODT for each EDM measurement cycle. We inducecoherent atomic precession in the controlled magnetic andelectric field with a polarizing laser pulse. The precessionfrequency is measured by firing a subsequent ‘detection’laser pulse and imaging the atom cloud photon absorptionfraction after a variable delay time δ (ms).We considered two methods for the sequencing of thepolarization pulse, detection pulse, and the electric fieldramping. In the first method, the polarizing laser pulseis fired after the electric field ramps on and the detec-tion pulse is fired before the field ramps off. This shiftsthe Ra ground state due to an interaction between theODT polarization and the DC electric field. In the secondmethod, the polarizing laser pulse is fired before rampingthe field on and after ramping the field off. This pulse se-quence avoids potential mixing of the excited state hyper-fine levels and suppresses atomic polarization from ODTand electric field interactions.The Ra EDM experiment uses the second method tomeasure atomic precession. We also consider precession3 ield angle response to di ff erent angular misalignments gap coordinate shift along vertical axis y ( µ m)angular misalignment (mrad) Figure 4: θ E as a function of the vertical distance y when theelectrodes are axially aligned for different angular misalignments.At y = 0, θ E is evaluated 0.5 mm below and centered on the topelectrode. perturbations caused by transient magnetic fields that aregenerated during electrode charging. This effect is sup-pressed if the ramping on and ramping off pulse shapesare symmetric. Even with zero charging field cancella-tion, this systematic will only become significant at the10 − e cm level [9]. Two identical electrodes make up the Ra EDM elec-trode pair. The primary surface, seen as the top surfacein Figure 1, is flat and 16 mm in diameter. The roundededges have 4 mm circular radial curvatures. We use plane-parallel electrodes (see Figure 2) so that the reversible fieldis uniform and symmetric as the electrodes alternate rolesas cathode and anode every EDM measurement cycle.The Ra EDM experiment requires an applied electricfield that is symmetric, uniform, and reversible in the cen-ter of the electrode gap where the precession frequency ofthe 50 µ m diameter radium cloud is measured. Our elec-trode geometry reliably meets these requirements at fieldstrengths of 12–30 kV/mm. In Section 1.5 we will usefinite element modeling to show that the electric field gen-erated by our electrodes matches that of the ideal infinite-plane capacitor in the atom cloud region.Systematic effects arising from asymmetric field reversalmust continue to be reduced as EDM statistical sensitivityimproves. In the current measurement scheme, one elec-trode is permanently grounded and the other electrodeis charged by a bipolar power supply. We will designa more symmetric apparatus that allows us to alternatethe charged and grounded electrodes using high voltageswitches and a unipolar 50 kV power supply in the nextphase of high voltage development. In addition, we will op-timize the electrode geometry to reduce field edge effectsusing the computational modeling described in Section 1.5. gap coordinate shift along vertical axis y ( µ m) θ E model with 1000 µ m axial misalignment
16 mrad angular misalignment
Figure 5: A straight line fit to the simulated polar angle of theelectric field for an angular misalignment of 16 mrad and an axialmisalignment of 1 mm. At y = 0, θ E is evaluated 0.5 mm below andcentered on the top electrode. One systematic that creates a “false” EDM-like signalscales with the sine of the angle between the electric fieldand the controlled uniform magnetic field we use for mea-suring the precession of the radium atoms. We modeledthe high voltage electrodes in the finite element analysissoftware
COMSOL Multiphysics (version 5.3) to study theelectrostatic behavior as the alignment is varied from per-fectly parallel, axially-centered electrodes. In the model,the electrodes are surrounded by a perfect vacuum. Theelectrode gap size is fixed at 1 mm and the top elec-trode is charged to −
30 kV for a nominal electric fieldof E = 30 kV/mm.Our simulations use the Extremely Fine settings with
Size Expression increased to 4 × − in the gap regionand Resolution increased to 200 along the upper curvedelectrode surface. We reduced the minimum mesh elementsize to 20 µ m, where we found that the electric field de-pendence on the mesh size converges to negligibly smallfluctuations.The coordinate system of the electrode pair electrostaticmodel is shown in Figure 3, with the origin defined as themidpoint between the two electrodes along their verticalaxis of the top electrode. We find that the vertical fieldstrength E y changes by less than 6 ppb per 100 µ m whenthe electrodes are perfectly aligned. The horizontal fieldmagnitude E ⊥ = (cid:112) E x + E z changes by less than 5 ppbper 100 µ m with respect to E within 0.5 mm of the origin.In practice, we align our electrodes to better than 4 mradin the high voltage test stand described in Section 3.1.We investigated the effect of misalignments betweenthe electrodes on the electric field angle, defined as θ E = arctan ( E y /E ⊥ ). There are two types of misalign-ments we consider. Angular misalignments, or tilts, areintroduced by rotating the bottom electrode about the z axis in the range 0–16 mrad. Axial misalignments, orshifts, translates the bottom electrode along the x axis4 able 2: Bulk material properties of electrodes. material Z φ (eV) strong magnetic density resistivity hardness outgas rateimpurity (%) a (kg/m ) ( µ Ω cm) b (kgf/mm ) (Torr nL s − cm − )niobium c
41 4.3 2 . × − . d
29 4.65 2 . × − . . e
22 4.33 5 . × − . f - 4.34 8 . × +1 . g
42 4.6 1 . × − . . a We define “strong magnetic impurities” as χ m / (10 − cm mol − ) > +1000, where χ m is the molar susceptibility. χ m (Nb) = +208. b Resistivity measured at 273 K. c Hardness measured at 473 K. Outgas rate estimated from the correlation between Cu, SS, and Nb desorption. d Hardness measured for single crystal (III) at 293 K. Outgas rate measured for unbaked OF high-conductivity after ten hours. e Hardness measured for iodide-annealed, 99.99% purity at 293 K. Outgas rate measured for unbaked OF high-conductivityafter ten hours. f Hardness measured for designation type 304. Outgas rate measured for unbaked, electropolished NS22S after ten hours. g Hardness measured at 293 K. and offsets the electrode centers. Shifts of up to 1 mmdisplacements are considered in this work. When the tiltand shift are zero, the electrodes are perfectly aligned and θ E = 0 near the center of the gap, corresponding to a uni-form vertical field.The electric field angle scales linearly with the angularmisalignments, as shown in Figure 4. We modeled thechange in θ E as a linear function of the position in boththe xy plane (Figure 5) and the xz plane. The linearmodel reproduces the change in the electric field angle toan accuracy of better than 1 µ rad in both planes up to1 mm from the center of the gap, even for large angularand axial misalignments.The vertical field strength is reduced minutely even forthe severe 16 µ rad tilt and 1 mm shift we’ve modeled inFigure 5. We find the vertical field strength fractionalchange ∆ E y /E ≈
230 ppm per 500 µ m from the origin.The electrode shift effectively changes the gap size nearthe origin, causing a constant offset in the vertical fieldstrength. For the case of a 16 mrad angular misalignmentand 1 mm axial misalignment, the offset in E y is 1 . y ) coordinate. Initiallyvertical ( θ E = 0) at the top surface of the electrode, thefield angle changes by 1% of the electrode tilt per 10 µ malong the y axis. The field angle is 8 mrad at the midplanehalfway between the electrodes and 16 mrad at the surfaceof the bottom electrode. If we scan horizontally in themidplane along the x axis towards the electrode edge, thepolar angle changes by 0 .
03% per 10 µ m.In the more realistic case of a 2 mrad tilt, we find that θ E changes by 0.2 µ rad per 100 µ m in the vertical plane and0.02 µ rad per 100 µ m in the midplane. EDM systematiceffects arising from field angle changes of this magnitudeare far below our current statistical sensitivity. We define discharge-conditioning as the process of ap-plying iteratively higher voltages to the electrodes to sup-press steady-state leakage current and discharge rates be-tween them. Leakage current refers to any current flow-ing between the electrodes detected by a picoammeter inseries with one of the electrodes, as shown in Figure 6.We differentiate our method from the standard “current-conditioning” method [23] because we characterize elec-trode performance by counting discrete discharges overtime and we use a periodic voltage waveform. In this pa-per we will interchangeably use the shorthand term “con-ditioning” when referring to discharge-conditioning.In the absence of surface particulate contamination,electrode discharges are caused by charge buildup on mi-croprotrusions on the electrode surfaces [24], which we willrefer to as charge emitters. We process and handle ourelectrodes in Class 100 or better environments to minimizeparticulate contamination. The height of charge emit-ters have been measured on the order of 1 µ m in bufferchemical-polished large-grain niobium electrodes preparedsimilarly to our electrodes [25]. If the charge emitter isnear the edge of the electrode, we expect the higher gra-dients will increase the likelihood of a discharge.Controlled discharges electrically polish away, or ablatecharge emitters over time, allowing the electrodes to per-form reliably at higher voltages [23]. As shown in Sec-tion 3, it may take tens to more than one hundred hoursof discharge-conditioning to suppress charge emitters. Weexpect the required conditioning duration may take longerif the surface is insufficiently polished or contaminated.Bulk properties, such as the work function, resistivity, orhardness of the electrode may also play a role in the con-ditioning time. These bulk properties are listed for a se-lection of commonly used electrode materials in Table 2.5 able 3: Ra EDM electrode inventory. The large-grain (LG) niobium electrodes have a residual resistance ratio (RRR) > ◦ C vacuum outgasbake. WB = 150–160 ◦ C water bake. USR = ultrasonic rinse after detergent bath. batch material pair surface processing recipe1 OF copper Cu
Simichrome → EP → USR → WB2 LG niobium Nb SiC → BCP → DPP → CSS → USR → VB · · ·· · · LPR → HPR2 LG niobium Nb SiC → BCP → USR → VB → HPR → resurface · · ·· · · BCP → HPR2 G2 titanium Ti SiC → HF → USR → VB → HPR2 G2 titanium Ti SiC → HF → EP → USR → VB → HPR3 LG niobium Nb
SiC → BCP → USR → HPR → WB3 LG niobium Nb SiC → BCP → USR → HPR a Legacy electrodes used for first two measurements [8, 9]. b Second generation electrodes described in this work and currently installed in the Ra EDM apparatus.
Four pairs of niobium electrodes and two pairs of ti-tanium electrodes were surface processed as described inTable 3. After high-pressure rinsing they are preservedin clean room environments of Class 100 (ISO 5) or bet-ter. We conditioned pairs of electrodes in a custom,Class 100-rated high voltage test station at MSU by ap-plying DC voltages as high as ±
30 kV at gap sizes inthe range 0.4–2.5 mm. Maximum fields of +52 . − .
2. Electrode Properties and Preparation
The first generation EDM measurements used a pair ofelectropolished oxygen-free copper electrodes [8, 9]. Their geometry is identical to the new electrodes discussed in thiswork (Figure 1). Surface processing of these electrodes,labeled as Cu , is detailed in Table 3.The legacy electrodes were conditioned at ANL with aunipolar −
30 kV power supply (Glassman PS/WH-30N15-LR) in a Macor holder at a 2 mm gap size in 2008. Theelectric field was reversed by turning the system off andmanually switching the power supply terminations at thehigh voltage feedthroughs. Voltage was increased from1–20 kV in 1 kV steps while monitoring the steady-stateleakage current. Conditioning was declared complete if theelectrodes could hold 20 kV with a steady-state leakagecurrent of <
100 pA for ten hours.Four pairs of electrodes total were tested in this manner,including two pairs of titanium electrodes and one pair ofcopper electrodes without electropolishing. The legacy ti-tanium electrodes all exhibited leakage current higher than100 pA at 20 kV. Flooding the test chamber with argongas and plasma discharge-conditioning the titanium elec-trodes was attempted without an observable benefit. Bothcopper electrode pairs were conditioned, with the elec-tropolished (EP) electrodes taking significantly less time.The legacy electrode pair Cu was mounted in a Macorholder at a 2.3 mm gap size and installed in the Ra EDMexperimental apparatus [26]. The pair was retested at20 kV / 2.3 mm = 8.7 kV/mm but exceeded the 100 pAlimit. This was remedied by reducing the electric field by25% to 6.5 kV/mm for the EDM measurement. We sus-pect that the primary surface of one or both of these legacyelectrodes was contaminated during installation. This wasa motivating factor in the development of the decontami-nation techniques for the new electrodes discussed in sub-sequent sections. We selected large-grain niobium and grade-2 titaniumfor testing after reviewing accelerator physics literature.6
1 2 4 5 6 9 8 3 7
Figure 6: MSU HV test apparatus. 1 ○ ○ Pfeiffer HiPace 80 turbomolecular pump withforeline Edwards nXDS10i A736-01-983 dry scroll rough pump and two valves 3 ○ Matheson 6190 Series 0.01 micron membrane filterand purge port 4 ○ Ceramtec 30 kV 16729-03-CF HV feedthroughs 5 ○ .
312 in . electrodes in PEEK holder (resistivity 10 MΩ cm)6 ○
20 AWG Kapton-insulated, gold-plated copper wire 7 ○ MKS 392502-2-YG-T all-range conductron/ion gauge 8 ○ Shielded protectioncircuit: Littelfuse SA5.0A transient voltage suppressor, EPCOS EX-75X gas discharge tube, Ohmite 90J100E 100 Ω resistor in series withKeithley 6482 2-channel picoammeter 9 ○ Ohmite MOX94021006FVE 100 MΩ resistors in series with Applied Kilovolts HP030RIP020 HV.
The bulk properties of these metals and other commonlyused high voltage metals are catalogued in Table 2. Ourgoal is to use the material that sustains the highest electricfield strength while minimizing leakage current and mag-netic impurities that could introduce EDM systematic ef-fects. Stainless steel was excluded from our testing due toits relatively high ferromagnetic content but its propertiesare nevertheless included for reference.Large-grain niobium electrodes with a cathode areaof 3170 mm have been tested to fields as high as18.7 kV/mm [25]. Fine-grain appears to perform slightlyworse, perhaps because the higher grain boundary den-sity increases particulate adherence to the electrode sur-face [27]. The highest reported electric field for gap sizesnear 1 mm that we found is 130 kV/mm using an asym-metric titanium anode and molybdenum cathode with aneffective area of 7 mm [28]. The effective area of theRa EDM electrode is 200 mm , approximately a factor ofthirty larger. There is evidence that larger stressed areasare prone to lower breakdown voltages, suggesting that aminiaturized Ra EDM electrode geometry could improvethe maximum stable electric field [29].In the presence of high electric fields, an oxide layeron an electrode surface could be a significant source ofparticle emission. Niobium oxidizes at a higher ratethan titanium and oxygen-free copper [30, 31, 32, 33, 34].However, significant oxidation rates for these materialshave only been observed at temperatures in excess of500 ◦ C [30, 34, 33, 35, 36]. The Ra EDM experimental ap-paratus is pumped to ultrahigh vacuum ( < − Torr) atroom temperature. We therefore expect that oxidationrates are negligibly low for any selection of the consideredelectrode materials. We have considered a potential EDM systematic arisingfrom magnetic impurities in the electrodes that changepolarization with each electric field reversal. A sufficientlyhigh concentration of such impurities could perturb themagnetic field in the radium cloud region. To address this,we measured the magnetization of copper, niobium, andtitanium electrode-sized pucks in a magnetically shieldedmu-metal enclosure with commercial fluxgates (BartingtonMag03IEL70). Titanium was the most magnetic, in agree-ment with the magnetic properties listed in Table 2. Acustom atomic vapor cell magnetometer with a 5 mm cubecell was also used to measure the magnetization of a pairof titanium electrodes to ≤ We fabricated four pairs of large-grain niobium elec-trodes and two pairs of grade-2 titanium electrodes in twoseparate batches. Surface treatment procedures for eachelectrode pair are catalogued in Table 3 (batches 2 and 3).Our target validation field strength was 15 kV/mm orbetter for this phase of the Ra EDM high voltage devel-opment. With this in mind, we used processing proce-dures informed by discussions with Jefferson Lab accelera-tor physicists and a review of the literature. All but one ofthe second generation electrode pairs are chemically pol-ished prior to HPR. Recently, centrifugal barrel polishinghas been shown to reduce the required conditioning timecompared to chemical etching [37]. This is an encouragingprospect for conditioning Ra EDM electrodes to signifi-cantly higher fields in a future phase of development.7 able 4: Surface decontamination comparison. P = rinse pressure, T = rinse time, CR = clean room, RR = rinse resistivity. Lab
P T
RR CR Ref.(psi) (min) (MΩ cm) (Class)CERN 1500 30 18 100 [38]JLab 1200 20 >
18 - [25]KEK 1100 5 80 100 [28]MSU 1200 20 18.1 100 This workThe four titanium electrodes (Ti , Ti , Ti , and Ti )were mechanically polished with silicon carbide after fabri-cation. Their mean surface roughness averages were mea-sured in the range 16–23 nm using a profilometer (Mi-croXAM) in a clean room. We electropolished pair Ti commercially and remeasured the electrode surfaces. Weobserved an increase in the surface roughness of the elec-tropolished titanium electrodes by ≈
50% and micropro-trusions in the range 1–10 µ m.We decontaminate the electrodes in clean rooms at theFacility for Rare Isotope Beams (FRIB) after polishing.The electrodes are cleaned with detergent and rinsed withpure water in an ultrasonic bath in a staging area. Theyare rinsed in a second ultrasonic bath with UPW insidea Class 100 clean room. The electrodes are then highpressure-rinsed with UPW at 1200 psi for twenty minutes.After HPR, the electrodes dry in the clean room for severaldays before being sealed in poly tubing backfilled with dry,filtered nitrogen. A summary of clean room and HPR pa-rameters from several high-gradient development groups isgiven in Table 4.
3. Electrode Discharge-Conditioning
A schematic of the MSU high voltage test station isshown in Figure 6. Electrode pairs are mounted to apolyether ether ketone (PEEK) holder inside a six-waycross vacuum chamber. We estimate current flowingthrough the PEEK holder (resistivity 10 Ω cm) is lim-ited to 10 − pA or less with an electrode voltage of 30 kV.The vacuum chamber is maintained at 10 − Torr with aturbomolecular pump (Pfeiffer Hipace 80). At this pres-sure the mean free path of residual gas molecules is overa meter, significantly larger than the dimensions of thechamber.The test station is frequently brought to atmosphericpressure for upgrades and electrode installations. Weperform this work in clean rooms that are validated toClass 100 or better with a NIST-calibrated particle counter(Lighthouse Handheld 3016). The chamber is backfilledwith dry, high-purity nitrogen through a 0.01 micron gasmembrane particle filter (Matheson 6190 Series) whileventing the chamber and after clean room operations.During initial evacuation the pump rate is controlled at
Table 5: Data acquisition and filtering settings. Usedfilters are bulleted. notch = band-rejection filter. setting Nb Nb Ti Nb sample rate (kHz) 16 16 30 30samples/point 8192 8192 8192 819225–35 Hz notch • • • • • • • • • • • • Chemical polishing removes thin layers of material froman electrode, minutely reducing its dimensions. We devel-oped an imaging system to measure electrode dimensionsand gap sizes without making contact with the electrode.The system uses a CMOS camera and bi-telecentric ma-chine lens (Thorlabs MVTC23024).The Ra EDM experiment requires a gap-measuring pre-cision of 0.1 mm or better. To test the electrodes at dif-ferent gap sizes, we adjust the gap size in situ by translat-ing the bottom electrode vertically with a high-precisionlinear drive (MDC 660002). We initially tested electrodeperformance over gap sizes ranging 0.4–2.5 mm before re-moving the linear drive and standardizing the gap size to1 . ± . µ m waist size and requires a minimum electrodegap size of 1.0 mm to avoid heating the electrode surface. A complete description of acquisition and filtering set-tings used for each tested pair of electrodes is given inTable 5. We record the power supply current, power sup-ply voltage, vacuum pressure, leakage current, and roughpump foreline pressure with a 16-bit, 250 kS/s data acqui-sition device (NI DAQ USB-6218) connected to an office8
Nb56 5 discharge rates (-HV) r a t e - b a s e li n e ( h r - ) Nb56 5 discharge rates (+HV) conditioning time (hr) -200204060 s i ze - b a s e li n e ( p A ) Nb56 5 discharge sizes (+HV) conditioning time (hr) -200204060
Nb56 5 discharge sizes (-HV) baseline = 118 hr -1 baseline = 59 hr -1 baseline = 42 pA baseline = 31 pA Figure 7: Discharge-conditioning timeline for Nb at a 1 mm gap size. model desktop PC. The analog signals are digitally filteredto remove 60 Hz outlet noise and mechanical vibrationsfrom the vacuum pumps. We initially sampled data at16 kHz but later increased the sample rate to 30 kHz af-ter upgrading the RAM and hard disk of the DAQ PC.The mean and standard deviation for each recorded datapoint is calculated from 8192 samples. We removed theoutlet noise filters after conditioning several pairs of elec-trodes because they introduced artificial shapes in the sig-nal waveform. Comparing the leakage current data of elec-trode pairs with different filtering settings, we found thatthe digital filters did not significantly affect the distribu-tion of the dataset discussed in Section 3.4. Discharges occur on a much shorter timescale thanthe integration time of the data acquisition, with a dis-charge lasting ≈ ≈
270 ms of integra-tion. Steady-state current data is primarily sensitive tochanges on the order of the integration time. On the otherhand, we have found that the sample standard deviationis effective for counting discrete discharges and estimat-ing discharge size. To illustrate, we can compare the dis-charges identified by the mean data and the standard de-viation in the third hour of the 19.9 kV conditioning shiftin Figure 7. We count a polarity-combined 54 dischargeswith the standard deviation data but only 2 discharges with the mean data over the same period. We thereforeidentify discharge rates and discharge sizes with the stan-dard deviation and characterize the slower, steady-stateleakage current with the mean.We condition the electrodes with DC voltages and alter-nate the polarity of the voltage every 60 s. The voltage isapplied to the top electrode. The periodic voltage wave-form is chosen to simulate the EDM measurement andis more challenging to stabilize than holding off a staticunipolar field. We usually observe the highest rates of dis-charges during the second and third hours of conditioning.For this reason, we condition our electrodes over five-hourshifts at a single voltage magnitude per shift.Our goal is maximize the electric field strength whileminimizing the discharge rate and discharge size. This isa complex function of the properties of the electrode pair,the time spent conditioning duration, and the chosen finaloperating voltage. In Figures 7, 8, 9, and 10, an estimateof the initial performance of each electrode pair is madeby calculating ‘baseline’ averages of the discharge rate anddischarge sizes.In the final conditioning phase we validate the electrodesat some fraction of the maximum voltage and verify thatthe discharge rate is suppressed. The validation voltage istypically 80–95% of the maximum tested voltage [23, 27].The leakage current is modeled reasonably well by a9
Nb78 5 discharge rates (-HV) r a t e - b a s e li n e ( h r - ) Nb78 5 discharge rates (+HV) conditioning time (hr)
Nb78 5 discharge sizes (-HV) conditioning time (hr) s i ze - b a s e li n e ( p A ) Nb78 5 discharge sizes (+HV) baseline = 58 hr -1 baseline = 347 hr -1 baseline = 24 pA baseline = 39 pA Figure 8: Discharge-conditioning timeline for Nb with a 1 mm gap size. Gaussian distribution. To test our choice, we fit Gaussianprofiles to the leakage current under positive and nega-tive high voltage for all the conditioning data presentedin Figures 7, 8, 9, and 10. The steady-state leakage cur-rent, discharge rates, and discharge amplitudes are usedto characterize the electrode performance. Any sample er-rors that are five standard deviations (5 σ ) greater than theGaussian average sample error are identified as discharges.We are sensitive to discharges as small as σ ≈ Nb The average discharge rate over the course of condition-ing the niobium electrode pair Nb is shown in the up-per panels of Figure 7. At each voltage, the dischargerates, expressed in discharges per hour (dph), tend to de-crease as we condition. There is a step-like increase indischarge rates when the voltage is increased. Nb wasvalidated at 20 kV / 1 mm with an average discharge rateof 98 ±
19 dph after approximately thirty hours of condi-tioning.At negative polarity, the discharge rate increases moreslowly with each voltage step. However, the overall curvedoes not flatten at a minimum count rate as it does atpositive polarity. This suggests that additional condition-ing could further suppress discharges at negative polarity.It’s also possible that the test station design facilitates ahigher discharge rate at negative polarity. We will explorethis in the near future by conducting conditioning testswhile the electrodes are removed from the test station.Nb discharge sizes are shown in the lower panels ofFigure 7. As we will see with all the discharge plots, thedischarge size behavior does not scale with the dischargerate. The largest median discharge size over the course ofconditioning is 60 pA, which is relatively small comparedto the typical discharge sizes of the other electrode pairs.In the last hour of conditioning the discharge sizes are20 pA smaller than the starting discharge sizes.10
20 40 60 80 100-1000010002000300040005000600070008000
Ti13 5 discharge rates (-HV) baseline = 134 hr -1 r a t e - b a s e li n e ( h r - ) Ti13 5 discharge rates (+HV) baseline = 143 hr -1 conditioning time (hr) -150-100-50050100150200250300 Ti13 5 discharge sizes (-HV) baseline = 101 pA conditioning time (hr) -150-100-50050100150200250300 s i ze - b a s e li n e ( p A ) Ti13 5 discharge sizes (+HV) baseline = 116 pA
Figure 9: Discharge-conditioning timeline for Ti at a 0.9 mm gap size. As mentioned in Section 2.1, the legacy copper elec-trodes were conditioned to 10 kV/mm but could onlybe operated at 6.5 kV/mm after installing them in theRa EDM apparatus. For the second generation electrodes,we made two major improvements to our technique toprevent a similar reduction in field strength. First, ourelectrodes are now preserved in Class 100 or better cleanroom environments during conditioning and transport asdescribed in Sections 2.3 and 3.1. Second, we used thenew, rigorous discharge-conditioning procedure describedSection 3.4 for Nb and the electrodes discussed in thesubsequent sections.Nb was installed in the Ra EDM apparatus using theclean room methods described in Section 1.6. They wererevalidated at 20 kV/mm after installation. This electrodepair will be used for upcoming second generation EDMmeasurements. Nb Discharge rates and sizes for the second pair of niobiumelectrodes Nb are given in Figure 8. We started condi-tioning Nb at 12 kV/ 1 mm, the same electric field asNb . The initial discharge rates are occasionally in excessof 1000 dph, or about once every three seconds for severalhours with discharge sizes of 50 pA. The high dischargerate coupled with low discharge size is an indication thatwe are operating at an optimized voltage for discharge- conditioning. During the last 10 hours of conditioning thedischarge rates decrease to less than the initial rates. Thefinal conditioning shift was performed at 17.8 kV/mm.These electrodes were packaged according to our proce-dure described in Section 1.6 and shipped to the Universityof Science and Technology of China, where they are beingused in an ytterbium EDM measurement. Ti We changed our data acquisition and digital filter set-tings for Ti and the pair that we will discuss in Sec-tion 3.8 (see Table 5). To reach electric fields higherthan 20 kV/mm, we conditioned the titanium electrodesfor ≈
110 hours, four times longer than the previous pairs.Discharge rates and sizes for the titanium electrodes areshown in Figure 9. We started conditioning the electrodesat 14.9 kV/ 0.9 mm = 16.5 kV/mm. The initial dischargesizes are approximately 100 pA, significantly higher thanNb and Nb . The discharge rates did not consistentlydecrease over the course of several shifts at 19.4 kV. Athour 12, we reduced the voltage to 0.7 kV for one shiftto verify that the discharge rates decrease before resumingtesting at higher voltages.The discharge rate increases from 290 dph to 5550 dphwhen stepping the voltage from − . − .
20 40 60 80 100 1200500100015002000250030003500
Nb23 5 discharge rates (-HV) r a t e - b a s e li n e ( h r - ) Nb23 5 discharge rates (+HV) conditioning time (hr)
Nb23 5 discharge sizes (-HV) conditioning time (hr) s i ze - b a s e li n e ( p A ) Nb23 5 discharge sizes (+HV) baseline = 190 hr -1 baseline = 131 hr -1 baseline = 30 pAbaseline = 35 pA Figure 10: Discharge-conditioning timeline for Nb at a 1 mm gap size. consistent with our expectations, given the physical pic-ture of conditioning we describe in Section 1.6. In prin-ciple, the emission sites, which may be thought of as mi-croprotrusions, are ablated after spending sufficient timeis spent discharge-conditioning the electrodes. The fac-tors influencing the required amount of time include thesmoothness of the high-gradient surfaces, the gap size, andthe applied voltage. We were unable to significantly reducethe discharge rates at 27.6 kV / 0.9 mm = 30.7 kV/mmdespite more than twenty hours of conditioning.During the final shift, we reduced the voltage to14.7 kV / 0.9 mm = 16.3 kV/mm and again observed thedischarge rates returning to the baseline. Ti can likelybe conditioned to perform stably at ≈
24 kV, or 85% ofthe maximum applied voltage with additional condition-ing. However, the concentration of magnetic impurities inour titanium electrodes (shown in Table 2) is likely toohigh to be used for an EDM measurement. Nb We first tested Nb at a 0.4 mm gap with fields as highas +52 . − . discharges rates are shown inFigure 10. The rates stay near the baseline, about 200 dphfor both polarities up to 20 kV. When we increased thevoltage from 20 to 22 kV, the discharge rates become ashigh as 3000 dph (about once every second). The dischargesizes were low, less than 500 pA, so we continued condi-tioning at this voltage. Despite conditioning the electrodesat 22 kV/mm for more than twenty hours, the dischargerate remained high. We expect that reducing the voltageby ≈ at30 kV/mm before a destructive discharge inhibited per-formance. We recovered 80% of the original electric fieldperformance by repolishing and reconditioning Nb . Chemical polishing and discharge-conditioning enabledus to reach electric fields significantly higher than10 kV/mm. We expect conditioning to further improvethe electrode surface quality, allowing the electric fieldstrength to scale faster than the discharge rates. Table 6compares the electric fields tested and discharge rates ob-12 able 6: Overall electrode conditioning comparison. E max = max field strength. E initial = initial field strength. E final = validated field strength ( E final ≤ E max ).DR = discharge rate. pair E max (kV/mm) E final E initial final DRinitial DRNb , the final polarity-averaged discharge rates werelower than the initial discharge rates. We tested the tita-nium electrodes at higher electric fields and triggered fieldemission sites, inflating the discharge rates.Of particular note is the polarity dependence of the elec-trode discharge rates. In all cases except for Nb , thenegative polarity discharge rates are significantly higherthan the discharge rates at positive polarity. Polarity-dependent discharge rates could be a feature of perma-nently grounding the bottom electrode and only chargingthe top electrode, as illustrated by Figure 6. In the future,we plan to design a more symmetric test station that willalternate the role of grounded and charged electrode tofurther investigate this effect.We plot the weighted average steady-state leakage cur-rent for each applied voltage for all the electrodes inFigure 11. Leakage offsets and drifts due to the picoam-meter, protection circuit, and power supply are suppressedby subtracting the leakage current measured at zero volt-age from the high voltage leakage current. For voltagesbelow 22 kV, the leakage current magnitude is higherat positive voltage than negative voltage. There is amodestly linear relationship with an ohmic resistance of40 kV /
10 pA ≈ Ω. We observe asymptotic leakagecurrents, correlated with high discharge rates, for Ti andNb beyond 22 kV.The steady-state leakage current must be less than100 pA to avoid systematics that could mimic an EDMsignal at our current statistical sensitivity. This crite-rion is similar to metrics used in other electrode develop-ment groups [25, 28]. As shown in Figure 11, we validatethe steady-state leakage current of Nb at 20 kV/mm to ≤
25 pA (1 σ ).
4. Conclusions and outlook
The Ra EDM experiment measures the atomic electricdipole moment of
Ra. During the measurement, theatoms precess between a pair of identical plane-parallelelectrodes that generate a uniform and stable DC electric -30 -20 -10 0 10 20 30-20-15-10-50510 l ea k a g e c u rr e n t ( p A ) steady-state leakage current Nb56Nb78Ti13Nb23-30 -20 -10 0 10 20 30 voltage (kV) -1 | l ea k a g e c u rr e n t | ( p A ) Figure 11: Weighted averages of the steady-state leakage current onlinear and log scales. Errors are on the order of 0.1 pA. field that reverses direction every measurement cycle. Weused a pair of oxygen-free copper electrodes that operatedat ± . . × − e cm in the first generation of measurements.For the second generation measurements, we will use a newpair of large-grain niobium electrodes whose systematiceffects have been evaluated to the 10 − e cm level.Two pairs of grade-2 titanium and four pairs of large-grain niobium electrodes were fabricated and polished ac-cording to surface preparation techniques that we modifiedfrom accelerator physics literature. We constructed a highvoltage test station to condition high voltage electrodes atgap sizes of 0.4–2.5 mm with a 30 kV bipolar power sup-ply at MSU. Procedures were developed to decontaminateelectrodes and preserve them in Class 100 environments.We discharge-conditioned three pairs of niobium elec-trodes and one pair of titanium electrodes, alternating thepolarity of the applied DC field every 60 s to mimic theEDM measurement. Electric fields were tested as high as+52 . − . ±
19 discharges per hour and a steady-state leakage less than 25 pA (1 σ ).The large-grain niobium electrodes (Nb ) were trans-ported to ANL and installed in the Ra EDM apparatus all13hile preserving the electrodes in Class 100 environments.After installation, the performance of Nb was revalidatedat 20 kV/mm. The improved electric field strength willcontribute an initial 3.1 enhancement factor in our EDMstatistical sensitivity.In the next phase of the Ra EDM high voltage develop-ment, we will design a more symmetric high voltage testchamber using a unipolar power supply that alternates thefield direction by switching connections between the elec-trodes. Our goal is to discharge-condition electrodes tooperate reliably at ±
50 kV/mm over a 1 mm gap.
Acknowledgments
The authors would like to thank: Matthew Poelker andthe Electron Gun group at Jefferson Lab for fabricat-ing our electrodes and advising on polishing techniques;TU Munich (TUM) for sharing their mu-metal prototypeenclosure for our electrode magnetization measurements;Zheng-Tian Lu and his EDM group at the University ofScience and Technology of China for sharing their elec-trode magnetization measurements; Laura Popielarski andDaniel Victory (FRIB) for helping us high-pressure rinseour electrodes; and Samuel Nash (NSCL) for advising uson clean room design and validation.We acknowledge support from: Michigan State Uni-versity; US DOE Office of Science, Office of Physicsunder DE-AC02-06CH11357; DOE Oak Ridge Institutefor Science and Education DE-SC0014664; DOE Na-tional Nuclear Security Administration through NSSCDE-NA0003180; and US DOE Office of Science, Office ofNuclear Physics under contract DE-SC0019455.
Appendix A. Magnetic Johnson noise
One source of magnetic field instability that could po-tentially limit the sensitivity of this experiment is theJohnson-Nyquist noise [39, 40] from conducting materi-als near the detection region. Thermal agitation (i.e. en-ergy fluctuations) of the charge carriers inside conductorsgive rise to this electronic noise with a nearly frequency-independent spectral power density of: dP n dν = 4 k B T (A.1)where k B is the Boltzmann constant and T is the temper-ature in Kelvin. A derivation of this equation as well as adiscussion of its frequency dependence is given in [41]. Bynoting that the power dissipated by a conductor is givenby P = I R , we can rewrite the noise spectrum in termsof the RMS current noise as: (cid:112) I n = (cid:114) dI n dν (∆ ν ) = (cid:114) k B T (∆ ν ) R (A.2)where R is the resistance and ∆ ν is the bandwidth. Thiscurrent noise generates a magnetic field noise spectrum that, in general, depends on the geometry of and distancefrom the conductor and the frequency.In general, it is quite onerous to follow the frequency-dependent prescription of Varpula & Poutanen [42] for ar-bitrary geometries. However, as pointed out by Lamoreux[43], calculating the noise density at zero frequency alwaysprovides a conservative upper limit for the noise density atall frequencies. In this case, called the quasistatic case, weignore the effect of eddy currents and are able to directlyapply the Biot-Savart Law to calculate the magnetic fieldfrom a steady state current distribution: d (cid:126)B ( (cid:126)r ) = µ π (cid:34) Id(cid:126)(cid:96) × ( (cid:126)r − (cid:126)u ) | (cid:126)r − (cid:126)u | (cid:35) (A.3)where (cid:126)B ( (cid:126)r ) is the magnetic field at the location (cid:126)r , I is thecurrent, and d(cid:126)(cid:96) is the line element in the direction of thecurrent at the location (cid:126)u . This integral over d(cid:126)(cid:96) is assumedto be zero for randomly fluctuating noise currents.On the other hand, the RMS magnetic field is not ex-pected to be zero and, for example, the y − component canbe written as: dB y = µ π (cid:34) { I x d(cid:96) x ( r z − u z ) − I z d(cid:96) z ( r x − u x ) } | (cid:126)r − (cid:126)u | (cid:35) = µ π (cid:34) I x ( d(cid:96) x ) ( r z − u z ) + I z ( d(cid:96) z ) ( r x − u x ) | (cid:126)r − (cid:126)u | (cid:35) (A.4)where I q is the current in the q direction and the subscripts q = x, y, z label the component of the vectors such thatˆ x × ˆ y = ˆ z .The randomly fluctuating noise currents in two differ-ent directions are assumed to be completely uncorrelated.Therefore, the cross term (i.e. I x I z ) is assumed to inte-grate to zero and only the quadratic terms (i.e. I x , I z )survive. The field noise density can be written in termsof the current noise density, which, in the q direction, isgiven by: dI n,q dν = 4 k B TR q = 4 k B Tρ (cid:18) dA q d(cid:96) q (cid:19) (A.5)where R q is the resistance in the q direction, d(cid:96) q is thelength in the q direction, and dA q is the cross sectional areanormal to the q direction. For example, for a randomlyfluctuating current in the x -direction, we have: dI n,x dν = 4 k B TR x = 4 k B Tρ (cid:18) dA x d(cid:96) x (cid:19) = 4 k B Tρ (cid:18) d(cid:96) z d(cid:96) y d(cid:96) x (cid:19) (A.6)Plugging this into Eqn. (A.4) and dropping the cross terms(as argued before), we find that the q component of thefield noise density is given by: dB n,q dν = (cid:18) µ k B T π ρ (cid:19) (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ( (cid:126)r − (cid:126)u ) × ˆ q | (cid:126)r − (cid:126)u | (cid:12)(cid:12)(cid:12)(cid:12) d u (A.7)14here d u = ( d(cid:96) x )( d(cid:96) y )( d(cid:96) z ) and the scale factor is µ k B T π ρ = (cid:20) .
989 pT √ Hz (cid:21) (cid:20) T
273 K (cid:21) (cid:20) ρ Cu (273 K) ρ ( T ) (cid:21) · cm(A.8)where ρ Cu (273 K) = 1 . × − Ω · m.For an infinite conducting plane of thickness d , Varpula& Poutanen [42] have found an analytic form for the noisevolume integral: (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ( (cid:126)r − (cid:126)u ) × ˆ y | (cid:126)r − (cid:126)u | (cid:12)(cid:12)(cid:12)(cid:12) d u = π y (cid:20) dd + y (cid:21) (A.9)where y is the distance from the surface of the conduct-ing plane. While a numerical integration of the noise vol-ume integral for the noise in the y -direction converges tothe analytic formula above, the same numerical integra-tion suggests that, contrary to the conclusion of Varpula& Poutanen, the three components of the magnetic fieldnoise due to an infinite plane are related by: (cid:32) dB n,x dν (cid:33) = (cid:32) dB n,z dν (cid:33) = 32 (cid:32) dB n,y dν (cid:33) (A.10)Finally, we model the two Ra EDM HV electrodes ascylinders with radius of 1.2 cm and height of 1.6 cm. As-suming a gap between electrodes of 1 mm, noise volumeintegrals at a location directly in between the electrodesare: (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ( (cid:126)r − (cid:126)u ) × ˆ x | (cid:126)r − (cid:126)u | (cid:12)(cid:12)(cid:12)(cid:12) d u = (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ( (cid:126)r − (cid:126)u ) × ˆ z | (cid:126)r − (cid:126)u | (cid:12)(cid:12)(cid:12)(cid:12) d u = 92 . − (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ( (cid:126)r − (cid:126)u ) × ˆ y | (cid:126)r − (cid:126)u | (cid:12)(cid:12)(cid:12)(cid:12) d u = 56 . − (A.11)For Niobium electrodes ( ρ Nb /ρ Cu = 9 .
85) that are heldat room temperature ( T = 298 K), we calculate a magneticfield noise density of: (cid:114) dB n,x dν = (cid:114) dB n,z dν = 3 .
17 pT √ Hz (cid:114) dB n,y dν = 2 .
48 pT √ Hz (A.12)
Appendix B. Code and data availability
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