2.1　Theory of the bunch purity deterioration

As mentioned before, parasitic electrons in the storage ring could be inherited from pre-accelerators. The sources could be the electron gun, the dark current from the buncher in the linac, or the Touschek scattering in the booster. Parasitic electrons could also be produced owing to the Touschek scattering in the main bunches of the storage ring [9-11]. The contribution ratios of these are different for different machines. At the BEPCII, a full energy linac is injected directly into the storage ring, and we hope to determine the relative ratios of parasitic electron contributions caused by the linac beam and the Touschek scattering in the storage ring.

Touschek scattering

The trajectories of electrons in the longitudinal phase space are described by the synchrotron oscillation equation [17]. When the momentum deviation is small, an electron undergoes a damping oscillation around the synchronous phase. An electron that gains a positive momentum larger than the height of the RF bucket exits the main bunch. The trajectories of the electrons in the longitudinal phase space are shown in Fig. 2. The electrons with positive momenta larger than the RF bucket height move toward the backward buckets +1, +2, +3 (+ represents backward buckets). Without the synchrotron radiation, the electrons thrown out from the 0^th bucket are never captured by the following buckets. However, radiation damping produces a momentum window Δp_i on the i-th bucket. Electrons with momenta corresponding to this window can be captured by the +1, +2, +3 buckets. The increasing rate of parasitic bunches owing to the Touschek scattering and recaptures in subsequent buckets is given by [10, 11]

Fig. 2Full-screen View

(Color online) Trajectories of electrons that are captured by the following buckets in the longitudinal phase space.

$\frac{{\text{dN}}_{\text{i}}}{\text{dt}}\text{=}\frac{{\text{N}}_{\text{0}}}{\text{2}}\text{(}\frac{\text{1}}{{\text{τ}}_{\text{t}}\left({\text{δ}}_{\text{i}}^{\text{min}}\right)}-\frac{\text{1}}{{\text{τ}}_{\text{t}}\left({\text{δ}}_{\text{i}}^{\text{max}}\right)})={\text{C}}_{\text{i}}^{\text{T}}{\text{N}}_{\text{0}}\text{,}$ (3)

where ${\text{δ}}_{\text{i}}^{\text{min}}$ and ${\text{δ}}_{\text{i}}^{\text{max}}$ are the minimal and maximal momenta, respectively, of the i-th bucket. The coefficient 1/2 is owing to the fact that one of the two electrons that take part in the collision loses momentum and is never captured by the backward buckets. The bucket height of the BEPCII is 0.97% for a cavity voltage of 2.60 MV; the typical parameters of the BEPCII are listed in Table 1 . By solving the synchrotron oscillation equation numerically, the RF acceptances with radiation damping are shown in Fig. 2, and the momentum windows (p_i =${\text{δ}}_{\text{i}}^{\text{min}}$ and △p_i = ${\text{δ}}_{\text{i}}^{\text{max}}-{\text{δ}}_{\text{i}}^{\text{min}}$) corresponding to different parasitic bunches are given in Table 2 . Electrons with momenta gains larger than 1.8% are expected to be lost in the vacuum chamber because they are outside the lattice transverse momentum acceptance. The Touschek lifetime τ_t(δ) was calculated using the Elegant code [18]. The simulated growth rates,${\text{C}}_{\text{i}}^{\text{T}}$, calculated using Eq. (3), are listed in Table 2 .

Table 2Full-screen View

Calculated relative momentum window, the Touschek lifetime, and the impurity growth rate, for the parasitic bunches obtained using the Elegant code with the nominal parameters from Table 1 .

i	pi (×10-2)	△pi (×10-5)	${\tau }_{\text{t}}({\delta }_i^{\min })$h	${\tau }_{\text{t}}({\delta }_i^{\max })$h	CTi (×10-5 h-1)
1	1.19508	4.23	143.367	144.460	2.6384
2	1.38351	3.68	197.103	198.257	1.4776
3	1.54988	3.30	252.858	254.045	0.9239
4	1.70056	3.02	310.390	311.605	0.6281
5	1.83937	2.79	369.541	370.796	0.4581
6	1.96876	2.62	430.202	431.477	0.3433

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The electrons in the parasitic bunches can be lost owing to the beam-gas or the Touschek scattering. Therefore, the growth rate of the electrons in a parasitic bunch is given by

$\frac{{\text{dN}}_{\text{i}}}{\text{dt}}\text{=}\left({\text{C}}_{\text{i}}^{\text{T}}{\text{+C}}_{\text{i}}^{\text{inj}}\right){\text{N}}_{\text{0}}\text{-}\frac{{\text{N}}_{\text{i}}}{{\text{τ}}_{\text{t}}{\text{(N}}_{\text{i}}\text{)}}\text{-}\frac{{\text{N}}_{\text{i}}}{{\text{τ}}_{\text{g}}}\text{,}$ (4)

where τ_g and τ_t are the lifetimes associated with the beam-gas scattering and the Touschek scattering, respectively. The time τ_g is independent of the number of electrons in a parasitic bunch, while τ_t is not. The Touschek lifetime of a parasitic bunch is very long owing to the low bunch current compared with that owing to the beam-gas scattering, and can therefore be neglected. The impurity growth rate dPu_i / dt = d(N_i/N₀) / dt can be described as

$\frac{{\text{dPu}}_{\text{i}}\text{(t)}}{\text{dt}}=\left({\text{C}}_{\text{i}}^{\text{T}}+{\text{C}}_{\text{i}}^{\text{inj}}\right)-\frac{{\text{Pu}}_{\text{i}}\text{(t)}}{{\text{τ}}_{\text{g}}}$ (5)

with the solution of

${\text{Pu}}_{\text{i}}\text{(t)}=\left({\text{C}}_{\text{i}}^{\text{T}}+{\text{C}}_{\text{i}}^{\text{inj}}\right){\text{τ}}_{\text{g}}{\text{(1-e}}^{\text{-}\frac{{\text{t-t}}_{\text{0}}}{{\text{τ}}_{\text{g}}}}\text{)}\text{.}$ (6)

2.2　Experimental setup

The purity measurement system is shown schematically in Fig. 3a. The visible light that came from a bending magnet was sent to a photon detector (Microchannel Plate-Photomultiplier Tube MCP-PMT, China Electronics Technology Group Corporation or PMA Hybrid 400, Picoquant [19]) after passing through a full reflection mirror, a focus lens, a variable attenuator, a grating spectrometer, and two adjustable apertures. It shared a photon beam transport system with a beam-profile measurement system [20]. Two slits and neutral density (ND) filters located in front of the photon detector were used for reducing the photon flux down to the level of one photon detection per approximately dozens of revolutions of the beam. A grating spectrometer, for rejecting the stray light, was set on the optical line to the detector. As presented in Fig. 3b, the output signal of the detector was sent to a multichannel picosecond event timer (HydraHarp 400, Picoquant or Constant Fraction Discriminator CFD, ORTEC935, time to amplitude converter TAC, ORTEC 567, and multi-channel analyzer MCA, ORTECTRUMP-PCI-2k) modules [21], where the time difference between the trigger signal and the photon detector signal was measured using a time-to-digital converter (TDC). The detector recorded the distribution of photons that corresponded to the distribution of electrons over time in the revolution period. The reference signal was provided by the event generator / event receiver from the timing system, which provided a revolution period signal of 1.2433 MHz in the SR mode. A digital delay generator (DG645, SRS) was used for adjusting the delay between the detection of photons and the reference signal.

Fig. 3Full-screen View

(Color online) a Schematic of the purity measurement optical system. b Sketch of the TCSPC experimental setup. c Purity measurement interface of HydraHarp 400.

Originally, the detector electronics were composed of a nuclear instrumentation module (NIM), and the reason for the recent upgrade to the HydraHarp module was a short dead time of 80 ns, allowing a maximal count rate of 12.5 × 10⁶ s^-1. The count rates were 10 counts/s for background noise (without synchrotron light) and 10⁶ counts/s for purity measurements. The maximal number of time bins was 2¹⁶ = 65536, which allowed the minimal time resolution of 16 ps for a revolution time of 800 ns. The count depth per time bin was 32 bits, corresponding to 2³² = 4,294,967,296 increments, which was very useful for high-purity measurements, compared with the PicoHarp300 module with the storage capability of 16 bits [19].

2.3　Purity measurements

Figure 3c shows the purity measurement results for the top-up operation at the BEPCII. The filling pattern was designed for pump-probe experiments [22]. The reading of the bunch current monitor (BCM) is shown in the inset of Fig. 3c; this provided the filling pattern information for operators. The total beam current of 250 mA was distributed equally across a single isolated bunch that was located at the center of a 200-ns-wide gap and another 99 bunches in the bunch train, with the bunch spacing of 6 ns.

The single-photon responses of the two detectors are shown in Fig. 4a. The rise time (10% to 90%) was 380 ps for the hybrid detector, and the response time of the MCP-PMT was at the same level but had a different tail. The single-bunch purity measurement results are shown in Fig. 5, both with a tail of several tens of nanoseconds. The bunch root mean square (RMS) length was approximately 50 ps, as measured by a streak camera. The full width at half maximum (FWHM) values of the bunch responses were 427 ps and 128 ps, respectively. The difference was mainly caused by the NIM’s response being much slower than that of HydraHarp. The width of a single bunch is determined by the instrument response function (IRF) of the system and the time structure of the bunch photons. The parasitic bunch signals are very easily smeared by spurious signals induced by the main bunch, especially for detectors with fast responses and high photon-electron conversion gains. This effect is alleviated for slow detectors, such as avalanche photo diodes (APDs). There are several different names and possible reasons for these spurious signals, such as ghost reflection [2] and after-pulsing [14]. The details of the hybrid detector’s measurement results near the main signal are shown in Fig. 4b. There are four spurious peaks, located at 0.93 ns, 1.34 ns, 2.94 ns, and 6.81 ns after the main bunch, compared with the MCP-PMT detector that revealed peaks at 4 ns and 26 ns, respectively (data not shown). Several reasons explain why these spurious peaks do not correspond to real electrons. First, they appear outside stable RF phase regions. Second, their characteristics do not depend on injections, and the ratio of spurious to the main peaks sometimes shifts slightly depending on the photon flux but not by the time after an injection, as shown in Fig. 6a. Third, they are not affected by the RFKO system. Finally, different labs have reported similar results, although different detectors yielded spurious peaks at different positions [2, 11]. Yet, baseline counts increase, hindering the accurate evaluation of parasitic branches. Fortunately, none of the spurious peaks were at the locations of the parasitic bunches for the PMA hybrid detector, as shown in Fig. 4b.

Fig. 4Full-screen View

(Color online) a Single-photon responses of the two detectors. b Spurious peaks of the PMA hybrid detector.

Fig. 5Full-screen View

(Color online) Single-bunch purity measurement results for (a) the MCP-PMT + NIM modules, and (b) Hybrid + HydraHarp.

Fig. 6Full-screen View

(Color online) a The bunch purity measurement results for a week-long operation, showing the details of a 14-ns-wide span around the main bunch, located at the center of a 200-ns-wide gap. b Evolution of the different parasitic bunches’ purities (ratios of the numbers of electrons in different parasitic branches to that in the main bunch) during the week-long operation.

Figure 6a shows the bunch purity measurement results at different times during a week-long top-up operation. The main bunch (No. 0, the first bunch in Fig. 3c) located at 635.2 ns is in the center of a 200-ns-wide gap. Besides the parasitic bunches located at 635.2 + 2 × i ns, there are three spurious peaks located at 636 ns, 638.5 ns, and 642 ns. These spurious peaks could not be cleaned by the bunch purification system. Therefore, they do not correspond to real electrons. One possible explanation for the appearance of these peaks is the inner reflection of the photon detector. We excluded the possibility of reflection photons created in the photon transport process. Further studies need to be done for confirming the origin of these spurious peaks. As shown in Fig. 6a, the counts associated with parasitic bunches increase gradually, but those associated with the spurious peaks do not. The ratios of the numbers of electrons in different parasitic branches to that in the main bunch are shown in Fig. 6b. The experimental results confirmed the Obina and Keil theory [10, 11]. Impurity growth was never observed in forward (No. -1) bunches. Impurity growth rates were different for parasitic bunches, and were mainly determined by the different momentum gain probabilities through the Touschek scattering. The whole-bunch purity deterioration could be seen clearly throughout the week-long measurement. Equation (6) suggests that purity approaches equilibrium with a rise time of τ_g. The typical gas lifetime τ_g for the BEPCII was 6.77 h, while the elastic and inelastic scattering gas lifetimes were 14.32 h and 12.84 h, respectively, for the pressure of 1.3 × 10^-8 Pa.

To elucidate the contributions of the injection and the Touschek scattering to the ratios, the average purities of the 99 bunches in the bunch train in the top-up and decay modes are shown in Fig. 7a. (Fig. 6b depicts the purity of a single isolated bunch.) The average purities in the 4-h-long decay mode were compared with that of the normal top-up operation under the same filling pattern.

Fig. 7Full-screen View

(Color online) a Purity results for the top-up and decay modes, with the fill pattern being the same as in Fig. 1c. b Fit results for Pu_2ns^Top-up, using Eq. (6).

The main bunch spacing was 6 ns, and there were two possible parasitic bunches between the two main bunches, each of which could be the contribution of two forward main bunches. The bunch purity (+2 ns) corresponded to C₁^T + C₄^T, while the bunch purity (+4 ns) corresponded to C₂^T + C₅^T. According to Eq. (6), we obtained Eqs. (7) and (10):

As shown in Fig. 7a, the difference between Pu_4ns^Top-up and Pu_4ns^Decay was very small, indicating a small C₂^inj compared with C₂^T. C₁^inj could be determined using Eq. (8) over Eq. (7). Thus, C₁^inj was (9.123 ± 0.204) × 10^-6 h^-1. The result of Pu_2ns^Decay over Pu_4ns^Decay was 1.32 ± 0.075, very close to the momentum window calculation result of 1.69, as shown in Table 2 . The fit result for Pu_2ns^Top-up is shown in Fig. 7b, indicating that C₁^inj was (1.38 ± 0.0504) × 10^-5 h^-1. Then, k = 2.22 × 10^-6 was determined from Eq. (2) with f = 0.2 % and τ_g = 6.77 h. The horizontal error bars in Fig. 7b represent the time spent on the purity measurement. The vertical error bars were calculated as follows: To calculate the average value of the background counts, including all the time bins (outside of the RF stable region), there were no electrons, and to calculate the average value of the peak counts for the main bunches, the uncertainty was determined by the ratio of the two counts. The final calculation results yielded the purity uncertainty in the 5 × 10^-6 – 7 × 10^-6 range, depending on the discrimination threshold setting of the electronic and filling patterns. The purity uncertainty evaluated by this method was almost the same as that shown in Fig. 6, and was larger than the special mode in Sect. 3, where there were only several bunches in the ring. There was no significant difference between Pu_4ns^Top-up and Pu_4ns^Decay after 2 h, and then the range C₂^inj ≤ 1.25× 10^-6 h^-1 – 1.75 × 10^-6 h^-1 was determined from the uncertainty.

3.1　The principle and experimental setup

The bunch-cleaning technique developed at SPring-8 [15] is shown schematically in Fig. 8. A sinusoidal signal comprising the N^th harmonics of the revolution frequency and the vertical betatron frequency as f = N×f_rev+ f_βy is mixed with a pseudo-square wave that defines the bunch fill pattern. The main bunches, which are represented by blue dots, are located at the middle of the rise or fall edge (also the zero-crossing point) of the pseudo-square signal, and all the other bunches (red dots) at both positive and negative polarities of the square signal are cleaned. The mixed signal is amplified and sent to a kicker to stimulate a beam. If the amplitude of the excited oscillations is sufficiently large, parasitic bunches are lost in a vacuum chamber. The main apparatus of the bunch-cleaning system, including the timing system, the kickers, and the amplifiers, is shared with a transverse bunch-by-bunch feedback system.

Fig. 8Full-screen View

(Color online) Schematic of the bunch-cleaning system.

The bunch-cleaning system in the BEPCII consists of an arbitrary waveform generator (UHFAWG, 1.8 GSa/s, 14 bit, 600 MHz, Zurich Instruments; the minimal time step of programming is 555 ps), a function/arbitrary waveform generator (33250A, Agilent), a digital delay generator (DG645), a solid-state amplifier (Amplifier Research 10 kHz-250 MHz 75 W), and a 60 cm strip-line transverse feedback kicker.

3.2　Bunch-cleaning experiments

Figures 9a and 9b show the typical signals generated by the bunch-cleaning system. Curve 1 in red shows the output of the UHFAWG arbitrary waveform generator, while Curve 2 in blue is a sinusoidal wave with a frequency of f_rev+ f_βy, produced by the Agilent arbitrary waveform generator. Curve 3 in green shows the waveform after mixing (the colors of the curves in Fig. 9 are the same as in Fig. 8). Curve 4 is not a real signal but the result of a mathematical operation (multiplication) on Curves 1 and 2. The maximal output Vpp of the UHFAWG was 750 mV, so there is a factor of 3 in Curve 4. To show this more clearly, the curves were translated along the vertical axis. Figure 9b shows the result of the scope in the single-shot mode, and Figs. 9c and 9d show the infinite persistence modes. The time scales were 100 ns and 4 ns per division, respectively. Fig. 9c shows the BPM (upper trace) and the strip-line kicker (lower trace) signals of the cleaning system. The BPM signal was used for monitoring the effect of cleaning, and the signal originating from the downstream port of the strip-line kicker was used for observing the temporal relations between the excitation signal and the bunches. First, we set the frequency of the sinusoidal wave far away from the resonance frequency, so the bunches located at the flat part of the kicker signal would not be cleaned, as shown in Fig. 9c. Several bunches were located at the flat of the square wave, and we then changed the delay to ensure that the main bunch was located at the lowest point, as indicated by the arrow in Fig. 9d. After finding the proper timing set, we changed the frequency of the sinusoidal wave to the right resonance frequency, which was cleaned in less than 100 ms. The lost electrons in the main bunch were ignored, and the perturbation was very small, demonstrating the possibility of using the system transparently during user operation.

Fig. 9Full-screen View

(Color online) a-b Typical signals generated by the bunch-cleaning system. c BPM (upper trace) and the strip-line kicker downstream port (lower trace) signals of the cleaning system. d Detailed view of the rise edge of the excitation signal. The arrow shows the safe point where the main bunch is located. The results in Fig. 9a and Fig. 9b were obtained in the single-shot mode, while the results in Fig. 9c and Fig. 9d were obtained in the infinite persistence mode.

After confirming the functionality of the bunch-cleaning system, a special fill pattern was designed for testing. Five bunches with a spacing of 100 ns shared a total current of 15 mA equally, and the bunches were not refilled during the entire test. Fig. 10a shows the purity results before and after the cleaning system came into effect. Purity before cleaning was measured 10 min after the injection, and the purity measurement process took several minutes. Fig. 10b shows the details of the remaining bunch. Only the third main bunch survived. Table 3 shows the purity results after and before the cleaning system was used. All of the parasitic bunch purities after cleaning reached 10^-7. The parasitic bunches after No.+2 were submerged in noise. The purity of the theoretical value given by C_i^T in Table 2 is shown in the last column of Table 3 . The purity results show that the cleaning system works well and the sensitivity of the purity measurement system is 10^-7, limited by the photon detector’s characteristics. Detection of visible light suffered from a relatively high background caused by the detector itself or by stray light. Both the spurious peaks and dark counts increased the uncertainty of the purity. Using the method mentioned in Sect. 2.3, the uncertainty was in the 8 × 10^-8 – 9 × 10^-8 range, much better than that for the normal operation.

Fig. 10Full-screen View

(Color online) a Purity results, after and before the cleaning system operation. The purity results before cleaning were obtained in 10 minutes after the injection. b Details around the remaining bunch.

The absolute uncertainty (or sensitivity) of the system was difficult to evaluate, but the factors affecting it could be identified, as follows. The impulse response of the photon detector, with spurious peak counts much higher than normal noise counts, is expected to strongly deteriorate the uncertainty. The effective dynamic range of the system, which is determined by the minimum across all components, is smaller for the photonic detector than for the electronic one for the purity system, and count rate above 10⁷ is expected to trigger the PMA Hybrid 400 detector self-protection. The overall current for the normal operation was 17 (20 for the bunch amount) times higher than that in Fig. 10, and the uncertainty was 70 times worse. The time spent on purity measurements is typically 4–8 min, and purity increases to the 9.8 × 10^-7 – 19.6 × 10^-7 range, according to the growth rate C₁^T = 2.64 × 10^-7 in Table 2 . The discrimination threshold setting of the electronic system also plays an important role. Quantification using the ratio of average background counts to the main bunch counts only provides a reference, but it is a good approximation.