Estimation of synchrotron-radiation background based on a real beam orbit

Some high-energy experiments have suffered from synchrotron-radiation background. As a measure, we have developed a new calculation method of synchrotron radiation based on a real beam orbit, aiming at quantitative estimations and the construction of a possible alarm system for the background. An explanation and a demonstration of our method are given.


INTRODUCTION
At the initial stage of the KEK B-factory (KEKB) experiment [1], synchrotron radiation (SR) from the circulating 8-GeV electron beam caused serious damage to the Belle silicon vertex detector (SVD) [2], which was located close to the interaction point (IP). Some of the preamplifiers mounted on the inner-most SVD layer died in about ten days just after start of the gain drop. Other high-energy experiments have also suffered from SR background.
We have developed a new method to calculate SR wattages based on a real beam orbit, aiming at quantitative estimations and the construction of a possible alarm system for SR background. Here, the real orbit is obtained by fitting measurements of beam-position monitors (BPMs). In the following sections, our method is explained together with some illustrations using the 8-GeV electron beams of KEKB High Energy Ring (HER). Finally, we present a simulation of the orbit and the gain drop at the time of the SVD gain-drop accident as a bench mark.

II. ORBIT CALCULATION
Beam orbits are calculated by a linear approximation using transfer matrices, which are shown in Appendix A. For numerical calculations, we coded a dedicated computer program in Fortran, where the correctness of the orbit calculation was checked by comparing with a calculation using SAD [3]. Furthermore, we confirmed that there was no significant effect from round-off errors by comparing the results of double-and quadruple-precision calculations. * E-mail address: tetsuo.abe@kek.jp

A. Fitting procedure
Orbit fitting is performed based on the optics, field strengths of the magnets, and BPM measurements. We require minimization of χ 2 , defined as where X ) the horizontal (vertical) position of the calculated orbit at BPM:j, and σj is the BPM:j resolution. There are four floating parameters, which are the orbit positions at the entrance and exit of the relevant accelerator section: X (in) , Y (in) , X (out) , Y (out) . Minimization of χ 2 is performed using the computer program MINUIT [4,5]. An example of the orbit-fitting results for KEKB HER is shown in Fig. 1. Here, we define the vertical and horizontal χ 2 to estimate the goodness of the fit for each direction separately: which lead to the following relations: where N df (X) (N df (Y ) ) indicates the number of degrees of freedom related to the horizontal (vertical) direction, and N df for both: N df = N df (X) + N df (Y ) . In the example shown in Fig. 1, the goodness of the fit is χ 2 X /N df (X) = 10.5 and χ 2 Y /N df (Y ) = 40.3 for a conservative BPM resolution of 100 µm. As is often the case, χ 2 is too large, and also the orbit at the IP is unrealistically deflected in the negative direction by about 3 mm. This is due to the fact that the absolute positions of magnets and BPMs are not known sufficiently well, leading to a need for the offset corrections.

B. Offset corrections
We modify the χ 2 formula (1) as follows: where ∆X ) the horizontal (vertical) offset of the real origin of the BPM:j measurement with respect to the real field center of QUAD:j. The index i runs for different orbits in the case of a global fitting, which is explained in the next paragraph.
In order to obtain the offset sizes (∆X ), we perform a global fitting for different orbits within a time period in which magnets are not expected to have moved. In case of KEKB, we make various measurements in machine studies using six different orbits in each of the horizontal and vertical directions, which are produced by making a small kick at one of the six steering magnets. We fit those six orbits simultaneously with not only the usual floating parameters (X (i = 1, 2, 3, 4, 5, 6)), but also the offsets in formula (6). We furthermore introduce an additional floating parameter: the correction factor of the field strength of the bending magnet (BEND), a, which is defined as θcorr = (1 + a) × θorg, where θorg and θcorr are the kick angles before and after the correction, respectively. The BPM and QUAD offsets as well as the correction factors of the BENDs are common among the six orbits. It is clear that the simple fitting procedure leads to failure because there are too many floating parameters (more than one hundred in our example).
Therefore, we first categorize the offset and correctionfactor parameters to be floating into three sets: the BPMrelated, QUAD-related, and BEND-related ones. We choose zero or one parameter to be floating in each category, and the other parameters are fixed at zero. Then, a global fitting is performed for the combination of the choice. We try all of the possible combinations (about 10 million in our example), and a large number of results are filtered, requiring the following criteria: • the minimization of χ 2 converges with a positivedefinite Hessian matrix, • the vertical and horizontal displacements of the fitted orbit at the IP with respect to the nominal IP are within 1 mm.
Then, the best (lowest) χ 2 sample is selected as the final one. Figure 2 shows the fitted orbit after the offset corrections.
With the corrections, the χ 2 is small, and the orbit passes close to the BPM measurements and the nominal IP. Choosing two offsets to be floating in each category, we obtained almost the same results. This method is to find a solution with a minimum modification within the scope of the numerical approach.
Planck constant (reduced), and K 2/3 and K 1/3 are modified Bessel functions. Figure 3 explains the two variables: φ and ψ. We regard ∆φ as being an infinitesimal quantity, and the integral form for formula (7) is approximated by the following three-dimensional integration: where the s axis is taken to be along the beam orbit, and ρ(s) is the radius of curvature at s. A wattage distribution in a plane can be obtained by taking the mapping of (s, ψ) → (X, Y ), where (X, Y ) is a point of intersection of the plane and the line with the angle ψ with respect to the tangent of the orbit at s. Integration on s and ψ corresponds to surface integration of the Poynting vector in the plane. We perform numerical integrations of formula (8) using the Monte-Carlo (MC) integration program BASES [7]. The implementation of the wattage calculation in our program was checked by comparing with the numerical calculation using SAD [8]. In the SAD calculation, the exact formula in electrodynamics [9] is used.
It is important to consider any beam-size effects in the SR simulation, especially for the SR from QUADs. The effects are included by adding four additional integration variables (X, X ′ , Y, Y ′ ), which indicate a phase-space point of the orbit at the IP. MC integration is performed together with the following Gaussian weight function to describe the beam profile: where (X f it , X ′ f it , Y f it , Y ′ f it ) indicates the phase-space point of the fitted orbit at the IP, σX (σY ) is the horizontal (vertical) beam size, and σ X ′ and σ Y ′ indicate the beam size in X ′ -Y ′ space. The beam sizes are calculated according to the following formulas: assuming α * = 0, where α * is one of the Courant-Snyder parameters at the IP in the notation: γx 2 + 2αxx ′ + βx ′2 = ǫ. In the above formulas, ǫX (ǫY ) indicates the horizontal (vertical) emittance, and β * X (β * Y ) is the horizontal (vertical) beta function at the IP. The orbits in the relevant magnets are calculated using the inverse transfer matrices from the IP. This implementation in our program has been checked by comparing the results with no beam size and with a beam size in the asymptotic behavior: ǫX → 0 and ǫY → 0.
In Fig. 4, an example of the wattage distributions at the IP is shown in the plane perpendicular to the beam axis. The SR in this example was a dominant source in the SVD gain-drop accident.

IV. BENCH-MARK TEST
In order to demonstrate the validity of our method, we adopt the SVD gain-drop accident as a bench mark. For radiation-dose calculations, we made a detector simulation based on EGS4 [10], which was improved for low-energy photons [11,12,13,14,15]. In this detector simulation, the exact geometry and materials of the relevant SVD components have been installed according to engineering drawings. SR photons, generated with the general-purpose event generator SPRING [7], are fed into the detector simulation, and energy deposits in the SVD are calculated. Photons are simulated down to 1 keV, and electrons to 20-keV kinematic energy. In  Tables I, II, and Fig. 7. The gain-drop plots are extracted from [17].
order to transfer the radiation dose into the gain drop, we use the result on the irradiation test to investigate the radiation hardness of the SVD preamplifiers [16].
We make two different sets of calculations for a more careful demonstration. One of them is on the radiation dose before starting the gain drop (period I), and the other is after the start (period II), as shown in Fig. 5. During each of the two periods, the beam orbit was almost unchanged. Table I gives the results of a radiation-dose calculation for period I. Obtaining information on the time of the period (4641 minutes), and on the average beam current (4.6 mA), the gain drop is estimated to be around or less than 1 %, which is consistent with the measurements ( 1 %). After starting the gain drop, the estimated doses are much higher in period II than those in period I, as shown in Table II, where the time of the period is 8071 minutes, and the average beam current is 6.7 mA. What changed significantly are the kick angle of the vertical BEND, which made a dominant contribution to the radiation dose in period II, and the orbit around the BEND, as shown in Fig. 6. The bending angle changed from 0.53 mrad with 1.7-keV critical energy (E crit ) to 0.66 mrad with E crit = 2.2 keV in the same direction. Figure 7 shows a comparison between the estimated and measured gaindrop values. We have obtained good agreements not only on the absolute size, but also for the azimuthal distribution.

V. CONCLUSIONS AND FUTURE PROSPECTS
We have developed a new method to calculate SR wattages, or radiation doses, based on a real beam orbit with reasonable offset corrections. Here, the orbit is   obtained by fitting BPM measurements. SR wattages are calculated based on the fitted orbit with the MC integration of the analytical formula. We have coded a dedicated computer program, where the correctness of the orbit and wattage calculations in our program has been confirmed by comparing with the results using SAD. Finally, we have successfully reproduced the SVD gain-drop accident quantitatively, so that the practicability of our method has been established.
This method can be applied to the construction of an online alarm system for SR background, leading to prior notice for detector damage and possibly more flexibility in accelerator operation. With this technique, a warning can be issued when the orbit is approaching dangerous areas, and furthermore the expected amount of SR background can be estimated before making certain changes in the orbit.
We have made a tentative version of the online program for KEKB operation. It takes about twenty seconds per cycle (getting the magnet information and BPM measurements, fitting the orbit, and MC integration of SR   wattages from all the magnets in the HER straight section near the IP) using the workstation with a 500MHz CPU. If we use a recent faster computer, the turnaround time could be comparable to the time of BPM measurements (several seconds).

APPENDIX A: TRANSFER MATRICES
The exact forms of the transfer matrices used in the orbit calculation are shown in this appendix. Here, L indicates the effective length of the component. k is defined as k = (B 0 /b)/(Bρ), where B 0 is the magnetic field strength at radius b, and Bρ is the magnetic rigidity of the central reference trajectory. θ x (θ y ) is the horizontal (vertical) kick angle in the BEND. R rot (α) indicates the rotation matrix for the rotation angle, α. The elements with a blank mean zero.