rf measurements and tuning of the 1-meter-long 750 MHz radio-frequency quadrupole for artwork analysis

The 750 MHz PIXE-RFQ (radio-frequency quadrupole), developed and built at CERN, provides 2 MeV protons over a length of one meter for proton-induced X-ray emission analysis (PIXE) of cultural heritage artwork. In this paper, we report low-power rf measurements and tuning of the PIXE-RFQ, which have been completed mid-2020. Using a novel algorithm based on direct measurements of the response matrix, field and frequency could tuned at the same time in only two steps to satisfying accuracy. Additionally, we report measurements of single modules, quality factors, and coupling strength. In all cases, very good agreement between rf measurement and design values was observed.


I. INTRODUCTION
After successful design, construction, and commissioning of the HF-RFQ, a compact 750 MHz radio-frequency quadrupole (RFQ) for medical applications [1][2][3][4][5][6][7][8], CERN initiated the development of a new RFQ operating at this high frequency in mid-2017. The so-called PIXE-RFQ [3,[8][9][10][11] will accelerate protons to 2 MeV over a length of one meter only. A CAD model is shown in Fig. 1 and the RFQ key parameters are listed in Table I. The PIXE-RFQ has been developed in the context of the MACHINA collaboration (Movable Accelerator for Cultural Heritage In-situ Non-destructive Analysis) between CERN and INFN [11,12]. The aim of the project is to build the first transportable system for proton-induced * hermann.winrich.pommerenke@cern.ch X-ray emission analysis (PIXE) of cultural heritage artwork, allowing employment in museums, restoration centers, or even in the field. Low-power rf measurements and tuning of the 750 MHz four-vane RFQ have been completed in mid-2020.
In any four-vane RFQ, field tuning plays an important role to achieve the desired transverse and longitudinal field distribution. Vane modulation, certain design choices (e.g. a piecewise-constant cross section, as implemented here [10]), as well as manufacturing imperfections and misalignments lead to local variations of the capacitance and inductance distribution. Consequently, the ideal quadrupole field of the TE 210 operating mode is perturbed [13,14]. The perturbation must be corrected by means of bead-pull measurements and movable piston tuners that allow for locally modifying the inductance. Many rf cavities and RFQs in particular are tuned using transmission line models (see e.g. Refs. [15][16][17][18][19]). Contrarily, we adopted the tuning algorithm developed for the HF-RFQ by Koubek et al. [5,6] based on Ref. [14], where the effects of idividual tuner movements on the field are directly measured and recorded in a response matrix. Corrective tuner movements are than obtained by matrix inversion. We augmented this algorithm such that both frequency and field could be tuned at the same time.
This paper covers low-power rf measurements and tuning procedure of the PIXE-RFQ. In the following, after reviewing preliminary considerations to the measurements (Section II), we report the single module measurements carried out on the individual mechanical modules in Section III. Section IV discusses both algorithm and execution of the tuning procedure. Lastly, quality factor measurements are reported in Section V.

II. PRELIMINARY CONSIDERATIONS
This section briefly describes the bead-pull measurement, lists the tuning goals, and discusses corrections of frequency measurements with respect to temperature and medium.

A. Bead-pull field measurement
In bead-pull measurement, a small object (the bead) is introduced into the rf cavity, effectively removing a small volume from the resonator. Following Slater's perturbation theorem [20,21] this can be observed as a change in resonance frequency of the cavity, proportional to the squared field amplitudes at the location of the bead. In RFQs the bead is typically introduced into the four quadrants, i.e. the regions occupied by the magnetic field.
In the present case, the bead-pull measurements were performed using the phase ∆φ = arg S 11 of the reflection S 11 = Γ measured through the single input power coupler (upper part of Fig. 1). The same bead-pull bench and pulley system as for the HF-RFQ was used [5,6]. The bead was threaded on a 0.3 mm fishing wire. The size of the perturbing bead must be small enough such that the frequency shift stays within the linear regime of the ∆φ(∆f ) curve, but large enough such that an acceptable signal-to-noise ratio (SNR) is achieved. Following the measurements of the HF-RFQ, a 7 mm×4 mm aluminum bead was used. While the PIXE-RFQ features the same frequency and a comparable quality factor, its length and thus volume are approximately halved. Therefore, the same bead introduces roughly double the frequency shift when inserted into the PIXE-RFQ. With ∆f 5 kHz, or ∆φ 17°, the shift induced by the bead still remained well within the linear regime.
From the raw phase measurements of the RFQ quadrants the field components were extracted and aligned by means of a few data processing steps detailed in Ref. [6] and a smoothing Savitzky-Golay filter [22,23]. With ∆φ proportional to the squared magnetic field, the relative quadrant amplitudes a 1 , . . . , a 4 arise by taking the square root and assigning the proper sign to account for the the alternating field orientation of the TE 210 mode. The field flatness is then quantified by one quadrupole two dipole components of orthogonal polarizations [14]: (1)

B. Tuning goals
By design, the inter-vane voltage of the PIXE-RFQ is constant at V 0 = 35 kV, corresponding to a magnetic field that is constant along the RFQ and equal in all four quadrants. The goal of the tuning process can thus be defined as Q = 100 % = const. and D S = D T = 0 at all sampling points. Errors of ±2 % with respect to the average quadrupole component are acceptable in each of the three field components from a beam dynamics point of view. These tolerances have been established with respect to the 2 m long 750 MHz HF-RFQ [5].
The PIXE-RFQ represents a stand-alone machine; no rf structures requiring frequency and phase stability are installed downstream of the RFQ. Therefore, frequency accuracy does not represent a critical tuning goal. A frequency shift is in fact foreseen by design during nominal operation in order to maintain a constant cooling water temperature [10]. Deviations from the design resonant frequency as much as few MHz are acceptable for the beam dynamics [10]. Nevertheless, the frequency was tuned to match the design value: 749.480 MHz under vacuum at 22°C with an error smaller than ±60 kHz. This tolerance emerges from the water temperature range of a typical cavity cooling system: ±5 K around the nominal value. The corresponding sensitivity of the PIXE-RFQ resonant frequency amounts to −13.3 kHz/K, as determined in Ref. [10].

C. Frequency correction
Most of the measurements were conducted under air since the bead-pull setup denies evacuating the RFQ cavity. Furthermore, the ambient temperature was not controlled and deviated significantly from the RFQ design reference temperature of 22°C. The measured frequency was therefore affected by two main aspects: (i) thermal expansion of the RFQ cavity, and (ii) the relative permittivity ε r of air.
The thermal behavior of the cavity during low-level rf measurements was dominated by the ambient temperature that changed during the day. Because of thermal expansion, the resonant frequency is anti-proportional to the bulk copper temperature: ∆f /f = −α∆T , where α = 1.66 × 10 −5 /K is the secant thermal expansion coefficient of the copper cavity and ∆T the difference between measured and reference temperature. The ±0.2 K precision of the used thermometers translates to a frequency uncertainty of ±2.6 kHz. This error is by an order of magnitude smaller than the error introduced by humidity uncertainty (see the following).
The speed of light and thus the resonant frequency in a medium are by a factor of √ ε r µ r lower than the vacuum values [24]. For air we assume µ r = 1 and ε r = 1.00058986 [25] at standard temperature and pressure (STP, 0°C, 1 atm, 900 kHz). The measured frequency f meas is thus corrected as follows: A major source of uncertainty is the dependence of ε r on the ambient humidity. Analytical expressions for this relationship have been formulated based on experimental studies and can be found in Refs. [26][27][28]. The influence can be mitigated by measuring the humidity and calculating the corresponding ε r . Alternatively, the cavity can be flooded with dry nitrogen (N 2 ). The latter represents a standard procedure for rf cavity measurements and has been done during the final tuning steps of the medical HF-RFQ [5]. However, there are no strict accuracy requirements for the PIXE-RFQ frequency. Hence, we accepted the humidity uncertainty and worked with the constant STP value of ε r . A humidity uncertainty of ±30 % translates to an error in frequency of approximately ±30 kHz. Additional, but much smaller errors originate in pressure and temperature dependence of ε r [29]. Thus, we expected that the PIXE-RFQ could be tuned under air to an accuracy of approximately ±30 kHz when measuring only the cavity temperature. If frequency stability was required, this error would still lie within the ±5 K tuning range of a typical water cooling system.
In the following, the frequency is exclusively reported in terms of the corrected value (the value under vacuum at 22°C).

III. SINGLE MODULE MEASUREMENTS
The PIXE-RFQ consists of two mechanical modules, in the following denoted as module 1 and module 2, which were brazed individually and then clamped together to form the full assembly. Both modules were measured individually to ascertain the manufacturing quality, and to determine if it would be necessary to use the vacuum pumping ports as "emergency" tuning features in addition to the piston tuners. By comparing both resonant frequency and field distribution to the simulation (eigenmode solver of CST Microwave Studio ® 2018 [30]) it was found that no special measures were required, as very close agreement was observed between measurement and simulation. Figure III shows module 1 mounted to the measurement bench for the single module measurement. No auxiliaries were installed and all ports were closed by aluminum flange covers. In the absence of coupler or diagnostic pickup antennas, a small makeshift antenna crafted from simple copper wire was mounted to one of the ports. Although the antenna was strongly undercoupled, a phase shift of 4°was observed upon introducing the bead into the cavity. With an SNR of roughly 45 dB this was considered as sufficient. As no end plates were present, the upstream and downstream ends of the module were terminated by aluminum extension tubes in order to obtain well-defined boundary conditions that could be reproduced in a 3D eigenmode simulation.  The deviations in the dipole-mode frequencies are smaller than 1 MHz.
A near-perfect agreement between measured and simulated quadrupole component Q [see Eq. (1)] was observed with an error less than 1 % of the average Q amplitude (Fig. 4). The measured dipole components D S , D T , which vanish in the simulation, reach amplitudes of up to ±4 %. Larger deviations were expected for the full assembly, as the clamped connection of the two modules introduces additional alignment errors.

IV. TUNING
After the assembly of the full structure-two modules, two end plates, pumping ports, power coupler, and diagnostic antennas-the PIXE-RFQ was tuned by means of sixteen movable piston tuners. In this section, the tuner tooling is described, reliability measurements are shown, the augmented tuning algorithm is derived, and the tuning steps are reported. Field and frequency after tuning and final tuner installation are compared to the initial

A. Tuner setup and tooling
The PIXE-RFQ features sixteen piston tuners, copper slugs with a conical tip [ Fig. 5d]. Four tuners each are mounted at four positions along the RFQ. Their designations are shown in Fig. 1.
FIG. 5. PIXE-RFQ on bead-pull bench with tuners mounted in guidance tubes (a). Closeup of tuner tooling is shown in (b). CAD model of the tuner tooling is shown in (c); taken from Ref. [5] with courtesy of B.
Koubek. Tip of tuner inside RFQ cavity seen from one of the bead-pull holes is shown in (d).
The nominal tuner length is given by the nominal tuner insertion into a perfect cavity while mounted flush in the so-called flange-to-flange position. Initially the tuners were machined with an over-length of 11 mm, allowing for a mechanical tuning range of ±11 mm. A final length for each tuner was determined by iterative tuner adjustments and bead-pull measurements. The tooling was originally developed for the HF-RFQ [5,6] and is shown in Figs. 5b and (c). During the tuning, the slugs were mounted on a threaded, spring-loaded piston, allowing for adjustment of the tuner penetration into the RFQ. The piston was mounted in a guidance tube fixed to the RFQ flange, which offered accurate transverse positioning. A scale with 10 µm graduation was used to adjust the tuner position. The penetration was confirmed by means of a caliper before the tooling was removed after the final tuning step. Then, the tuners were remachined to their final lengths and installed flange-to-flange with copper gaskets.
During the tuning procedure, the field quality was assessed by measuring the quadrupole and dipole components of the TE 210 eigenmode, Q, D S , and D T [Eq. (1)] at discrete points. Each sampling point corresponds to an interval of the continuous field profile over which the data were averaged to reduce noise (Fig. 6). Brackets visualize the windows over which the measurement is averaged to obtain the discrete sampling points.

B. Reliability measurements
Before the tuning process, the measurement error was estimated by means of reliability measurements. The authors differentiate between two distinct concepts: (i) repeatability, the error observed between multiple beadpull measurements taken without any changes to the RFQ itself, in particular no tuner movements, and (ii) reproducibility, the error introduced by tuner movements and mechanical hysteresis effects. The errors arising from these two aspects pose the ultimate limit for the tuning accuracy.

Repeatability of field measurement
To assess repeatability, thirteen bead-pull measurements were carried out over several days without moving any tuners. The deviations between the repeated measurements were larger than 0.5 % for one quadrant. Such an error would limit the dipole-component tuning accuracy to more than 1 % (Fig. 7). The deviations comprise systematic errors caused by change of ambient parameters such as temperature and humidity. However, they were expected to be negligible in this case, being several orders of magnitude smaller than what could be resolved by bead-pull measurement. A significantly larger error source is random noise introduced by vibration and slippage of the wire and by the vector network analyzer (VNA) itself.
Therefore, we studied to which extent the error could be reduced by averaging over several repetitions. Based on the data from the thirteen runs, all possible combinations of three, six, or ten measurements where formed. The average of each combination was calculated at each sampling point, generating new artificial sets of measurements. The resulting error was calculated as δX = max |X − X |, where X ∈ {Q, D S , D T } and · indicates averaging over all thirteen available measurements. The results are summarized in Fig. 7, given as a percentage of the quadrupole component ( Q = 100 %). Averaging over three measurements guarantees a measurement error < 0.5 % for D S , D T and < 0.2 % for Q, which lies well within the tuning requirements. Especially for sampling points with large spread, the Q error could be reduced by nearly a factor of two. With respect to the considerable effort for carrying out these measurements, it was decided to repeat all tuning-related bead-pull measurements three times. dipole components (δD S , δD T ) when averaging over three, six, and ten measurements, compared to no averaging (one measurement). The source data was obtained from thirteen measurements carried out over several days.

Reproducibility with respect to tuner position
A second study was performed to assess errors that originate from tuner movements, i.e. the measurement reproducibility. Several tuners were moved from 0 mm (nominal position in perfect RFQ) to 3 mm, 6 mm, and back to nominal, after which the sequence was repeated. Results for tuner 4 are shown in Fig. 8 (without any averaging as discussed above). The error observed in this study is virtually the same as the one observed during the repeatability study (Fig. 7), indicating that the overall measurement error was dominated by the bead-pull setup itself, while mechanical hysteresis played a negligible role.
From Fig. 8, an estimate of the influence of a single tuner on the field distribution could be obtained. An insertion by 3 mm reduced the Q component by roughly 3 % in the vicinity of the tuner, and modifies D T by roughly 6 %. Contrarily, it has almost no influence on D S -as one would expect because of the dipole component definitions, each incorporating two opposite quadrants [Eq. (1)]. In the bottom plot at z ≈ 700 it becomes clear that the relationship is nonlinear; the influence of a tuner increases with its insertion depth.

C. Augmented tuning algorithm
The PIXE-RFQ was tuned with an improved version of the algorithm developed for the HF-RFQ by Koubek [5,6], which is based on the tuning scheme for four-vane RFQs described in Ref. [14]. For the PIXE-RFQ, the response matrix was augmented by the authors to also include the frequency, such that it could be tuned at the same time as the field.

Previous tuning algorithm used for HF-RFQ
Koubek [5] defined (in a slightly different notation) (3) as the vector of quadrupole and dipole field amplitudes Q i , D S i , D T i currently measured in the RFQ at the discrete field sampling points i = 1, . . . , n. The corresponding target values are summarized in the vector Similarly, the current tuner positions t j and target tuner positions t trg,j , j = 1, . . . , m are collected in By moving each tuner individually by some distance (all other tuners remain in nominal position), the response matrix R = ∂x cur /∂t cur ∈ R 3n×m is obtained. Each matrix entry quantifies the effect of one tuner on Q, D S , D T at one sampling point in first-order approximation. In a perfectly tuned constant-voltage RFQ the quadrupole component equals unity (Q trg,i = 1 ∀i) whereas the dipole components vanish (D S trg,i = D T trg,i = 0 ∀i) at all sampling points. Thus, the arising system of equations reads The new tuner positions required to correct the field distortion are obtained as where R † identifies the pseudo-inverse of R.

Including the frequency
In this work, we augmented the presented system of equations to include also the frequency. The measured frequency is introduced as a dimensionless quantityf = w f f such that it can be combined with the likewise dimensionless measured field amplitudes Q i , D S i , D T i by appending it to x cur . Analogously, the target frequencyf trg = w f f trg is appended to x trg . The normalizing weight w f (in units of 1/Hz) can be used to control the influence of the frequency compared to the field amplitudes. More precisely, w f determines the contribution of the frequency deviation f trg − f to the residual ||∆x − R∆t||, which is minimized when solving the overdetermined system by means of the least-squares method. Larger w f translate to higher importance. The frequency is incorporated by appending a new row to R: The algorithm presented in Ref. [5] forces the normalized quadrupole component to equal unity, Q trg,i = 1∀i, which might not lead to an optimum solution when including the frequency. Instead, we use the relaxed condition Q trg,i =Q trg ∀i, which only requires all quadrupole component samples to be equal to some valueQ trg , not necessarily unity, but close. As an unknown quantity, Q trg must be brought to the right-hand side. This is accomplished by normalizing the tuner positions by a weight w t , i.e. replacing the t i byt i = w t t i , and t trg,i byt trg,i = w t t trg,i . R is augmented with a cor-responding new column, and the system emerges, where ∆f =f trg −f = w f ∆f , and all quantities are dimensionless.

Matrix inversion by SVD
Eq. (9) must be solved to obtain the tuner corrections. One possible solution is given by where R † = (R T R) −1 R denotes the Moore-Penrose inverse [31] (pseudo-inverse) of the generally non-square R.
Koubek [5,6] pointed out that R is potentially illconditioned and proposed to use a special method based on singular value decomposition (SVD) [31] to compute the inverse.
To simplify the notation, we identify N = 3n + 1 and M = m + 1 corresponding to a setup with n longitudinal field sampling points and m tuners. The SVD of R ∈ R N ×M is given as where U ∈ R N ×N , V ∈ R M ×M are orthonormal matrices, and Σ ∈ R N ×M is a rectangular diagonal matrix whose diagonal entries σ 1 , . . . , σ M are the singular values of R in descending order. Note that M ≤ N is an essential requirement of the algorithm, which can always be achieved by increasing the number of sampling points. The pseudo-inverse can then be constructed as R † = VΣ † U T , where Σ † is obtained by inverting each diagonal entry σ i of Σ and transposing the result. For ill-conditioned, almost singular R, the reciprocals 1/σ i of the smaller singular values (larger i) approach infinity. This can lead to invalid solutions, i.e. tuner adjustments that lie outside of the physical tuner movement range. Koubek [5,6] proposed to circumvent this problem by consecutively setting the largest 1/σ i of Σ † to zero. We define as the matrix Σ † where the k largest entries have been set to zero, with k = 0, . . . , (M − 1). Note that Σ † 0 = Σ † is the initial Moore-Penrose inverse, whereas k = M leads to R † = 0 and no tuner movements at all. The matrices give M possibly useful solutions Naturally, solutions with tuner adjustments outside the physical movement range must be discarded. The remaining solutions are checked by computing the predictionx trg,k for corrected field and frequency using the original response matrix: The tuner movement for the current tuning step is chosen as that ∆t k whose correspondingx trg,k best fulfills the requirements: a quadrupole component as flat as possible, dipole components and frequency deviation as close to zero as possible.
After applying one tuner adjustment, field and frequency are measured again and a new vector x cur emerges. The process is repeated until the measured field components match the requirements to a desired accuracy [5,6,14]. Koubek showed that it is sufficient to use only the initial R and choose a different solution [Eq. (13)], i.e. a different value for k, for each tuning iteration [5,6]. This way, time-consuming remeasurement of R for each tuning iteration is avoided.

D. Tuning steps
The tuning procedure can be structured into two phases: at first, the response matrix R was measured by means of individual tuner movements. Then, two corrective tuner movements were carried out.

Measurement of response matrix
The entries of R from Eq. (9) were determined by means of spectrum and bead-pull measurements. Each matrix column was obtained by moving the corresponding tuner a certain length while all other tuners remained in nominal position. The probing tuner movements ∆t j should resemble the anticipated corrective movement as close as possible as R contains only firstorder approximations of the de facto nonlinear responses: ∂X i /∂t j ∆X i /∆t j = const., where X ∈ {Q, D S , D T }. From Fig. 8 it can be seen that a movement of 3 mm affects D T to an amount similar to the detuning in D T . Therefore, the derivatives in R were approximated as difference quotients where ∆t j = 3 mm, i.e. each tuner was retracted by 3 mm. Naturally, the tuner normalization constant was chosen as w t = (3 mm) −1 .
In Fig. 9a, a visual representation of the response matrix is shown. The last row and column are omitted here in order to show only the field sensitivities. All four tuners at the same longitudinal position (e.g. tuners 1 to 4) have approximately the same effect on Q, where tuners located at the extremities of the RFQ induce slightly stronger field tilts than those near the center. Each tuner has a strong effect on the dipole component comprising the quadrant in which the tuner is located, while the other dipole component is influenced only marginally [Eq. (1)]. Two tuners located opposite of each other (e.g. tuners 1 and 3, or 2 and 4, see Fig. 1) have opposite effects on the respective dipole component. The frequency-an integral quantity proportional to the total field energy-is affected by each tuner with roughly the same sensitivity of −50 kHz/mm [ Fig. 9b]. Individual deviations are attributed to the inhomogeneous capacitance and inductance distributions caused by vane modulation, misalignments, and design choices, which in fact motivate the RFQ tuning. A negative derivative means that the frequency is lowered when the tuner is retracted, as the total cavity inductance is increased.
In each tuning step, the frequency was enforced to equal the nominal frequency of 749.480 MHz to an accuracy ensuring that no significant frequency error would be introduced by the tuning algorithm itself. We selected w f = (10 4 Hz) −1 such that the predicted frequency obtained fromx trg,k [Eq. (14)] matched the desired frequency with an error smaller than 0.1 kHz, while still keeping the weight as small as possible.

Corrective tuner movements
With Eq. (14), M possible solutions ∆t k [Eq. (13)] yield M predictionsx trg,k of the field distribution after the tuning step, shown in Fig. 10a for the first tuning step. Corresponding amplitude errors are reported in Fig. 10b. Solutions with lower k (less singular values eliminated) generally deliver better corrections. In Refs. [5,6], Koubek reported that-with the original version of the algorithm-many of the possible solutions with small k yielded tuner adjustments that were outside the physical tuner movement range. In the present case this could not be confirmed; all k = 1, . . . , M − 1 solutions provide physically possible tuner movements. This is attributed to the fact that the augmented version of the algorithm [Eq. (9)] includes the frequency, which would strongly disagree with the desired value for unphysically large tuner movements and thus acts in a dampening manner. Furthermore, the response matrix of the HF-RFQ was more affected by measurement noise, as no averaging was performed [5,6]. Nevertheless, it is still advantageous to make use of the truncated SVD in the improved algorithm and select a solution "by eye," as solution k does not strictly provide a better correction than solution k + 1.
The PIXE-RFQ was tuned in only two steps after the measurement of R. For the first step, k = 1 was chosen as the option that provides very small errors for all three components while equally correcting D S , D T . The corresponding field prediction is shown in Fig. 11 (red dashed line). After applying the tuner adjustments, the frequency error improved from 624 kHz to 46 kHz (Fig. 12). A significant deviation between field prediction and measurement (red solid line) was observed, in particular for D S , where an error of nearly ±3 % remained. The reason is found in the nonlinear relation between field and tuner position, that is represented in the response matrix only by linear approximation.
A second tuning step with the same response matrix R was carried out, however, this time we chose k = 3. Predicted and measured fields after the second tuning step are reported in Fig. 11 (yellow lines). D S could be suppressed to an error of 1.3 %, and the frequency deviation was improved to 1.5 kHz. This error is of the same order of magnitude as the error arising from the thermometer precision (±2.6 kHz). The second tuning step already saw a slight worsening in D T . This indicates that an accuracy limit given either by the overall noise level or the linear approximation was reached. The errors in frequency and field after the second tuning step fulfilled the requirements listed in Section II B: δQ = ±0.6 %, δD S = ±1.3 %, and δD T = ±0.6 %. As a third iteration delivered no satisfying predictions of improvement, it was decided to stop the tuning procedure after two steps.

Tuner recutting
Final tuner positions are reported in Fig. 13. Note that on average the tuners were retracted, as the initial measured frequency was too high and the total cavity inductance had to be increased. All adjustments are considerably smaller than the maximum foreseen movement range of ±11 mm. The final tuner lengths were determined both from the scale on the tuner tooling [ Fig. 5b], which was used during the tuning process itself, and from an additional measurement using a caliper. The errors between the two measurements read up to 60 µm, likely originating in small inclinations of the tuners within the guidance tubes. The average of both values was used for remachining.
Following remachining, a bead-pull measurement was carried out with the tuners fixed to the corresponding flanges without seals as a mean of validation, after which they were installed with vacuum gaskets and a final measurement was performed. The results are reported in Fig. 14 in comparison to the initial field and the field after the second tuning step. Corresponding errors are summarized in Table II. A small deviation, larger than measurement noise, was observed between the field after the second tuning step and the field after they were remachined and fixed. This deviation likely originates in the finite accuracy of length measurement and material cutting.
The final frequency measured under vacuum was 23.5 kHz above the target value. This error was achieved measuring under air, solely correcting for temperature and constant air permittivity (STP value). The deviation can be explained by the fact that response matrix and initial frequency were measured on a rainy day (≈ 100 % rel. humidity), while the tuning steps were performed under sunny weather (≈ 50 % rel. humidity). It is likely that the remaining error could have been significantly reduced by correcting for the measured humidity or using a dry flooding gas. Nevertheless, the final frequency error is by more than a factor of two smaller than the tuning range given by a typical water cooling system (±60 kHz). Step

V. QUALITY FACTOR MEASUREMENT
After tuning, the quality factors of the PIXE-RFQ were measured under vacuum to validate the design and provide information on the input power reflection originating in potential over-or under-coupling.
The raw measurement data were the complex reflection coefficient S 11 = Γ measured in a bandwidth of 1 MHz symmetrically around the TE 210 eigenfrequency. The reflection coefficient describes a circle in the complex plane in very good approximation, from which unloaded (Q 0 ), external (Q ex ), and loaded quality factor (Q ) can be obtained, where Additionally, the coupling coefficient emerges. The parameters were extracted by fitting an ideal circle to Γ following the method described in Refs. [32][33][34][35]. Contrarily to e.g. the three-point method [36,37], the circle-fitting method performs implicit averaging by solving a heavily over-determined system of equations, and is thus significantly less susceptible to measurement noise.
The technique is based on the Möbius transformation withω = 2(ω/ω 0 −1), which describes a circle in the complex plane characterized by the parameters b 1 , b 2 , b 3 . For ω → ±∞, Γ → b 1 /b 3 approaches the detuned reflection, whereasω = 0 delivers Γ(∆ω = 0) = b 2 as the reflection coefficient of the loaded resonator. Their distance equals the circle diameter D = |b 1 /b 3 − b 2 |, and the coupling coefficient can be determined as β c = (2/D − 1) −1 . Furthermore, Q = {b 3 }, from which the two remaining quality factors can be calculated using Eqs. (15) and (16) [33]. The VNA measures Γ(ω i ) at N discrete sampling points ω i . Thus, Eq. (17) can be written in matrix form: a heavily overdetermined system of equations with each row representing one measurement point. The curve described by Γ is not a perfect circle-deviations become stronger with increasing distance from ω 0 (or |ω| 0). Furthermore, equidistant frequency sampling implies lower density of measurement points around the critical ω 0 . Therefore, a weighting matrix W is introduced, which reduces the significance of measurement points further away from ω 0 : In Table III, measured quality factors and coupling coefficient are compared to the design values [10]. Very close agreement was observed: the measured Q 0 agrees with the design value with an error smaller than 0.5 %. This indicates not only the reliability of the power loss calculation carried out at the design stage [10], but also the manufacturing quality of the PIXE-RFQ. A larger deviation of 6 % was observed in Q ex . This translates to a comparable error in β c and an error of 3 % in Q . The errors could be introduced by imperfect machining, brazing, and alignment of the physical coupler. A 6 % error in β c could correspond to a mechanical error of 300 µm in the coupling loop size. As the coupler represents a complex geometry composed of multipole components assembled by means of bolts, we cannot exclude such an error. (In contrast, different simulation models generally yield errors in Q ex which are by an order of magnitude smaller.) All measurements reported in this section were conducted under vacuum (approximately 10 −6 mbar). An interesting observation was made in that the measured coupling increased by approximately 2 % (from 1.16 to 1.18) when the RFQ cavity was evacuated. A possible explanation is found in the mechanical structure of the power coupler: The inner conductor is only supported by the coupling loop and a polyether ether ketone (PEEK) rf window [10]. Subjected to the pressure difference between vacuum and exterior atmosphere, the PEEK window could deform and push inwards the inner conductor. This would slightly increase the effective area of the coupling loop, resulting in a larger coupling coefficient.
The discrepancy highlights that it is important to design the coupler with an over-coupling margin of few ten percent. Although the PIXE-RFQ coupler features a rotatable flange that allows for fine-tuning the coupling coefficient by means of measurements, we decided not to take advantage and rather accepted an over-coupling of 18 % (β c = 1.18). This translates to an input reflection of −22 dB. If the RFQ cavity consumes 65.0 kW of peak power in terms of surface losses [10], an input power of 65.5 kW is required, and 0.5 kW, i.e. roughly 0.7 % of the input power, are reflected back towards the generator. These values are well acceptable for operation.

VI. SUMMARY
In this paper, the low-power rf measurements and tuning procedure carried out at CERN on the one-meter-long 750 MHz PIXE-RFQ are reported.
Before full assembly, bead-pull measurements were performed on mechanical modules to confirm that no severe errors occured during machining and brazing. The field showed near-perfect agreement with the simulation, whereas the frequency error was equal or smaller than 600 kHz in both cases. The small observed deviations are well within the capabilities of the tuning system.
The main part of this paper is dedicated to the RFQ tuning procedure. Longitudinal and transverse field tilts were compensated by means of sixteen movable piston tuners. The aim of the tuning was to achieve the most flat possible quadrupole component and vanishing dipole components. Furthermore, the operating mode frequency was tuned to the design frequency of 749.480 MHz under vacuum at 22°C.
Reliability measurements were carried out before the actual tuning procedure in order to assess the accuracy limits given by bead-pull measurement and tuner tooling. The bead-pull measurement error could be improved to 0.5 % by performing each measurement thrice. No significant mechanical hysteresis effects were observed.
The tuning algorithm used for HF-RFQ [5] was augmented for the PIXE-RFQ such that frequency and field could be tuned at the same time. The results suggest that including frequency and corresponding weights also leads to a more stable convergence of the tuning iterations. Af-ter measuring the response matrix, the PIXE-RFQ was tuned in only two steps. Subsequently, the tuners were recut to their final lengths. The measurement under vacuum revealed a remaining frequency error of 23.5 kHz, which can be explained with the measurement uncertainty with respect to air humidity. The error lies well within the ±60 kHz tuning range of a typical water cooling system (±5 K) and is in any case acceptable since no rf accelerating structures are present downstream of the RFQ.
The three quality factors and the coupling coefficient were measured using a circle-fitting method. Excellent agreement with an error smaller than 0.5 % was observed in Q 0 . Q ex was measured to be 6 % higher than the design value. The discrepancy could be attributed to mechanical manufacturing error in the coupler of roughly 300 µm. The larger Q ex corresponds to a measured coupling coefficient of 1.18 compared to a design value of 1.25. The measured over-coupling of 18 % requires an extra 0.5 kW (0.7 %) of rf peak input power. It was decided to not make use of the possibility of rotating the coupler to closer approach critical coupling.
With the low-power rf measurements completed in mid-2020, the PIXE-RFQ has been transported to INFN, Florence, Italy for high-power rf conditioning and beam measurements. The world's smallest RFQ is expected to commence operation towards the end of 2020.