Source
string | Question
string | Answer
string | Question_type
string | Referenced_file(s)
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string | expert_annotation
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Expert
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What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
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Carilli_2024.pdf
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III. EXPERIMENTAL SETUP The Xanadu optical bench setup at the ALBA synchrotron light source was the same as that used in Torino & Iriso (2016), including aperture mask location, reimaging optics to achieve far-field equivalence, narrow band filters centered at 538 nm with a bandwidth of 10 nm, and CCD camera imaging. The distance from the mask to the target source, which is used to relate angular size measurements to physical size of the electron beam, was 15.05 m. The optical extraction mirror is located 7 mm above the radiation direction (orbital plane of the electrons), at a distance of 7 m from the electron beam, implying an off-axis angle of $0 . 0 5 7 ^ { o }$ We employ multiple aperture masks. Figure 1 shows the full mask on the optical bench, with the illumination pattern from the synchrotron. The full mask had 6 holes. Aperture masks of differing number of holes were generated by simply covering various holes for a given measurement. The geometry of the 6-hole mask is shown schematically in Figure 1. The mask was machined in the ALBA machine shop to a tolerance we estimate to be better than 0.1 mm in hole position and size, based on measurements of the fringe spacings in the intensity images, and coordinates of the u,v points in the visibility plane.
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augmentation
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NO
| 0
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Expert
|
What is a stable closure phase?
|
A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
|
Carilli_2024.pdf
|
The challenge is that one cannot independently determine the gain of hole 5 from the measurements, as was done for the five hole non-redundant data, since one cannot separate the gain factor and source size from the decoherence due to redundancy. As a start to the analysis, we investigate the time variability of the visibility amplitudes. We expect the variability of the redundantly sampled baselines should be higher than for non-redundant baselines given phase fluctuations and implied decoherence. Figure 28 shows the results for a few of the visibility baselines for the 6-hole data. Shown are the two redundantly sampled baselines ([0-1 + 2-5] and [0-2 + 1-5]), and two non-redundant baselines that are similar in length and orientation to these redundant baselines (2-4 and 1-3, respectively). Also shown are the two longest baselines (rising and falling diagonals 1-2 and 0-5; see Figure 2). We note that the visibility amplitudes for the non-redundant baselines in the 6-hole data are typically within $1 \\%$ of the same visibilities measured with the 5-hole mask. A number of features are clear in Figure 28. First, the stability of the redundant baselines is much worse. The rms fluctuations with time are a factor 3 to 6 larger than for the non-redundant baselines. Second, the amplitudes of the non-redundant baselines that are similar in length and orientation to the redundant baselines are lower than for non-redundant baselines (red vs. blue and black vs. green; although better treatment of decoherence including the gains is given in Figure 29). Third, the time variations for the two redundant samples (black and red) are correlated. All these phenomena are consistent with decoherence of the redundantly sampled baselines due to aperture-based phase fluctuations.
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augmentation
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NO
| 0
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Expert
|
What is a stable closure phase?
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A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
|
Carilli_2024.pdf
|
$$ \\mathrm { D e c o h e r e n c e } = \\mathrm { V } _ { \\mathrm { 6 H m e a s u r e d } } / ( \\gamma _ { \\mathrm { 0 , 1 } } \\mathrm { G } _ { \\mathrm { 0 } } \\mathrm { G } _ { \\mathrm { 1 } } + \\gamma _ { \\mathrm { 0 , 1 } } \\mathrm { G } _ { \\mathrm { 2 } } \\mathrm { G } _ { 5 } ) $$ Figure 29 shows the decoherence time series for the two redundantly sampled visibilities. Again, the scatter is substantial, as seen in Figure 28. The mean and rms values for visibility $[ 0 \\mathrm { - 2 ~ + ~ } 1 \\mathrm { - 5 } ] = 0 . 9 3 1 \\pm 0 . 0 5 2$ , while those for [0-1 + 2-5] are $0 . 9 4 2 \\pm 0 . 0 4 7$ . Note that the maximum decoherence ratio reaches a value of unity, as expected for no phase decoherence, ie. when the two redundant visibilities are in-phase. This maximum of unity lends some confidence in the assumed gain for hole 5.
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augmentation
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NO
| 0
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Expert
|
What is a stable closure phase?
|
A consistent phase sum around aperture triangles, indicating low noise and beam symmetry.
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Fact
|
Carilli_2024.pdf
|
To check if some of the decoherence of 3 ms vs 1 ms data could be caused by changing gain solutions in the joint fitting process, in Figure 25 we show the illumination values for the 5-holes derived from 1 ms vs. 3 ms data from the source fitting procedure. The illumination is defined as $\\mathrm { G a i n ^ { 2 } }$ , which converts the voltage gain from the fitting procedure into photon counts (ie. power vs. voltage). We then divide the 3 ms counts by 3, for a comparison to 1ms data (ie. counts/millisecond). Figure 25 shows that the derived illuminations are the same to within 2%, at worst, which would not explain the 5% to $1 0 \\%$ larger coherences for 1!ms data. In Section VII we consider the effect of averaging time on all the data, including 2-hole and 3-hole measurements. E. Bias subtraction We have calculated the off-source mean counts and rms for data using 2, 3, and 5-hole data, and for 1mÀú s to 3 ms averaging, and for 3 mm and 5 mm holes. The off-source mean ranged from 3.43 to 3.97 counts per pixel, with an rms scatter of 5 counts in all cases. We have adopted the mean value of 3.7 counts per pixel for the bias for all analyses. The bias appears to be independent of hole size, number of holes, and integration time, suggesting that the bias is
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augmentation
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NO
| 0
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IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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The principle of the laser wire profile measurement is based on photoionization, as shown in Eq. 1. $$ H ^ { - } + \\gamma H ^ { 0 } + e ^ { - } $$ When the Laser intercepts with the $\\mathrm { H } ^ { - }$ beam at a certain wavelength, it causes electrons to detach from the $\\mathrm { H } ^ { - }$ . The density of the $\\mathrm { H } ^ { - }$ beam can be determined by measuring the density of the detached electrons [3]. Figure 1 illustrates the Laser wire scanner system. It primarily involves a laser wire system, a deflecting magnet to divert the detached electrons away from the $\\mathrm { H } ^ { - }$ beam, and a Faraday cup (FC) to collect those electrons. There are 13 laser wire stations located along the $2 0 0 \\mathrm { m }$ long SCL, first at $2 . 1 \\mathrm { M e V }$ and last at $8 0 0 { \\mathrm { M e V } } .$ The FC stands out as a straightforward, easily handled, and cost-effective solution for accurately measuring absolute beam current. Careful consideration of the FC's physical properties is key to minimizing signal losses. Factors such as geometry, material nature, and target wall thickness are crucial in the FC design. Further, challenges arise due to phenomena like secondary electron emission and backscattering, which can lead to inaccuracies in beam current measurements. Therefore, understanding these phenomena and optimizing the FC design accordingly is essential for achieving accurate measurements [4-6].
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augmentation
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NO
| 0
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IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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File Name:WIRE_SCANNER_ASSESSMENT_OF_TR_ANSVERSE_BEAM_SIZE_IN_THE.pdf WIRE SCANNER ASSESSMENT OF TRANSVERSE BEAM SIZE IN THE FERIMLAB SIDE-COUPLED LINAC\\* E. Chen, R. Sharankova, J. Stanton, Fermi National Accelerator Laboratory, Batavia, USA Abstract The Fermilab Side-Coupled Linac contains seven 805 MHz modules accelerating H- beam from $1 1 6 \\mathrm { M e V }$ to 400 MeV. Each module contains at least one wire scanner, yielding beam intensity at positions along a transverse direction. These wire scanners each contain three wires, mounted at different angles: "X", "Y", and $4 5 ^ { \\circ }$ between "X" and "Y" to analyze coupling. Recently, a significant amount of transverse X-Y coupling was identified within wire scanner data from the Side-Coupled Linac, which has been present in data from the past decade. This realization has prompted an investigation into the wire scanner's utility as a diagnostic tool in the Fermilab Linac. This work presents efforts to better characterize the wire scanners' limitations and the phenomenon occurring in the Side-Coupled Linac. INTRODUCTION The Fermilab Side-Coupled Linac (SCL) was added in a Linac upgrade during the 1990s [1]. During the upgrade, the last four drift tube tanks were taken out and replaced with ${ 8 0 5 } \\mathrm { M H z }$ side-coupled cavities. In the SCL, there are seven main modules (Modules 1-7) with four sections in each module, as well as a transition section (Module 0) composed of a buncher (B) and Vernier (V) cavity, Fig. 1.
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augmentation
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NO
| 0
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IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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Table: Caption: Table 1: Scanning Magnet Parameters Body: <html><body><table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Width</td><td>385 mm</td></tr><tr><td>Length (mechanical)</td><td>250 mm</td></tr><tr><td>Effective Length</td><td>327 mm</td></tr><tr><td>Gap</td><td>75 mm</td></tr><tr><td>Weight</td><td>140 kg</td></tr><tr><td>Electrical resistance (4 coils in series)</td><td>0.23 Ohm</td></tr><tr><td>Inductance (4 coil in series)</td><td>43 mH</td></tr></table></body></html> SCANNING PROCEDURE SIMULATION The scanning procedure was modelled in three steps. In the first step the beam dynamics LINAC code [5] was employed to propagate the beam in vacuum within the accelerator up to the end of the extraction pipe. In the second step the LINAC file with the computed output coordinates was given as input to the Montecarlo code TRIM [6] to simulate the beam propagation in the delivery line, from the entrance of the $2 5 ~ { \\mu \\mathrm { m } }$ Ti window up to the target position with no deflection applied. In this way the energy loss and lateral straggling caused by the interaction with air, windows and He and the final size of the pencil beam are evaluated. Finally, a home-made MATLAB code was used in the third stage to simulate the scanning procedure, starting from the computed pencil beam. The goal was to assess the maximum scanned area that could be achieved with no beam losses during propagation, determine the corresponding magnet settings, and verify the uniformity of particle distribution in the irradiation field. The simulations discussed herein were carried out using a $7 1 \\mathrm { M e V }$ beam.
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augmentation
|
NO
| 0
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IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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The overall dimension of the IR-FEL facility is $1 1 \\mathrm { m }$ $\\times 1 0 \\mathrm { m }$ . Compared with FELiChEM facility [5], the primary difference is that this FEL leaves out the magnetic compressor (chicane) and redesign the buncher section while retaining the feasibility of output a high-peak-current electron beam. On the basis of the requirement of FEL physics, the norminal parameters of the electron beam are listed in Table 1. Table: Caption: Table 1: Norminal Parameters of the Electron Beam Body: <html><body><table><tr><td>Parameter</td><td>Specification</td></tr><tr><td>Energy (E)</td><td>12-60 MeV</td></tr><tr><td>Energy spread (△E) Emittance (εxn)</td><td><0.5% <40 mm ·mrad</td></tr><tr><td>Charge (Q)</td><td>1nC</td></tr><tr><td>Peak current (Ip)</td><td>>95 A</td></tr><tr><td>Micropulse Pulse length (FWHM)</td><td>4-10 ps</td></tr><tr><td>Reptition rate</td><td></td></tr><tr><td></td><td>119, 59.5, 29.75 MHz</td></tr><tr><td></td><td></td></tr><tr><td>Pulse length Macropulse</td><td>3-10 um</td></tr><tr><td>Max. avg. current (I)</td><td>>100 mA</td></tr><tr><td>Reptition rate</td><td>10 Hz</td></tr></table></body></html> THE INJECTOR DESIGN As the electron source, the thermionic triode electron gun can emit $1 0 0 \\mathrm { k e V }$ energy and 1 ns pulse width electron beam with $1 \\mathrm { n C }$ bunch charge. The micro-pulse repetition rate is optinal according to subharmonic of the RF frequency. The pre-buncher is a standing wave cavity operating at $4 7 6 \\mathrm { M H z }$ with a gap voltage of $4 0 \\mathrm { k V }$ , which will give the beam en energy chirp and then make the beam compressed in the following drift space. The electron pulse could be compressed by about 17 times in $2 4 \\mathrm { c m }$ long drift space. The buncher section is a travelling wave tube operating at $2 8 5 6 \\mathrm { M H z }$ , which is comprised of input and output couplers, five low-beta cells (phase velocity of 0.63, 0.8, 0.9, 0.95 and 0.98) and 19 cells at phase velocity of 1.0. With a maximum gradient of $9 . 0 \\mathrm { M V / m }$ , the electron beam could be further compressed to 4-10 ps in FWHM and also be accelerated to about 6 MVat the end of buncher. Two 2-meter-long travelling wave linacs are also operating at $2 8 5 6 \\mathrm { M H z }$ , each of which consists of 57 cells, the input and output couplers. According to the analysis of beam loading effect in Ref. [5], one linac can offer about $3 0 \\mathrm { M e V }$ energy gain with $2 0 \\mathrm { M W }$ power input for a $3 0 0 \\mathrm { m A }$ current beam.
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augmentation
|
NO
| 0
|
IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
|
medium changes, thus adjusting the dispersion and the pulse width. This mechanism allows for the UV pulse width to be regulated from 145 fs (FWHM) to 210 fs (FWHM), as shown in Fig. 2. The starting width of 145 fs is due to the minimum thickness of $3 \\ \\mathrm { m m }$ of the optical wedge. However, the smallest achievable adjustment in pulse width is about 5 fs, limited by the precision of the measurement equipment and the adjustability of the optical wedges. ANALYSIS OF ULTRAFAST ELECTRON DIFFRACTION BEAM The layout of MeV UED is shown in Fig. 3. UV lasers irradiate a photocathode to produce an electron beam, which is then accelerated by a 1.4-cell electron gun [8]. A solenoid magnet is positioned at the exit of the electron gun to focus the beam transversely. A THz resonator is used to compress the longitudinal length of the electron beams, with the THz signal generated by optical rectification. The ASTRA [9] particle tracking software is employed to simulate the beamline. The main parameters of the simulation are shown in Table 2. Table: Caption: Table 2: Simulation Parameters Body: <html><body><table><tr><td>Parameters</td><td>Value</td><td>Unit</td></tr><tr><td>Laser spot size (rms)</td><td>200</td><td>μm</td></tr><tr><td>Laser pulse width (rms)</td><td>35</td><td>fs</td></tr><tr><td>Bunch charge</td><td>100</td><td>fC</td></tr><tr><td>Bunch charge after the THz buncher</td><td>96.5</td><td>fC</td></tr><tr><td>Beam kinetic energy</td><td>2.96</td><td>MeV</td></tr><tr><td>PeakTHz buncher field</td><td>782</td><td>MV/m</td></tr><tr><td>Cathode position</td><td>0</td><td>m</td></tr><tr><td>Sample position</td><td>1.2</td><td>m</td></tr></table></body></html>
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augmentation
|
NO
| 0
|
IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
|
Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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In this paper, we will describe our recent upgrade of the laser wire system. The previously used Q-switched laser was replaced by a customized laser system consisting of fiber-based seeders and diode-pumped solid-state amplifiers. The laser pulse width can be selected over a wide range from a few picoseconds to over 100 ps, which enables measurements in the longitudinal domain as well as timeresolved beam diagnostics. We have also implemented several modifications in the laser wire chamber and detection scheme to improve the measurement dynamic range. LASER SYSTEM The original light source for the laser wire measurements was a commercial flash-lamp pumped Q-switched laser with the pulse width of $\\cdot \\sim 7$ ns at $3 0 \\mathrm { H z }$ . Such a laser system has high pulse energy, excellent reliability, reasonable beam quality, and is generally insensitive to the phase jitter since the pulse width is much wider than the $\\mathrm { H } ^ { - }$ beam bunch width (10 – 100 ps). A major drawback is its relatively long pulse width which produces excessive exposure on the vacuum window, causes background noise due to the reflection, and is unsuitable for beam diagnostics in the longitudinal domain.
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augmentation
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NO
| 0
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IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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Table: Caption: Table 2: Fiber Laser Parameters Body: <html><body><table><tr><td>LaserWavelength</td><td>1054 nm</td></tr><tr><td>Laser Powerat Beamline</td><td>Up to 1 W</td></tr><tr><td>Laser Pulse Frequency</td><td>162.5 MHz</td></tr><tr><td>Laser Pulse Width (FWHM)</td><td>12 ps</td></tr></table></body></html> To enhance signal detection, the laser pulses could be amplitude-modulated to control the detection of the photoionized electrons at the same modulation frequency using lock-in amplifier techniques. A detailed diagram of the prototype may be found in Fig. 2. The fiber laser itself was locked to the $8 ^ { t h }$ harmonic of the accelerator, $1 . 3 \\ : \\mathrm { G H z }$ , and generated pulses at $1 6 2 . 5 \\mathrm { M H z }$ . Each pulse passed through an acousto-optic (AO) modulator to modulate a pulse train at $2 1 . 4 \\mathrm { M H z }$ . Lastly, the laser pulses were passed through the PriTel amplifier stages reaching a maximum power of $1 \\mathrm { \\Delta W }$ . The amplified light was then transported via a single-mode polarization maintaining fiber into the beamline vacuum chamber. Laser Prototype Results Initial beamline measurements were conducted without amplitude modulation of the laser pulses. This allowed investigation of the background electron signals in the PIP2IT MEBT that were generated by $H ^ { - }$ intra-beam stripping and collected via a Faraday cup. This phenomenon produced background electrons with the same kinetic energy as the photoionized electrons. The laser profiler was operated both with the laser off and then on to obtain an estimate of the background. The data indicated a $2 \\mathrm { n A }$ signal with the laser off and a $6 ~ \\mathrm { { n A } }$ signal with the laser on. The substantial background signal constrained the dynamic range of the broadband direct current profile measurement, suggesting that a narrowband lock-in amplifier might eliminate the incoherent background.
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augmentation
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NO
| 0
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IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
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Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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Gun to Sample Design 200fC Using the results in Fig. 2, an optimised solution at $2 0 0 \\mathrm { f C }$ , including solenoidal lenses, has been designed at a gun phase of $+ 5 ^ { \\circ }$ and laser pulse length of 5 ps, and is shown in Fig. 4. The solution uses a gun solenoid lens $0 . 4 2 \\mathrm { m }$ from the gun. This solenoid position must be tuned for both the imaging and diffraction lines, and is currently under active optimisation. One or two further condenser solenoids are used to match into the objective lens, which is positioned at $3 . 6 2 5 \\mathrm { m }$ from the cathode. The optimised solution uses one condenser lens for ease of optimisation, however only minor resolution improvements were found using two condenser lenses since the chromatic emittance increase is relatively small in these lenses. The final estimated resolution for this mode, not including the projector section, is $< 1 \\mathrm { n m }$ . $2 0 \\mathbf { p C }$ The same methodology has been used for the $2 0 \\mathrm { p C }$ design, with a gun phase of $+ 5 ^ { \\circ }$ and laser pulse length of 11 ps, and shown in Fig. 5. Only one solenoidal condenser lens is used. The results show a spot size of $3 0 \\mu \\mathrm { m }$ , rather than the design value of $1 0 \\mu \\mathrm { m }$ and this corresponds with a normalised emittance value of $4 5 \\mathrm { n m }$ ‚àírad and a $\\beta$ -function value of ${ \\sim } 1 \\mathrm { m m }$ at $2 \\mathrm { M e V }$ . Achieving spot sizes smaller than this at $2 0 \\mathrm { p C }$ bunch charge is challenging. In Fig. 6 we show the calculated resolution and charge transmission as a function of sample radii at specific locations in the beamline, demonstrating the significant effect of the chromatic emittance increase due to the solenoidal focusing lenses and the relative increase in geometric emittance from decelerating the electron bunch to $2 \\mathrm { M e V }$ .
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augmentation
|
NO
| 0
|
expert
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
|
Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
|
assume a specific shape (e.g., Gaussian) of the distribution, asymmetries, double-peaks, or halos of the distribution can be reconstructed (an example is shown in Appendix C). Properties of the transverse phase space including, transverse emittance in both planes, astigmatism and Twiss parameters can be calculated from the reconstructed distribution. To obtain the full 4D emittance, cross-plane information, such as correlations in $x - y ^ { \\prime }$ or $x ^ { \\prime } - y$ need to be assessed. For this purpose, the phase advance has to be scanned independently in both planes. This can be achieved with a multiple quadrupole scan as explained for instance in [20,21] but is not achieved by measuring beam projections along a waist, as the phase advance in both planes is correlated. The presented phase space reconstruction algorithm could also be adapted to use two-dimensional profile measurements from a screen at different phase advances to characterize the four-dimensional transverse phase space. The python-code related to the described tomographic reconstruction technique is made available on github [22]. A. Reconstruction of a simulated measurement To verify the reconstruction algorithm, we generate a test distribution and calculate a set of wire scan projections (nine projections along different angles at seven locations along the waist). The algorithm then reconstructs the distribution based on these simulated projections. For this test, we choose a Gaussian beam distribution with Twiss parameters $\\beta _ { x } ^ { * } = 2 . 0 ~ \\mathrm { c m }$ , $\\beta _ { y } ^ { * } = 3 . 0 ~ \\mathrm { c m }$ and a transverse emittance of 200 nm rad in both planes. An astigmatism of $- 1 \\ \\mathrm { c m }$ (longitudinal displacement of the horizontal waist) is artificially introduced. Moreover, noise is added to the simulated wire scan profiles to obtain a signal-to-noise ratio similar to the experimental data show in Sec. IV. The Gaussian kernel size for the reconstruction $\\rho _ { x , y }$ [see Eq. (2)] is $8 0 \\mathrm { n m }$ , which is around one order of magnitude smaller than the beam size in this test. Figure 4 compares the original and reconstructed transverse phase space at $z = 0$ cm. Good agreement ( $\\text{‰}$ error) is achieved for the emittances and astigmatism, which is manifested as a tilt in the $x - x ^ { \\prime }$ plane. For this numerical experiment, the algorithm terminates according to the criterion described in Appendix B after around 100 iterations. The run-time on a single-core of a standard personal computer is around two minutes. Parallelizing the computation on several cores would reduce the computation time by few orders of magnitude.
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2
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NO
| 0
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expert
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
|
Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
|
Table: Caption: TABLE I. Normalized emittance $\\varepsilon _ { n }$ , Twiss $\\beta$ -function at the waist $\\beta ^ { * }$ , and corresponding beam size $\\sigma ^ { * }$ of the reconstructed transverse phase space distribution. Body: <html><body><table><tr><td></td><td>εn (nm rad)</td><td>β*(cm)</td><td>0* (μm)</td></tr><tr><td></td><td>186±15</td><td>3.7 ± 0.2</td><td>1.04 ± 0.06</td></tr><tr><td>y</td><td>278±18</td><td>3.7 ±0.2</td><td>1.26 ± 0.05</td></tr></table></body></html> Figure 7 shows the beam size evolution around the waist. We quantify the normalized emittance and $\\beta$ -function of the distribution by fitting a 2D Gauss function to the distribution in the $( x , x ^ { \\prime } )$ and $( y , y ^ { \\prime } )$ phase space. The 1- $\\mathbf { \\sigma } \\cdot \\sigma _ { \\mathbf { \\lambda } }$ ellipse of the fit is drawn in blue in all subplots of Fig. 6. We use the following definition for the normalized emittance: $$ \\varepsilon _ { n } = \\gamma A _ { 1 \\sigma } / \\pi , $$ where $A _ { 1 \\sigma }$ is the area of the $_ { 1 - \\sigma }$ ellipse in transverse phase space. The values for the reconstructed emittance, minimal $\\beta$ -function $( \\beta ^ { * } )$ and beam size at the waist are summarized in Table I. The measurement range $( 8 \\mathrm { c m } )$ along the waist with $\\beta ^ { * } = 3 . 7$ cm covers a phase advance of around $9 0 ^ { \\circ }$ .
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1
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NO
| 0
|
IPAC
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
|
Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
|
Other heating mechanisms are, for instance, ohmic heating due to currents flowing through the wire or electromagnetic discharge between the wire and the accelerator components. Cooling The two principal cooling mechanisms are radiative and thermionic emissions. The heat capacity of the wire plays an important role for the fast scan of high-brightness beams [3]. Due to small cross section of the wire, the conductive cooling is usually negligible. DUCTILE DAMAGE The scanner studied here [4] is located on the ${ 5 9 0 } \\mathrm { M e V }$ proton beam of Ultra Cold Neutron beamline at the PSI High Intensity Proton Accelerator facility . The beam is produced in 8 s long pulses with $1 . 8 \\mathrm { m A }$ current. The beam size in scanner position is $6 . 2 \\mathrm { m m }$ in the direction of scan (horizontal) and $1 . 3 \\mathrm { m m }$ in vertical. The scanner uses $2 5 \\mu \\mathrm { m }$ molybdenum wire stretched on a C-shaped fork with a prestress of approximately $4 0 0 \\mathrm { { M P a } }$ . The scan speed is $6 \\mathrm { c m / s }$ .
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5
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NO
| 1
|
expert
|
What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
|
5 µm
|
Fact
|
Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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At the SLAC Final Focus Test Beam experiment a laserCompton monitor was used to characterize a $7 0 \\ \\mathrm { n m }$ wide beam along one dimension [25]. The cost and complexity of this system, especially for multiangle measurements, are its main drawbacks. Concerning radiation hardness of the nanofabricated wire scanner, tests with a single wire and a bunch charge of $2 0 0 \\ \\mathrm { p C }$ at a beam energy of $3 0 0 \\mathrm { M e V }$ at SwissFEL did not show any sign of degradation after repeated measurements [9]. VI. CONCLUSION In summary, we have presented and validated a novel technique for the reconstruction of the transverse phase space of a strongly focused, ultrarelativistic electron beam. The method is based on a series of wire scans at different angles and positions along the waist. An iterative tomographic algorithm has been developed to reconstruct the transverse phase space. The technique is validated with experimental data obtained in the ACHIP chamber at SwissFEL. The method could be applied to other facilities and experiments, where focused high-brightness electron beams need to be characterized, for instance at plasma acceleration or DLA experiments for matching of an externally injected electron beam, emittance measurements at future compact low-emittance FELs [3], or for the characterization of the final-focus system at a high-energy collider test facility. For the latter application, the damage threshold of the free-standing nano-fabricated gold wires needs to be identified and radiation protection for the intense shower of scattered particles needs to be considered. Nevertheless, the focusing optics could be characterized with the presented method using a reduced bunch charge.
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What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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Countermeasures The remedies to wire breakage are: reduction of prestress, usage of di!erent wire material (e.g. carbon fiber will withstand the beam intensity and the prestress), and the scan speed increase. The solution currently applied is the installation of a thinner wire, which will lead to lower scan temperatures. A wire with a diameter of $1 3 \\mu \\mathrm { m }$ was successfully used to scan the beam. However, the calculations show that it will su!er from the same breakage but after a higher number of scans. OTHER DAMAGE MECHANISMS Material melting was observed, for example, in the case of LEP beryllium wires [9], where the wires were heated due to coupling to the beam RF field. Electrical discharges between the tungsten wires and a support stud were found to be responsible for the damage observed on the SLAC scanners [10]. In vacuum, the vapor pressure of the materials is very low, leading to high sublimation rates. The case of carbon fiber was studied in a series of measurements at CERN [11]. Due to the stabilizing e!ect of the thermionic emission on temperature, it is possible to gradually sublimate the wire material. Extreme sublimation, down to $4 \\mu \\mathrm { m }$ (more than $9 0 \\%$ of wire material), has been reported [3]. The decrease in diameter leads to smaller heating and higher cooling mechanism performance, which makes carbon fiber a particularly good target. The sublimation process is relatively well understood and good agreement has been reported between predictions and measurements [12]. A new damage mechanism that leads to "blowing" of carbon nanotube wires was recently observed [2]. The reason was tracked to the presence of iron impurities in the wire structure.
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What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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APPENDIX B: TERMINATION CRITERIONFOR RECONSTRUCTION ALGORITHM The algorithm to reconstruct the phase space from wire scan measurements iteratively approximates the distribution that fits best to all measurements (see Sec. III). The iteration is stopped when a criterion based on the relative change from the current to the previous iteration is reached. We define $p _ { k }$ as the average probability for a particle to be added or removed to the ensemble in iteration $k$ . $$ p _ { k } = \\frac { 1 } { n _ { p } n _ { \\theta } n _ { z } } \\sum _ { i , \\theta , z } \\lvert \\Delta _ { z , \\theta } ^ { i } \\rvert $$ The iteration terminates when the relative change of $p _ { k }$ reaches a tolerance limit $\\tau$ : $$ \\frac { \\left| p _ { k } - p _ { k - 1 } \\right| } { \\left| p _ { k } \\right| } < \\tau $$ For the case of the presented data set $\\tau = 0 . 0 0 5$ is found to provide stable convergence and a consistent solution. Around 110 iterations are required to reach the termination criterion.
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What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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2. Amplitude errors Jitter to the BLM signal is introduced by read-out noise of the PMT $( < 1 \\% )$ , charge fluctuations of the machine and halo-particles scattering at other elements of the accelerator. The charge measured by the BPMs fluctuated by $1 . 3 \\%$ (rms) during the measurement. The signal-to-noise ratio (SNR) of the measurements varies from 25 to 45 depending on the respective projected beam size. We define the SNR as: $s _ { \\mathrm { m a x } } / \\sigma _ { \\mathrm { n o i s e } }$ , where $s _ { \\mathrm { m a x } }$ is the maximum of the signal and $\\sigma _ { \\mathrm { n o i s e } }$ refers to the standard deviation of the background. 3. Uncertainty of the reconstruction Due to the error sources mentioned above the measured projections are not fully compatible with each other, i.e., the reconstructed distribution cannot match to all measured data points. The error of the reconstructed phase space density and the derived quantities is estimated by a procedure similar to the main reconstruction algorithm. The reconstructed distribution is now taken as input. Instead of averaging over all projections, the iteration is performed for each projection individually. Hence, a set of $n _ { z } \\times n _ { \\theta }$ distributions is generated, in which each distribution matches best to one measured projection. All derived quantities, such as the emittance or $\\beta$ -function, are computed for each distribution and the error is taken as the standard deviation of this set.
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What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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V. DISCUSSION The reconstructed phase space represents the average distribution of many shots, since shot-to-shot fluctuations in the density cannot be characterized with multishot measurements like wire scans. Errors induced by total bunch charge fluctuations and position jitter of the electron beam could be corrected for by evaluating beam-synchronous BPM data. Since the BPMs in the Athos branch were still uncalibrated, their precision was insufficient to correct orbit jitter in our measurement. This issue is considered further in Appendix A. The expected waist is located at the center of the chamber $z = 0 \\mathrm { c m } )$ , whereas the reconstructed waist is found $6 . 2 \\mathrm { c m }$ downstream. In addition, the $\\beta$ -function at the waist $( \\beta ^ { * } )$ was measured to be around $3 . 6 \\ \\mathrm { c m }$ in both planes, which is in disagreement with the design optics $( \\beta _ { x } ^ { * } = 1 ~ \\mathrm { c m }$ , $\\beta _ { y } ^ { * } = 1 . 8 ~ \\mathrm { c m } ,$ ). This indicates that the beam is mismatched at the chamber entrance and improving the matching of the electron beam to the focusing lattice could provide even smaller (submicrometer) beams in the ACHIP chamber.
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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The reconstructed normalized emittances are up to a factor of two larger than the normalized emittances measured after the second bunch compressor. This emittance increase can be attributed to various reasons. Within a distance of $1 0 3 \\mathrm { ~ m ~ }$ the electron beam is accelerated from $2 . 3 { \\mathrm { G e V } }$ (conventional emittance measurement) to around $3 . 2 { \\mathrm { ~ G e V } }$ and is directed to the Athos branch with a fast kicker and a series of bending magnets. Chromatic effects in the lattice, transverse offsets in the accelerating cavities or leaking dispersion from dispersive sections in the switch-yard can lead to a degradation of the emittance along the accelerator. These effects were not precisely characterized and corrected before the measurement, since the priority was to validate a new method for transverse phase space characterization of a strongly focused ultrarelativistic electron beam. Another possible explanation for the discrepancy of the emittances: the conventional emittance measurement uses the horizontal and vertical beam profiles measured for different phase advances (quadrupole currents) with a scintillating screen (single-shot). A Gaussian fit to the beam profiles at each phase advance is used to estimate the emittance [15]. In contrast, the tomographic wire scan technique presented here reconstructs the transverse phase space averaged over many shots. Afterwards, a Gaussian fit estimates the area of the distribution in the transverse phase space. Both large shot-to-shot jitter and non-Gaussian beams can give rise to differences between the results of the two techniques. The wire scan acquisition time could be reduced by using fewer projection angles. This could be done, if less detailed information on the beam distribution is acceptable, e.g., if only projected beam sizes are of interest, two projection angles are sufficient. The optimal number of angles depends on the internal beam structure and the beam quantities of interest.
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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$$ \\sigma ( z ) = \\sqrt { \\beta ( z ) \\varepsilon _ { n } ( z ) / \\gamma ( z ) } , $$ where $\\beta$ denotes the Twiss (or Courant-Snyder) parameter of the magnetic lattice, $\\gamma$ is the relativistic Lorentz factor of the electrons and $\\varepsilon _ { n }$ is the normalized emittance of the beam. With an optimized lattice a minimal $\\beta$ -function of around $1 \\ \\mathrm { c m }$ in the horizontal and $1 . 8 ~ \\mathrm { c m }$ in the vertical plane is expected from simulations [11,12]. In order to reduce chromatic effects of the focusing quadrupoles [14], we minimize the projected energy spread by accelerating the beam in most parts of the machine close to on-crest acceleration. From simulations, we expect an optimized projected energy spread of $4 2 \\mathrm { k e V }$ for a $3 { \\mathrm { G e V - } }$ beam with a charge of $1 \\ \\mathrm { p C }$ [11], which corresponds to a relative energy spread of $1 . 4 \\times 1 0 ^ { - 5 }$ . For this uncompressed and low-energy-spread beam we expect chromatic enlargement of the focused beam size on the order of $0 . 1 \\%$ .
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What is a typical diameter of a wire in a wire scanner at a free electron laser facility?
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Fact
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Hermann_et_al._-_2021_-_Electron_beam_transverse_phase_space_tomography_using_nanofabricated_wire_scanners_with_submicromete.pdf
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The ensemble of particles is iteratively optimized so that their projections match with the set of measured projections. The algorithm starts from a homogeneous particle distribution. One iteration consists of the following operations. (i) Transport $T ( z )$ (ii) Rotation $R ( \\theta )$ (iii) Histogram of the transported and rotated coordinates (iv) Convolution with wire profile (v) Interpolation to measured wire positions (vi) Comparison of reconstruction and measurement (vii) Redistribution of particles In the case of ultrarelativistic electrons transverse space charge effects can be neglected since they scale as $\\mathcal { O } ( \\gamma ^ { - 2 } )$ and hence $T ( z )$ becomes the ballistic transport matrix: $$ T ( z ) = { \\left( \\begin{array} { l l } { 1 } & { z } \\\\ { 0 } & { 1 } \\end{array} \\right) } $$ for $( x , x ^ { \\prime } )$ and $\\left( { y , y ^ { \\prime } } \\right)$ . The rotation matrix is then applied to $( x , y )$ : $$ R ( \\theta ) = { \\binom { \\cos \\theta } { - \\sin \\theta } } \\quad \\sin \\theta { \\Big ) } .
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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File Name:MEASUREMENTSOFLONGITUDINALLOSSOFLANDAUDAMPINGIN.pdf MEASUREMENTS OF LONGITUDINAL LOSS OF LANDAU DAMPING IN THE CERN PROTON SYNCHROTRON L. Intelisano∗1, H. Damerau, I. Karpov, CERN, Geneva, Switzerland 1also at Sapienza Università di Roma, Rome, Italy Abstract Landau damping represents the most efficient stabilization mechanism in hadron synchrotron accelerators to mitigate coherent beam instabilities. Recent studies allowed expanding the novel analytical criteria of loss of Landau damping (LLD) to the double harmonic RF system case above transition energy, providing an analytical estimate of the longitudinal stability. The threshold has a strong dependence on the voltage ratio between the harmonic and the main RF systems. Based on that, measurements of single bunch oscillations after a rigid-dipole perturbation have been performed in the CERN Proton Synchrotron (PS). Several configurations have been tested thanks to the multi-harmonic RF systems available in the PS. Higher-harmonic RF systems at $2 0 \\mathrm { M H z }$ and $4 0 \\mathrm { \\ : M H z }$ , both in phase (bunch shortening mode) and in counter-phase (bunch lengthening mode) with respect to the principal one at $1 0 \\mathrm { M H z }$ , have been measured. INTRODUCTION Landau damping [1] represents the most effective way to maintain the beam stable from uncontrolled coherent oscillations in hadron synchrotrons. In the longitudinal plane, the spread of synchrotron frequencies of individual particles caused by the non-linear voltage of the RF system establishes this stabilization mechanism, which was studied for many years [2–11]. Hence, employing a double harmonic RF system is a common way to modify the synchrotron frequency spread (Fig. 1) and to improve beam stability [12]. An analytic expression for the loss of Landau damping (LLD) threshold in the single harmonic RF case has been derived [13] using the Lebedev equation [2], which was confirmed by numerical calculations with the code MELODY [14] and macroparticle simulation with BLonD [15]. The predictions were also consistent with available beam measurements. The beam response to a rigid-dipole perturbation was also analyzed and shown to be strongly affected by Landau damping. Recently these studies were extended to a specific configuration of the double harmonic RF system and a new analytic expression was proposed [16].
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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CONCLUSION AND PLANS We conducted computational investigation of the higher order mode suppression in a C-band high gradient accelerating structure with distributed coupling. The suppression is achieved with four damping manifolds running the length of the structure. We computed the Q-factors and the products of kick factor and damped Q-factors for the dipole modes in the frequency range from 5 to $4 0 \\mathrm { G H z }$ using CST and Omega3p. We optimized the geometry of damping manifolds to achieve the best HOM suppression. We concluded that the $3 4 ~ \\mathrm { m m }$ waveguide length with the two tapers resulted in the lowest Q-factors and the wakefield kick factors. In the future, we will continue optimizations of geometry and study various damping materials to provide strong absorption to all dipole modes. ACKNOWLEDGMENT This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). Parallel computations in this work used resources of the National Energy Research Scientific Computing (NERSC) Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02- 05CH11231.
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ After the emission of a photon, the action of our single electron is $$ \\begin{array} { r c l } { { J _ { y } ^ { \\prime } } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\gamma _ { y } y ^ { 2 } + \\alpha _ { y } y p _ { y } \\left( 1 - \\frac { d p } { P _ { 0 } } \\right) + \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } \\left( 1 - \\frac { d p } { P _ { 0 } } \\right) ^ { 2 } } } \\\\ { { } } & { { } } & { { } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\gamma _ { y } y ^ { 2 } + \\alpha _ { y } y p _ { y } - \\alpha _ { y } y p _ { y } \\frac { d p } { P _ { 0 } } + \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } - 2 \\cdot \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } \\frac { d p } { P _ { 0 } } + \\frac { 1 } { 2 } \\beta p _ { y } ^ { 2 } \\left( \\frac { d p } { P _ { 0 } } \\right) ^ { 2 } . } } \\end{array}
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $P _ { 0 }$ is the momentum of the reference particle. Since $\\vec { p }$ and $d \\vec { p }$ are collinear, the same relation can be written for the $y$ component of $\\vec { p }$ , $p _ { y }$ $$ p _ { y } ^ { \\prime } \\approx p _ { y } \\left( 1 - \\frac { d p } { P _ { 0 } } \\right) . $$ In order to analyze the effect on the beam, it becomes appropriate to transition to a more beneficial set of coordinates. Specifically, we will use the action and angle variables $J _ { y }$ and $\\varphi _ { y }$ . It is essential to underscore that these coordinates are not arbitrary choices; they too are canonical variables. Their significance lies in their ability to offer a more structured view into the dynamics of the entire beam. The action $J _ { y }$ is, by its definition $$ J _ { y } = \\frac { 1 } { 2 } \\gamma _ { y } y ^ { 2 } + \\alpha _ { y } y p _ { y } + \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } .
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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Why is the North Pole the preferred site for this experiment? What will be the circumference of this storage ring? Calculate the radiated power per electron! How would the situation change if we used protons with the same momentum? I.10.7.9 Permanent magnet undulators Which options exist to tune the photon energy of coherent radiation emitted by a permanent magnet undulator (give two options)? How is the critical photon energy from each dipole in the undulator affected by these two tuning methods? What are the consequences? I.10.7.10 Superconducting undulators Which options exist to tune the photon energy of coherent radiation emitted by a superconducting undulator (give two options)? I.10.7.11 Undulator An undulator has a length of $5 . 1 \\mathrm { m }$ and a period $\\lambda _ { u } = 1 5 \\mathrm { m m }$ . The pole tip field is $B _ { t } = 1 . 2 \\ : \\mathrm { T }$ . For a gap of $g = 1 0 \\ : \\mathrm { m m }$ , calculate: – The peak field on axis $B _ { 0 }$ , – The undulator parameter $K$ . The undulator is installed in a storage ring with an electron beam energy of $E = 3 { \\mathrm { G e V } } .$ Assume electron a beam current of ${ 5 0 0 } \\mathrm { m A }$ , beam emittances of $\\varepsilon _ { x } = 1 \\mathrm { n m }$ and $\\varepsilon _ { y } = 1 \\mathrm { p m }$ , alpha functions $\\alpha _ { x } = \\alpha _ { y } = 0$ , beta functions of $\\beta _ { x } = 3 . 5 \\mathrm { m }$ and $\\beta _ { y } = 2 \\mathrm { m }$ , and calculate:
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $E _ { \\mathrm { n o m } }$ is the nominal beam energy and $$ C _ { \\gamma } = \\frac { e ^ { 2 } } { 3 \\varepsilon _ { 0 } ( m _ { e } c ^ { 2 } ) ^ { 4 } } . $$ We define the following integral as the second synchrotron radiation integral $$ I _ { 2 } : = \\oint \\frac { 1 } { \\rho ^ { 2 } } d s . $$ From the energy lost per turn $U _ { 0 }$ and the critical photon energy $E _ { c }$ , we can calculate an average number of photons to be approximately $$ \\langle n _ { \\gamma } \\rangle \\approx \\frac { 1 6 \\pi } { 9 } \\alpha _ { \\mathrm { f i n e } } \\gamma , $$ where $\\alpha _ { \\mathrm { f i n e } } \\approx 1 / 1 3 7$ is the fine structure constant. This is a relatively small number; we will therefore have to consider the quantum nature of the radiation, and we will see later how this quantum nature ultimately defines the beam emittance in an electron storage ring.
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– The required magnetic field in the dipoles, – The losses through synchrotron radiation per particle per turn. How would the situation change for electrons of the same energy? Calculate the required magnetic field in the dipoles, and the synchrotron losses! I.10.7.43 A particle accelerator on the Moon Imagine a particle accelerator around the circumference of a great circle of the Moon (Fig. I.10.19) [8]. Assuming a circumference of $1 1 0 0 0 { \\mathrm { k m } }$ , and a dipole field of $2 0 \\mathrm { T }$ , what center-of-mass energy could be achieved – For electrons? – For protons? For simplicity, assume the same dipole filling factor as in LHC, i.e. $67 \\%$ of the circumference is occupied by dipole magnets. What is the fundamental problem with the electron accelerator? (hint: calculate the synchrotron radiation power loss per turn, and compare to the space available for acceleration. Which accelerating gradient would be required?) Calculate the horizontal damping time for proton beams circulating in this ring. What are the implications for the operation? I.10.7.44 Diffraction Why is diffraction often used in place of imaging when using X-rays? What is the phase problem in X-ray diffraction?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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applications in Section I.10.6. The total radiated power per particle, obtained by integrating over the spectrum, is $$ P _ { \\\\gamma } = \\\\frac { e ^ { 2 } c } { 6 \\\\pi \\\\varepsilon _ { 0 } } \\\\frac { \\\\beta ^ { 4 } \\\\gamma ^ { 4 } } { \\\\rho ^ { 2 } } . $$ The energy lost by a particle on a circular orbit, i.e. in an accelerator consisting only of dipole magnets, is $$ U _ { 0 } = \\\\frac { e ^ { 2 } \\\\beta ^ { 4 } \\\\gamma ^ { 4 } } { 3 \\\\varepsilon _ { 0 } \\\\rho } , $$ where we have used $T = 2 \\\\pi \\\\rho / c$ , assuming $v \\\\approx c$ . Of course, real accelerators contain also other types of magnets. The energy lost per turn for a particle in an arbitrary accelerator lattice can be calculated by the following ring integral $$ U _ { 0 } = \\\\frac { C _ { \\\\gamma } } { 2 \\\\pi } E _ { \\\\mathrm { n o m } } ^ { 4 } \\\\oint \\\\frac { 1 } { \\\\rho ^ { 2 } } d s ,
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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The scattering amplitude $I _ { 0 }$ can be calculated from classic electromagnetism. For non-relativistic electrons, it is sufficient to consider the electric component of the incoming wave. The Thomson scattering cross section is equal to $$ \\sigma _ { T } = { \\frac { 8 \\pi } { 3 } } \\left( { \\frac { e ^ { 2 } } { 4 \\pi \\varepsilon _ { 0 } m _ { e } c ^ { 2 } } } \\right) ^ { 2 } = 0 . 6 \\cdot 1 0 ^ { - 2 8 } \\mathrm { { m } } = 0 . 6 \\mathrm { { b a r n } , } $$ independent of the wavelength of the incoming photon. This is in contrast to Compton scattering, where we consider photons with an energy above a few $1 0 \\mathrm { k e V . }$ In this case, we have to consider quantum mechanical effects, and the photon transfers energy and momentum to the electron. The wavelength change of the scattered photon can be determined from the conservation of energy and momentum $$ \\Delta \\lambda = \\frac { h } { m _ { e } c } ( 1 - \\cos { \\vartheta } ) ,
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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File Name:Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation.pdf Chapter I.10 Synchrotron radiation Rasmus Ischebeck Paul Scherrer Institut, Villigen, Switzerland Electrons circulating in a storage ring emit synchrotron radiation. The spectrum of this powerful radiation spans from the far infrared to the $\\boldsymbol { \\mathrm { \\Sigma } } _ { \\mathrm { X } }$ -rays. Synchrotron radiation has evolved from being a mere byproduct of particle acceleration to a powerful tool leveraged in diverse scientific and engineering fields. Indeed, synchrotrons are the most brilliant X-ray sources on Earth, and they find use in a wide range of fields in research. In this chapter, we will look at the generation of radiation of charged particles in an accelerator, at the influence of this on the beam dynamics, and on the physics behind applications of synchrotron radiation for research. I.10.1 Introduction It is difficult to overstate the importance of X-rays for medicine, research and industry. Already a few years after their discovery by Wilhelm Conrad Röntgen, their ability to penetrate matter established Xrays as an important diagnostics tool in medicine. Experiments with $\\mathrm { \\Delta X }$ -rays have come a long way since the inception of the first $\\mathrm { \\Delta } X$ -ray tubes. The short wavelength of $\\mathrm { \\Delta X }$ -rays allowed Rosalynd Franklin and Raymond Gosling to take diffraction images that would lead to the discovery of the structure of DNA. Today, X-ray diffraction is an indispensable tool in structural biology and in pharmaceutical research. Industrial applications of X-rays range from cargo inspection to sterilization and crack detection.
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ \\\\lambda _ { n } = \\\\frac { \\\\lambda _ { u } } { 2 n \\\\gamma ^ { 2 } } \\\\left( 1 + \\\\frac { K ^ { 2 } } { 2 } \\\\right) , $$ while we have destructive interference for even harmonics $n = 2 m , m \\\\in \\\\mathbb { N }$ . If we observe the radiation under a small angle $\\\\vartheta$ from the beam axis, the emission is slightly red-shifted $$ \\\\lambda = \\\\frac { \\\\lambda _ { u } } { 2 \\\\gamma ^ { 2 } } \\\\left( 1 + \\\\frac { K ^ { 2 } } { 2 } + \\\\vartheta ^ { 2 } \\\\gamma ^ { 2 } \\\\right) . $$ As you can see, since $\\\\vartheta ^ { 2 } \\\\gamma ^ { 2 } > 0$ , the wavelength increases the further away from the axis it is observed. the angular width $\\\\Delta \\\\vartheta$ of the radiation cone is inversely proportional to the distance $L$ traveled by the radiation: $\\\\begin{array} { r } { \\\\Delta \\\\theta \\\\propto \\\\frac { 1 } { L } } \\\\end{array}$ . This occurs because the wavefronts from different points of the trajectory become more aligned the farther they travel, effectively narrowing the observed radiation cone.
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ and $$ \\vec { A } ( \\vec { x } , t ) = \\frac { 1 } { 4 \\pi \\varepsilon _ { 0 } C ^ { 2 } } \\int d ^ { 3 } \\vec { x } ^ { \\prime } \\int d t ^ { \\prime } \\frac { \\vec { j } ( \\vec { x } ^ { \\prime } , t ) } { | \\vec { x } - \\vec { x } ^ { \\prime } | } \\delta \\left( t ^ { \\prime } + \\frac { \\vec { x } - \\vec { x } ^ { \\prime } } { c } - t \\right) . $$ Solving the wave equation in this most general sense is quite elaborate. The derivation can be found in Jackson [1], Chapter 6. Here, we just cite the result: the intensity of the radiation per solid angle $d \\Omega$ and per frequency interval $d \\omega$ is given by $$ \\frac { d ^ { 3 } I } { d \\Omega d \\omega } = \\frac { e ^ { 2 } } { 1 6 \\pi ^ { 3 } \\varepsilon _ { 0 } c } \\left( \\frac { 2 \\omega \\rho } { 3 c \\gamma ^ { 2 } } \\right) ^ { 2 } \\left( 1 + \\gamma ^ { 2 } \\vartheta ^ { 2 } \\right) ^ { 2 } \\left[ K _ { 2 / 3 } ^ { 2 } ( \\xi ) + \\frac { \\gamma ^ { 2 } \\vartheta ^ { 2 } } { 1 + \\gamma ^ { 2 } \\vartheta ^ { 2 } } K _ { 1 / 3 } ^ { 2 } ( \\xi ) \\right] ,
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Longitudinal gradient bends: these are dipole magnets whose magnetic field varies along their length. By providing a variable field strength along the bend, longitudinal gradient bends (LGBs) concentrate the highest magnetic field in the middle, where the dispersion reaches a minimum. This further reduces the horizontal emittance, making diffraction-limited designs possible even for small storage rings; – Reverse bends: these are dipoles that have the opposite magnetic field of the regular dipoles, effectively bending the beam outwards. By carefully configuring the reverse bends, designers can disentangle horizontal focusing from dispersion matching, achieving a net reduction in beam dispersion. Combining MBAs with LGBs and reverse bends, designers can achieve a lower horizontal emittance. For the case of SLS 2.0, the reduction in emittance is a factor 25. The combination of longitudinal gradient bends with reverse bends is shown in Fig. I.10.8. Technical and beam dynamics considerations for diffraction-limited storage rings: – Magnet design: DLSRs require a significantly more complex magnetic lattice compared to conventional storage rings. The magnetic elements in these lattices, including bending magnets, quadrupoles, and sextupoles, are not only more numerous but also often feature higher magnetic field strengths. The quadrupoles and sextupoles are therefore built with a smaller inner bore. Energy-efficient magnet designs employ permanent magnets for the basic lattice and use electromagnets only where tuning is necessary;
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ If the phase of the radiation wave advances by $\\pi$ between $A$ and $B$ , the electromagnetic field of the radiation adds coherently3. The light moves on a straight line $\\overline { { A B } }$ that is slightly shorter than the sinusoidal electron trajectory $\\widetilde { A B }$ $$ { \\frac { \\lambda } { 2 c } } = { \\frac { \\widetilde { A B } } { v } } - { \\frac { \\overline { { A B } } } { c } } . $$ The electron travels on a sinusoidal arc of length $\\widetilde { A B }$ that can be calculated as $$ \\begin{array} { r l } { \\overrightarrow { A B } } & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\sqrt { 1 + \\left( \\frac { \\mathrm { d } x } { \\mathrm { d } \\xi } \\right) ^ { 2 } } \\mathrm { d } z } \\\\ & { \\approx \\displaystyle \\int _ { 0 } ^ { \\infty } \\left( 1 + \\frac { 1 } { 2 } \\left( \\frac { \\mathrm { d } x } { \\mathrm { d } \\xi } \\right) ^ { 2 } \\right) \\mathrm { d } z } \\\\ & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\left( 1 - \\frac { K ^ { 2 } } { 2 \\sqrt { 3 \\xi _ { 0 } ^ { 2 } \\gamma ^ { 2 } } } \\mathrm { e } ^ { \\mathrm { i } \\xi } \\mathrm { d } \\xi \\right) \\mathrm { d } z } \\\\ & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\left( 1 - \\frac { K ^ { 2 } } { 2 \\sqrt { 3 \\xi _ { 0 } ^ { 2 } \\gamma ^ { 2 } } } \\mathrm { e } ^ { \\mathrm { i } \\xi } \\mathrm { d } \\xi \\right) \\mathrm { d } z } \\\\ & { = \\displaystyle \\frac { \\lambda _ { 0 } } { 2 } \\left( 1 + \\frac { K ^ { 2 } } { 4 ( 3 \\xi _ { 0 } ^ { 2 } \\gamma ^ { 2 } ) } \\right) } \\\\ & { \\approx \\displaystyle \\frac { \\lambda _ { 0 } } { 2 } \\left( 1 + \\frac { K ^ { 2 } } { 4 \\gamma ^ { 2 } } \\right) . } \\end{array}
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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Synchrotron sources, with their intense and coherent X-ray beams, play a crucial role in both tomographic imaging and ptychography. They provide the necessary beam brightness and coherence, enabling the capture of high-resolution data and facilitating the reconstruction processes. I.10.7 Collection of exercises The subsequent section collects an assortment of problems discussed in tutorials, and used in the written exams at JUAS. Note that the exams were open-book exams, where personal notes and course material, as well as reference booklets were allowed. You can find solutions to these exercises at https://ischebeck.net/juas/book/solutions. pdf I.10.7.1 Energy and momentum An electron is accelerated by a DC voltage of 1 MV. What is its total energy? a) $E = 1 \\mathrm { M e V }$ b) $E = 1 \\mathrm { M e V } + 5 1 1 \\mathrm { k e V } = 1 . 5 1 1 \\mathrm { M e V }$ c) $E = \\sqrt { 1 ^ { 2 } + 0 . 5 1 1 ^ { 2 } } \\mathrm { M e V } = 1 . 1 2 3 \\mathrm { M e V }$ d) This depends on the particle trajectory. What is the momentum of the particle?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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in matter? I.10.7.49 Diffraction A scientist wants to record a diffraction pattern of crystalline tungsten at a photon energy of $2 0 \\mathrm { k e V . }$ What is the optimum thickness of the crystal, that maximizes the intensity of the diffracted spot? Derive the formula for the intensity $I$ of the diffracted spot as a function of thickness $z$ , and solve for $d I / d z = 0$ . Material constants can be found in the X-Ray Data Booklet. I.10.7.50 X-ray absorption spectroscopy X-Ray Absorption Spectroscopy can be used to determine. . . (more than one answer is possible) $a$ ) the presence of elements that occur in very low concentration $b$ ) the chemical state of atoms in the sample $c$ ) the transverse coherence of the X-ray beam $d { \\ ' }$ ) the doping of semiconductors I.10.7.51 Ptychography Ptychography. . . (more than one answer may be correct) a) . . . allows to reconstruct the entire skeleton structure from a three-dimensional scan of the fossils of a Quetzalcoatlus Northropi (a pterosaur found in North America and one of the biggest known flying animals of all time) $b$ ) . . . combines measurements taken from the same angle, but at different wavelengths $c$ ) . . . requires precise positioning and rotation of the sample
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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Two important aspects: – The photon energy is proportional to the square of the energy of the electrons; – The photon energy decreases with higher magnetic field.4 We are looking at spontaneous radiation, thus the total energy loss of the electrons is proportional to the distance travelled. Consequently, the total intensity of the radiation grows proportionally to the distance travelled. The width of the radiation cone for the fundamental wavelength decreases inversely proportional to the distance, therefore the central intensity grows as the square of the undulator length. The radiation is linearly polarized in $x$ direction. Undulators thus make use of the coherent enhancement of the radiation of each electron individually, which leads to a substantial increase in brillance (Equation I.10.1). This coherence occurs at specific wavelengths, which can be tuned by adjusting the strength of the magnetic field5, and occurs in a very narrow angle around the forward direction. Free electron lasers achieve an additional coherent enhancement from multiple electrons in each microbunch, which results in another supercalifragilisticexpialidocious enhancement in the peak brilliance. To compute the brillance of the radiation from an undulator, one first has to determine the flux $\\dot { N } _ { \\gamma }$ and the effective source size $\\boldsymbol { \\sigma } _ { ( x , y ) \\mathrm { e f f } }$ and divergence $\\sigma _ { ( x ^ { \\prime } , y ^ { \\prime } ) , \\mathrm { e f f } }$ . These are given by the electron beam size $\\sigma _ { ( x , y ) }$ and divergence $\\sigma _ { ( x ^ { \\prime } , y ^ { \\prime } ) }$ , and the diffraction limit for the radiation. Electron beam size and divergence can be calculated from the Twiss parameters $\\beta$ and $\\gamma$ , and the emittance $\\varepsilon$ of the beam. The diffraction limits for the radiation $\\sigma _ { r }$ and $\\sigma _ { r ^ { \\prime } }$ can be calculated, considering the length of the source (which is equal to the undulator length) $L$
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.6.2 Spectroscopy Spectroscopic methods are used for investigating the electronic structure, chemical composition, and dynamic properties of matter. X-ray absorption spectroscopy (XAS) techniques, including X-ray absorption near edge structure (XANES), Extended X-ray Absorption Fine Structure (EXAFS) and Near Edge X-ray Absorption Fine Structure (NEXAFS), use the sudden change in absorption near edges (Section I.10.5.2) to probe the local atomic structure and electronic states of specific elements within a material (see Fig. I.10.14). Absorption edges, related to the ionization potential of inner-shell electrons in an atom, have a very small dependence on the chemical configuration of the atom in a molecule, as this shifts the energy levels slightly. X-ray fluorescence (XRF) is based on the principle that when a material is irradiated with Xrays, electrons from the inner shells of the atoms in the material can be ejected, leading to the emission of fluorescence $\\mathrm { \\Delta X }$ -rays as electrons from higher energy levels fill these vacancies. The energy of the emitted fluorescence $\\mathrm { \\Delta } X$ -rays is characteristic of each element, thus enabling qualitative and quantitative analysis of the elemental composition of the sample (see Fig. I.10.15). Similarly, X-ray photoelectron spectroscopy (XPS) measures the kinetic energy and the number of electrons that are emitted from the sample upon X-ray irradiation. Since the mean free path of free electrons in solids is only a few molecular layers, this technique enables the study of surface chemistry. Angular-resolved photoelectron spectroscopy (ARPES) allows reconstructing the momentum of the electrons in the solid, which is used to reconstruct the electronic band structure of the material.
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where a dimensionless undulator parameter has been introduced, $$ K = \\frac { e B _ { 0 } } { m _ { e } c k _ { u } } . $$ The electron follows a sinusoidal trajectory $$ x ( z ) = - \\frac { K } { k _ { u } \\gamma \\beta _ { z } } \\sin ( k _ { u } z ) . $$ Synchrotron radiation is emitted by relativistic electrons in a cone with opening angle of approximately $\\frac { 1 } { \\gamma }$ (Equation I.10.7). In an undulator, the maximum angle of the particle velocity with respect to the undulator axis $\\begin{array} { r } { \\alpha = \\arctan ( \\frac { v _ { x } } { v _ { z } } ) } \\end{array}$ is always smaller than the opening angle of the radiation, therefore the radiation field may add coherently. Consider two photons emitted by a single electron at the points $A$ and $B$ , which are one half undulator period apart (see Fig. I.10.5) $$ \\overline { { A B } } = \\frac { \\lambda _ { u } } { 2 } .
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What is radiation damping?
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The process by which synchrotron radiation emission causes a reduction in transverse and longitudinal emittances.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ The critical angle is defined as $$ \\vartheta _ { c } = \\frac { 1 } { \\gamma } \\left( \\frac { \\omega _ { c } } { \\omega } \\right) ^ { 1 / 3 } . $$ Higher frequencies have a smaller critical angle. For frequencies much larger than the critical frequency, and for angles much larger than the critical angle, the synchrotron radiation emission is negligible. The total spectrum, integrated over all emission angles, is given by $$ \\frac { d I } { d \\omega } = \\int \\int _ { 4 \\pi } \\frac { d ^ { 3 } I } { d \\omega d \\Omega } d \\Omega = \\frac { \\sqrt { 3 } e ^ { 2 } } { 4 \\pi \\varepsilon _ { 0 } c } \\gamma \\frac { \\omega } { \\omega _ { c } } \\int _ { \\omega / \\omega _ { c } } ^ { \\infty } K _ { 5 / 3 } ( x ) d x . $$ It is shown in Fig. I.10.4. Unlike cyclotron radiation, emitted by non-relativistic electrons, synchrotron radiation has a broadband spectrum, shifted towards higher photon energies with the cube of the Lorentz factor $\\gamma$ . In the Swiss Light Source, the Lorentz factor $\\gamma$ is approximately 5000. As a result, the critical frequency of the radiation emitted by the dipole magnets is in the exahertz range, corresponding to the $\\mathrm { \\Delta X }$ -ray spectrum. The overall spectrum of synchrotron radiation covers infrared, visible, UV and X-ray wavelengths. While coherent beams in or near the visible spectrum can be conveniently generated by lasers, synchrotrons are widely used in research that requires X-ray photons. We will look at some typical
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ \\eta _ { s } = \\alpha _ { s } - 1 / \\gamma ^ { 2 } $$ where it is clear to see that if $\\eta _ { s } < 0$ , the particles that have higher momentum will have a higher revolution frequency, and if $\\eta _ { s } > 0$ the particles that have lower momentum will have a lower revolution frequency therefore at transition the revolution frequency of the particles is independent of the particles energy. This effect, in turn, reduces the bunch length increasing the peak current and space charge effects. At transition, $\\eta _ { s } = 0$ and the bunch length is at a minimum. The synchrotron tune begins to slow and the beam becomes nonadiabatic as transition is approached. The adiabticity condition, $$ \\Omega = \\frac { 1 } { \\omega _ { s } ^ { 2 } } \\left| \\frac { d \\omega _ { s } } { d t } \\right| \\ll 1 $$ where is the angular frequency and $t$ is time [12]. In Equ. 4, it is clear that there is no change in the action provided that $\\Omega \\ll 1$ [13]. The time period in which the beam becomes nonadiabatic is defined as [14]
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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The term diffraction-limited refers to a system, typically in optics or imaging, where the resolution or image detail is primarily restricted by the fundamental diffraction of light rather than by imperfections or aberrations in the source, or in imaging components. In such a system, the performance reaches the theoretical physical limit dictated by the wave nature of the radiation. Achieving diffraction-limited performance means maximizing image sharpness and detail by minimizing all other sources of distortion or blurring to the extent that diffraction becomes the overriding factor in limiting resolution. The horizontal emittance, conversely, is typically an order of magnitude larger. The X-ray beams are thus not diffraction-limited in this dimension. Diffraction-limited storage rings (DLSRs) overcome these constraints by minimizing the horizontal emittance to a level such that horizontal and vertical beam sizes in the undulators are similar. The diffraction limit of the X-ray beam thus becomes the defining factor for the source size, which leads to beams that are transversely fully coherent. This increased coherence translates into improved resolution and contrast in experimental techniques like X-ray imaging and scattering. The implications of achieving diffraction-limited performance are profound. The significantly improved coherence of the X-ray beams allow scientists to use the full beam for diffraction experiments, opening doors to previously intractable scientific questions. We will now see how this is achieved, and discuss briefly the challenges for design, construction and operation.
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ for any natural number $n$ . $$ A B = B C = { \\frac { d } { \\sin \\vartheta } } $$ and $$ \\ A C = { \\frac { 2 d } { \\tan \\vartheta } } , $$ from which follows $$ A C ^ { \\prime } = A C \\cos \\vartheta = { \\frac { 2 d } { \\tan \\vartheta } } \\cos \\vartheta = { \\frac { 2 d } { \\sin \\vartheta } } \\cos ^ { 2 } \\vartheta , $$ and we conclude $$ \\begin{array} { r c l } { { n \\lambda } } & { { = } } & { { \\displaystyle \\frac { 2 d } { \\sin \\vartheta } - \\frac { 2 d } { \\tan \\vartheta } \\cos \\vartheta = \\displaystyle \\frac { 2 d } { \\sin \\vartheta } \\big ( 1 - \\cos ^ { 2 } \\vartheta \\big ) = \\displaystyle \\frac { 2 d } { \\sin \\vartheta } \\sin ^ { 2 } \\vartheta } } \\\\ { { } } & { { = } } & { { \\displaystyle 2 d \\sin \\vartheta , } } \\end{array}
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– The wavelength of radiation emitted on axis, – The relative bandwidth, – The photon flux (hint: if your calculator cannot evaluate Bessel functions, you may read the value of $Q _ { n } ( K )$ from the plot in the lecture), – The electron beam size and divergence, – The effective source size and divergence, – The brilliance of the radiation at the fundamental wavelength. I.10.7.12 Undulators The energy of a synchrotron is increased by $10 \\%$ , keeping the beam optics (i.e. the lattice) and the current constant. The synchrotron has an undulator. Assume that the synchrotron radiation integral $I _ { 2 }$ along the undulator is negligible in comparison to the total integral around the ring, and that the dispersion is zero in the undulator: $D _ { x } = D _ { x ^ { \\prime } } = 0$ . We will initially assume that the undulator period, the pole tip field, and the gap are unchanged. – By how much is the horizontal beam emittance changed? – By how much is the photon energy of the fundamental radiation from the undulator changed? – By how much is the brilliance of the undulator radiation changed? Assume that the effective source size is dominated by the radiation in the vertical plane, and by the electron beam phase space in the horizontal plane.
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– The required magnetic field in the dipoles, – The losses through synchrotron radiation per particle per turn. How would the situation change for electrons of the same energy? Calculate the required magnetic field in the dipoles, and the synchrotron losses! I.10.7.43 A particle accelerator on the Moon Imagine a particle accelerator around the circumference of a great circle of the Moon (Fig. I.10.19) [8]. Assuming a circumference of $1 1 0 0 0 { \\mathrm { k m } }$ , and a dipole field of $2 0 \\mathrm { T }$ , what center-of-mass energy could be achieved – For electrons? – For protons? For simplicity, assume the same dipole filling factor as in LHC, i.e. $67 \\%$ of the circumference is occupied by dipole magnets. What is the fundamental problem with the electron accelerator? (hint: calculate the synchrotron radiation power loss per turn, and compare to the space available for acceleration. Which accelerating gradient would be required?) Calculate the horizontal damping time for proton beams circulating in this ring. What are the implications for the operation? I.10.7.44 Diffraction Why is diffraction often used in place of imaging when using X-rays? What is the phase problem in X-ray diffraction?
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Photoelectric absorption: absorption by electrons bound to atoms; – Thomson scattering: elastic scattering, i.e. scattering without energy transfer between the X-ray and a free electron; – Compton scattering: inelastic scattering, where energy is transferred from the X-ray photon to an electron. I.10.5.1 Interaction of X-rays with free electrons Thomson scattering occurs when photons with an energy that is much lower than the binding energy of electrons in atoms interact with free or loosely bound electrons. The incident photons are then scattered elastically, i.e. there is no energy transfer between the X-ray photon and the electron. The photon wavelength is inversely proportional to its energy, thus it remains constant in an elastic process. A collision, however, implies in general a change in direction, thus the momentum $\\vec { k }$ of the photon will change its direction. We can describe Thomson scattering by classical electromagnetism, considering a free electron that encounters an electromagnetic wave. The electron will start oscillating and radiate in all directions except the direction of the oscillation, with an intensity given by $I = I _ { 0 } \\cos ^ { 2 } \\vartheta$ . This re-radiated light has the same frequency as the incident light because the electron’s oscillation frequency is driven by the frequency of the electromagnetic wave, and there’s no energy loss in the system.
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Elastic scattering: in contrast to Thomson scattering, which occurs for free electrons, Rayleigh scattering describes the scattering on bound electrons, incorporating the quantum mechanical nature of the atoms; – Inelastic scattering: while we looked at free electrons previously, Compton scattering can still occur with weakly bound electrons in heavier atoms where the binding energy is much lower than the energy of the incident X-ray photon; – Photoelectric effect: when the energy of the incoming photon is greater than the binding energy of the electron in the atom, it can be completely absorbed, ejecting the bound electron (now referred to as a photoelectron) from the atom. The energy of the photoelectron is equal to the energy of the incident X-ray photon minus the binding energy of the electron in its original orbital; – Absorption edges: the requirement that X-rays have a minimum energy to ionize an electron in a given orbital leads to the formation of absorption edges. These edges are specific to each element, and are widely used to characterize samples; – Fluorescence: when an inner-shell electron is ejected (as in the photoelectric effect), an electron from a higher energy level falls into the lower energy vacancy, emitting an $\\mathrm { \\Delta } X$ -ray photon with a characteristic energy specific to the atom;
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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How would you measure this radiation? I.10.7.27 Superconducting undulators What is the advantage of using undulators made with superconducting coils, in comparison to permanentmagnet arrays? What are drawbacks? I.10.7.28 In-vacuum undulators What are the advantages of using in-vacuum undulators? What are possible difficulties? I.10.7.29 Instrumentation How would you measure the vertical emittance in a storage ring? I.10.7.30 Top-up operation What are the advantages of top-up operation? What difficulties have to be overcome to establish top-up in a storage ring? (give one advantage and one difficulty for 1P.; give one more advantage and one more difficulty for $\\begin{array} { r } { 1 \\mathrm { P } . \\star . } \\end{array}$ .) I.10.7.31 Fundamental limits The SLS 2.0, a diffraction limited storage ring, aims for an electron energy of $2 . 4 \\mathrm { G e V }$ and an emittance of $1 2 6 \\mathrm { p m }$ . How far is this away from the de Broglie emittance, i.e. the minimum emittance given by the uncertainty principle? I.10.7.32 Applications Why are synchrotrons important for science? I.10.7.33 Applications What applications for industry are there to synchrotrons? I.10.7.34 Orbit correction Which devices are used to measure and correct the orbit inside a synchrotron?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– What range can be reached with the fundamental photon energy? – What brilliance can be reached at the fundamental photon energy? – Is there a significant flux higher harmonics? I.10.7.20 Emittance and energy spread The equilibrium emittance of an electron bunch in a storage ring occurs when factors increasing $\\varepsilon$ are compensated by those reducing $\\varepsilon$ . – Which effect increases the horizontal emittance $\\varepsilon _ { x }$ ? – Which effect decreases the horizontal emittance $\\varepsilon _ { x }$ ? – Which effect increases the vertical emittance $\\varepsilon _ { y }$ ? – Which effect decreases the vertical emittance $\\varepsilon _ { y }$ ? I.10.7.21 Swiss Light Source The Swiss Light Source (SLS) is a storage ring optimized for synchrotron radiation generation, located at PSI in Switzerland. An upgraded lattice has been calculated in view of an upgrade10. Design values for this lattice are given below (the synchrotron radiation integrals have been numerically integrated around the design lattice, including undulators and superbends for radiation generation): Table: Caption: SLS Upgrade Lattice Body: <html><body><table><tr><td>Circumference</td><td>290.4 m</td></tr><tr><td>Electron energy</td><td>2.4 GeV</td></tr><tr><td>Horizontal damping partition jx</td><td>1.71</td></tr><tr><td>Vertical damping partition jy</td><td>1</td></tr><tr><td>Longitudinal damping partition jz</td><td>1.29</td></tr><tr><td>Second synchrotron radiation integral I2</td><td>1.186 m-1</td></tr><tr><td>Fourth synchrotron radiation integral I4</td><td>-0.842 m-1</td></tr></table></body></html>
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ When deriving the equations for the beam dynamics in the horizontal phase space, we need to consider: – Change in momentum: the emission of radiation leads to a recoil of the electron. This change in momentum is the same that we considered in the vertical phase space; – Dispersion: the emission of radiation results in a change in the energy deviation, denoted as $\\delta$ . This deviation brings about subsequent changes in the horizontal coordinate $x$ and its associated momentum $p _ { x }$ . When we explored the beam dynamics in the vertical phase space, we ignored the second factor, as we assumed that the vertical dispersion is zero. This assumption streamlined the analysis, but it can certainly not be made in the horizontal dimension. While the details of the interplay between the emission of synchrotron radiation and the damping of the emittance are unique to each plane, the outcomes are similar. The horizontal emittance decays exponentially $$ \\frac { d \\varepsilon _ { x } } { d t } = - \\frac { 2 } { \\tau _ { x } } \\varepsilon _ { x } $$ $$ \\Rightarrow \\varepsilon _ { x } ( t ) = \\varepsilon _ { x } ( 0 ) \\exp \\left( - 2 \\frac { t } { \\tau _ { x } } \\right)
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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Why is the North Pole the preferred site for this experiment? What will be the circumference of this storage ring? Calculate the radiated power per electron! How would the situation change if we used protons with the same momentum? I.10.7.9 Permanent magnet undulators Which options exist to tune the photon energy of coherent radiation emitted by a permanent magnet undulator (give two options)? How is the critical photon energy from each dipole in the undulator affected by these two tuning methods? What are the consequences? I.10.7.10 Superconducting undulators Which options exist to tune the photon energy of coherent radiation emitted by a superconducting undulator (give two options)? I.10.7.11 Undulator An undulator has a length of $5 . 1 \\mathrm { m }$ and a period $\\lambda _ { u } = 1 5 \\mathrm { m m }$ . The pole tip field is $B _ { t } = 1 . 2 \\ : \\mathrm { T }$ . For a gap of $g = 1 0 \\ : \\mathrm { m m }$ , calculate: – The peak field on axis $B _ { 0 }$ , – The undulator parameter $K$ . The undulator is installed in a storage ring with an electron beam energy of $E = 3 { \\mathrm { G e V } } .$ Assume electron a beam current of ${ 5 0 0 } \\mathrm { m A }$ , beam emittances of $\\varepsilon _ { x } = 1 \\mathrm { n m }$ and $\\varepsilon _ { y } = 1 \\mathrm { p m }$ , alpha functions $\\alpha _ { x } = \\alpha _ { y } = 0$ , beta functions of $\\beta _ { x } = 3 . 5 \\mathrm { m }$ and $\\beta _ { y } = 2 \\mathrm { m }$ , and calculate:
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ which is Bragg’s law. Note that contrary to the diffraction on a two-dimensional surface, which is often considered fir visible light, $\\mathrm { \\Delta } \\mathrm { X }$ -rays diffract on a three-dimensional crystal lattice. In this case, not only the exit angle matters, but also the incoming angle must fulfill the resonance condition! X-ray diffraction is one of the key techniques to resolve molecular structure in samples that can be crystallized. In the following section, we will look at different applications of synchrotron radiation in science, medicine and industry. I.10.6 Applications of synchrotron radiation Synchrotron radiation is used in a wide range of scientific and industrial applications, and over 60 synchrotron radiation sources are operating around the world. New facilities are under construction, reflecting the growing demand in research and industrial applications. I.10.6.1 Diffraction Coherent diffraction on crystals has been used before the emergence of synchrotrons, at the time enabled by X-ray tubes. The renowned Photo 51, recorded by Rosalind Franklin and her student Raymond Gosling, found its way (through dubious ways) into the hands of James Watson and Francis Crick, who used it to decipher the double helix structure of DNA (see Fig. I.10.12). Why do scientists use diffraction in place of imaging to determine the structure of molecules? Would it not be easier to simply magnify the X-ray image onto a detector, as we do in transmission electron microscopes?
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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The natural energy spread $\\Delta E / E$ is given by $$ \\frac { \\Delta E } { E } = \\sqrt { \\frac { C _ { q } \\gamma ^ { 2 } } { 2 j _ { z } \\langle \\rho \\rangle } } , $$ where $\\langle \\rho \\rangle$ is the average radius of curvature in the storage ring. Finally, in the vertical phase space of accelerators, the dynamics are somewhat different than in the horizontal or longitudinal phase spaces. This is primarily because of typically negligible dispersion in the vertical plane, and because this phase space is typically not coupled to the other dimensions. This means that, under normal conditions, variations in the energy of a particle do not significantly affect its vertical position. However, that does not exempt the vertical phase space from the effects of quantum excitation. Three effects remain that counterbalance radiation damping even in the vertical plane: – A (small) vertical component of the emitted photon, – Intra-beam scattering, – A remnant coupling between the horizontal and vertical plane. In most accelerators, the last point usually dominates, despite a careful set-up of the accelerator lattice that avoids coupling terms. It is worth noting that quantum effects determine macroscopic effects such as the beam size in a synchrotron. In fact, the value of Planck’s constant $\\hbar$ has just the right magnitude to make practical the construction of large electron synchrotrons [3].
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ with a horizontal damping time $$ \\tau _ { x } = \\frac { 2 } { j _ { x } } \\frac { E _ { 0 } } { U _ { 0 } } T _ { 0 } . $$ All effects related to the dispersion are summarized in the horizontal partition number $j _ { x }$ $$ j _ { x } = 1 - { \\frac { I _ { 4 } } { I _ { 2 } } } . $$ The second synchrotron radiation integral is defined in Equation I.10.12. For the sake of completeness, we now define all five synchrotron radiation integrals $$ \\begin{array} { r c l } { { I _ { 1 } } } & { { = } } & { { \\displaystyle \\oint \\frac { \\eta _ { x } } { \\rho } d s } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 2 } } } & { { = } } & { { \\displaystyle \\oint \\frac { 1 } { \\rho ^ { 2 } } d s } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 3 } } } & { { = } } & { { \\displaystyle \\oint \\frac { 1 } { | \\rho | ^ { 3 } } d s } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 4 } } } & { { = } } & { { \\displaystyle \\oint \\frac { \\eta _ { x } } { \\rho } \\left( \\frac { 1 } { \\rho ^ { 2 } } + 2 k _ { 1 } \\right) d s , \\qquad k _ { 1 } = \\frac { e } { P _ { 0 } } \\frac { \\partial B _ { y } } { \\partial x } } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 5 } } } & { { = } } & { { \\displaystyle \\oint \\frac { \\hat { \\mathcal { H } } _ { x } } { | \\rho | ^ { 3 } } d s , ~ \\qquad \\mathcal { H } _ { x } = \\gamma _ { x } \\eta _ { x } ^ { 2 } + 2 \\alpha _ { x } \\eta _ { x } \\eta _ { \\rho x } + \\beta _ { x } \\eta _ { p x } ^ { 2 } . } } \\end{array}
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ \\vec { B } ( 0 , 0 , z ) = \\vec { u } _ { y } B _ { 0 } \\sin ( k _ { u } z ) , $$ where $k _ { u } = 2 \\pi / \\lambda _ { u }$ with $\\lambda _ { u }$ the period of the magnetic field, $B _ { 0 }$ is the maximum field and $\\vec { u } _ { y }$ is the unit vector in $y$ direction. Due to the Maxwell equations, the curl and divergence of the static magnetic field vanish in vacuum, $\\vec { \\nabla } \\times \\vec { B } = 0$ and $\\vec { \\nabla } \\cdot \\vec { B } = 0$ . Thus, the field acquires a $z$ component for $y \\ne 0$ $$ \\begin{array} { r c l } { { { \\cal B } _ { x } } } & { { = } } & { { 0 } } \\\\ { { { \\cal B } _ { y } } } & { { = } } & { { { \\cal B } _ { 0 } \\cosh ( k _ { u } y ) \\sin ( k _ { u } z ) } } \\\\ { { { \\cal B } _ { z } } } & { { = } } & { { { \\cal B } _ { 0 } \\sinh ( k _ { u } y ) \\cos ( k _ { u } z ) . } } \\end{array}
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $P _ { 0 }$ is the momentum of the reference particle. Since $\\vec { p }$ and $d \\vec { p }$ are collinear, the same relation can be written for the $y$ component of $\\vec { p }$ , $p _ { y }$ $$ p _ { y } ^ { \\prime } \\approx p _ { y } \\left( 1 - \\frac { d p } { P _ { 0 } } \\right) . $$ In order to analyze the effect on the beam, it becomes appropriate to transition to a more beneficial set of coordinates. Specifically, we will use the action and angle variables $J _ { y }$ and $\\varphi _ { y }$ . It is essential to underscore that these coordinates are not arbitrary choices; they too are canonical variables. Their significance lies in their ability to offer a more structured view into the dynamics of the entire beam. The action $J _ { y }$ is, by its definition $$ J _ { y } = \\frac { 1 } { 2 } \\gamma _ { y } y ^ { 2 } + \\alpha _ { y } y p _ { y } + \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } .
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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The scattering amplitude $I _ { 0 }$ can be calculated from classic electromagnetism. For non-relativistic electrons, it is sufficient to consider the electric component of the incoming wave. The Thomson scattering cross section is equal to $$ \\sigma _ { T } = { \\frac { 8 \\pi } { 3 } } \\left( { \\frac { e ^ { 2 } } { 4 \\pi \\varepsilon _ { 0 } m _ { e } c ^ { 2 } } } \\right) ^ { 2 } = 0 . 6 \\cdot 1 0 ^ { - 2 8 } \\mathrm { { m } } = 0 . 6 \\mathrm { { b a r n } , } $$ independent of the wavelength of the incoming photon. This is in contrast to Compton scattering, where we consider photons with an energy above a few $1 0 \\mathrm { k e V . }$ In this case, we have to consider quantum mechanical effects, and the photon transfers energy and momentum to the electron. The wavelength change of the scattered photon can be determined from the conservation of energy and momentum $$ \\Delta \\lambda = \\frac { h } { m _ { e } c } ( 1 - \\cos { \\vartheta } ) ,
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expert
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What is synchrotron radiation?
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Synchrotron radiation is electromagnetic radiation emitted by charged particles when they are accelerated radially, typically within a synchrotron or storage ring.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $\\vartheta$ is the angle at which the photon is scattered. I.10.5.2 Scattering of $\\mathbf { X }$ -rays on atoms In the case of photon energies less than a few keV, the wavelength is longer than the size of the atom. The scattering is then coherent, i.e., the phases of the scattered waves from different parts of the electron cloud add up constructively. The electric field amplitude of the scattered wave is then proportional to the total number of electrons in the atom $Z$ , and the scattered intensity is proportional to $Z ^ { 2 }$ . The total cross section is then $$ \\sigma = Z ^ { 2 } \\sigma _ { T } . $$ The $Z ^ { 2 }$ dependence makes the scattering cross section for heavier atoms much larger compared to lighter ones, significantly influencing how $\\mathrm { \\Delta X }$ -rays are used in science and medicine. When we increase the photon energy, the wavelength becomes smaller than the size of the electron cloud of an atom, and decoherence between the scattered waves reduces the scattering cross section. As an approximation, the cross-section drops off as $1 / E _ { \\gamma } ^ { 2 }$ . The precise drop-off can be described by the atomic form factor $f ^ { 0 }$ , which depends on both the scattering angle and the photon wavelength. It can be parametrized as
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IPAC
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$$ \\left\\{ \\begin{array} { l l } { \\nu ^ { B } } & { = \\gamma \\cdot G , } \\\\ { \\nu ^ { E } } & { = \\beta ^ { 2 } \\gamma \\cdot \\left( \\frac { 1 } { \\gamma ^ { 2 } - 1 } - G \\right) . } \\end{array} \\right. $$ The “frozen spin” condition, by requiring the spin-rotation relative to the momentum rotation to be zero, implies that the B-field spin-rotation must be compensated by an E-field one; hence the storage ring arc-elements realizing this condition are no usual dipoles, but rather cylindrical Wien-filters $\\mathbf { E + B }$ elements), as in Fig. 1. The building of a brand-new facility dedicated to the experimental program is thus required. Apart from this inconvenience, there are further motives to look for an alternative. One might ask oneself, whence came the frozen spin (FS) condition in the first place? As mentioned above, it is grounded in the idea of measuring the rate of the beam polarization vector’s vertical component buildup. This buildup is the linear part of the general spin-precession described by the T-BMT equation. The immediate observable being the angle-of-rotation of the polarization vector, i.e. the phase $\\Theta$ of the $P _ { V } ( t ) = P _ { 0 }$ sin $\\Theta ( t ) \\approx P _ { 0 } ( \\omega t + \\Theta _ { 0 } )$ process, the method intended here is of the so-called “phase” or “space domain” variety. The FS-condition for such methods is the condition-of-possibility, both with respect to the linearization of the observable $P _ { V } ( t )$ and also with respect to systematic errors (the so-called “geometrical phase” error due to the non-commutativity of spin-rotations).
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$$ The equation for $\\nu _ { 0 }$ follows from the requirement of the existence of a nonzero solution of equation (4): $$ d e t \\big \\{ \\widehat { D } ( k , \\nu _ { 0 } ) \\big \\} = 0 $$ The uniqueness of the solution of eq. (6) is provided by the additional initial conditions. In the presence of a highly conductive outer wall, they look like this: ùúàùúà0 ‚Üí ùëóùëóùëöùëö,ùëñùëñ $\\left( J _ { m } ( j _ { m , i } ) = 0 \\right)$ at $\\omega 0$ for TM modes and $\\nu _ { 0 } \\to \\nu _ { m , i }$ $\\left( J _ { m } ^ { \\prime } \\big ( \\nu _ { m , i } \\big ) = 0 \\right)$ at $\\omega 0$ for TE modes [2]. RELATIONSHIP WITH RADIATION The frequency distributions of the wake fields (impedances) in the presence of a particle are also calculated using the partial area method [4]. In contrast to the calculations of the eigenvalues, here, in addition to the general solutions of the homogeneous Maxwell equations containing indefinite weight factors, particular solutions of the non-uniform Maxwell equations are also required. The result is a system of linear equations with non-zero right-hand sides, which in this case is reduced to a system of four inhomogeneous equations [4] relative to the weighting factors to be determined. As a partial solution, as a rule, charge fields in free space [5] of the solution for the ideal waveguide [4] are used. The phase factors of the particular and general solutions must be consistent.
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$$ Intuitively, at very short time, we would expect the fields generated by a given particle to look like free-space radiation, allowing us to further break up $\\mathbf { E } _ { c }$ into $$ \\mathbf { E } _ { c } ( \\mathbf { r } , t ) = \\mathbf { E } _ { 0 } ( \\mathbf { r } , t ) + \\mathbf { E } _ { \\mathrm { q i } } ( \\mathbf { r } , t ) , $$ where $\\mathbf { E } _ { 0 } ( \\mathbf { r } , t )$ is the field generated by a point particle at position $\\mathbf { r } ^ { \\prime }$ and moving with velocity $\\mathbf { v } ( t _ { R } )$ , where $t _ { R } ~ =$ $| \\mathbf { r } - \\mathbf { r } ^ { \\prime } | / c$ is the so-called ’retarded’ time. This field is simply the standard Lienerd-Wiechert potential. The second term, ${ \\bf E } _ { \\mathrm { q i } }$ in Eq. (3) are the contributions due to the first few wall reflections where the evanescent modes still contribute.
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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File Name:SEXTUPOLE_MISALIGNMENT_AND_DEFECT_IDENTIFICATION_AND.pdf SEXTUPOLE MISALIGNMENT AND DEFECT IDENTIFICATION AND REMEDIATION IN IOTA ∗ J. N. Wieland †1, A. L. Romanov, Fermilab, Batavia IL, USA 1also at Michigan State University, East Lansing MI, USA Abstract The nonlinear integrable optics studies at the integrable optics test accelerator (IOTA) demand fine control of the chromaticity using sextupole magnets. During the last experimental run, undesirable misalignments and multipole composition in some sextupole magnets impacted operations. This report outlines the beam-based methods used to identify the nature of the misalignments and defects, and the subsequent magnetic measurements and remediation of the magnets for future runs. INTRODUCTION The IOTA ring at Fermilab was constructed primarily for studies of nonlinear integrable optics of the type proposed by Danilov and Nagaitsev [1, 2]. The DN system places tight restrictions on the linear matching section outside of the nonlinear insert. In addition to the tight lattice requirements, the DN system requires accurate compensation of the chromaticity [3] which requires the installation of sextupoles in the matching section. IOTA has six families of two sextupoles each. See Fig. 1 for the geometry of IOTA and the sextupole placement. There are three designs of sextupoles in IOTA - A Fermilab constructed prototype and two designs based on the prototype parameters constructed by Elytt. The only distinction between the two Elytt designs are different lengths for packaging requirements in the ring. During the last electron run at IOTA, it was recognized that exciting the sextupoles caused undesirable distortions in the closed orbit, indicating a dipole term. While these distortions can be compensated for a fixed sextupole configuration, the sextupoles must be adjusted to various configurations for nonlinear dynamics studies. During the run beam-based methods were used for in situ alignment. After the run a particularly troublesome sextupole was removed for evaluation on a magnetic test stand. The European multipole convention will be used in this report, Eq. (1) [4].
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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File Name:ARBITRARY_TRANSVERSE_AND_LONGITUDINAL_CORRELATION.pdf ARBITRARY TRANSVERSE AND LONGITUDINAL CORRELATIONGENERATION USING TRANSVERSE WIGGLER AND WAKEFIELDSTRUCTURES G. Ha→, Northern Illinois University, Dekalb, IL, USA Abstract Transverse wigglers and wakefield structures are promising candidates for imparting arbitrary correlation on transverse and longitudinal phase spaces. They provide sinusoidal electromagnetic fields that become building blocks for Fourier synthesis. We present the progress of arbitrary correlation generation using transverse wiggler and wakefield structures. INTRODUCTION A particle beam’s correlation can be considered part of a periodic function or a small fraction of a function with a large domain because the beam only exists locally. This implies that most of the beam’s correlations can be approximated by the Fourier series or the summation of arbitrary cosines (i.e., cosine fitting). The development of methods or tools imparting sinusoidal modulation in the phase space would enable the generation of arbitrary two-dimensional correlations. Ref. [1] introduced a transverse wiggler as the tool to enable such manipulations. The transverse wiggler is a $9 0 ^ { \\circ }$ -rotated wiggler providing a vertical magnetic field along the $\\mathbf { \\boldsymbol { x } }$ -direction [1, 2]. This magnet array can provide arbitrary modulation amplitude, period, and phase. Especially the modulation amplitude can be easily adjusted by controlling the gap, and the phase can be controlled by the wiggler’s relative position to the beam axis. This flexibility of the transverse wiggler possibly realizes the generation of complex correlations such as the pattern in Fig. 1a.
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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3 Alternating Phase Focusing DLA 3.1 Principle and Nanophotonic Structures The advantage of high gradient in DLA comes with the drawback of non-uniform driving optical nearfields across the beam channel. The electron beam, which usually fills the entire channel, is therefore defocused. The defocusing is resonant, i.e. happening in each cell, and so strong that it cannot be overcome by external quadrupole or solenoid magnets [29]. The nearfields themselves can however be arranged in a way that focusing is obtained in at least one spatial dimension. Introducing phase jumps allows to alternate this direction along the acceleration channel and gives rise to the Alternating Phase Focusing (APF) method [30]. The first proposal of APF for DLA included only a twodimensional scheme (longitudinal and horizontal) and the assumption that the vertical transverse direction can be made invariant. This assumption is hard to hold in practice, since it would require structure dimensions (i.e. pillar heights) that are larger than the laser spot. The laser spot size, however, needs to be large for a sufficiently flat Gaussian top. Moreover, if the vertical dimension is considered invariant, the beam has to be confined by a single external quadrupole magnet [30, 31], which requires perfect alignment with the sub-micron acceleration channel. While these challenges can be assessed with symmetric flat double-grating structures with extremely large aspect ratio (channel height / channel width), it is rather unfeasible with pillar structures where the aspect ratio is limited to about 10-20. For these reasons, the remaining vertical forces render the silicon pillar structures suitable only for limited interaction length.
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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Efficient operation of the DLA undulator requires a design with optimized cell geometry to maximize the interaction of the electron beam with the laser field. Figure 11 shows simulation results for a parameter scan of the tilt angle $\\alpha$ and the fill factor $r _ { \\mathrm { f } }$ which is the tooth width divided by the grating period. The tooth height is kept constant at $h = 1 . 5 \\mu \\mathrm { m }$ . The Fourier coefficient $\\boldsymbol { e } _ { 1 }$ at the aperture center indicated by the red line as defined in Eq. (1.2) is a figure of merit for the interaction strength. For a DLA structure with reasonable aperture $\\Delta y = 1 . 2 \\mu \\mathrm { m }$ and tilt angle $\\alpha \\approx 2 5$ degrees the available structure constant is $\\left| e _ { 1 } \\right| / E _ { 0 } \\approx 0 . 4$ . At $2 \\ \\mu \\mathrm { m }$ , a reasonably short (three digit fs) laser pulse provides at the damage threshold of silica a maximum field strength of $E _ { 0 } \\approx 1 \\ldots 2 \\mathrm { G V / m }$ .
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$$ K _ { \\mathrm { z } } = a _ { \\mathrm { z } } { \\frac { k _ { \\mathrm { x } } } { k _ { \\mathrm { u } } } } = { \\frac { q } { m _ { 0 } c ^ { 2 } } } { \\frac { k _ { \\mathrm { z } } } { k k _ { \\mathrm { u } } } } \\left| e _ { 1 } \\left( \\alpha \\right) \\right| \\tan \\alpha \\ . $$ Figure $1 3 \\mathrm { a }$ ) shows the dependency of the undulator parameter $K _ { \\mathrm { z } }$ on the grating tilt angle $\\alpha$ and the undulator wavelength $\\lambda _ { \\mathrm { u } }$ in a vertically symmetric opposing silica grating structure. For this purpose the synchronous harmonic $\\boldsymbol { e } _ { 1 }$ at the center of the structure was determined as function of the tilt angle. The undulator parameter $K _ { \\mathrm { z } }$ shows a local maximum at an tilt angle of $\\alpha \\approx 2 5$ degrees. Furthermore, $K _ { \\mathrm { z } }$ increases linearly with the undulator wavelength $\\lambda _ { \\mathrm { u } }$ . We investigate a design using $\\lambda _ { \\mathrm { u } } = 4 0 0 \\lambda _ { \\mathrm { z } }$ which corresponds to an effective undulator parameter of $K _ { \\mathrm { z } } \\approx 0 . 0 4 5$ . In Fig. 13 b) the detuning with respect to the synchronous operation $k - \\beta k _ { \\mathrm { z } }$ determines the transversal oscillation amplitude $\\hat { x }$ and the energy of the generated photons $E _ { \\mathrm { p } }$ . For $0 . 2 5 \\%$ deviation from synchronicity, the silica DLA undulator induces a $\\hat { x } \\approx 3 0 \\mathrm { n m }$ electron beam oscillation and a wavelength of [55]
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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Key to the high gradients in DLA is the synchronization of optical near fields to relativistic electrons, expressed by the Wideroe condition $$ \\lambda _ { g } = m \\beta \\lambda $$ where $\\lambda _ { g }$ is the grating period, $\\lambda$ is the laser wavelength, and $\\beta = \\nu / c$ is the the electron velocity in units of the speed of light. The integer number $m$ represents the spatial harmonic number at which the acceleration takes place. The zeroth harmonic is excluded by means of the Lawson-Woodward theorem [7] as it represents just a plane wave; the first harmonic $( m = 1 )$ is most suitable for acceleration, as it usually has the highest amplitude. Phase synchronous acceleration (fulfilling Eq. 1.1) at the first harmonic can be characterized by the synchronous Fourier coefficient $$ e _ { 1 } ( x , y ) = \\frac { 1 } { \\lambda _ { g } } \\int _ { \\lambda _ { g } } E _ { z } ( x , y , z ) e ^ { 2 \\pi i z / \\lambda _ { g } } \\mathrm { d } z
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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In fact, with a full control over amplitude and phase seen by the particles the question becomes how to best optimize the drive pulse. At this purpose, it is envisioned adopting machine learning algorithms to optimize the 2D mask which gets applied to the laser drive pulse by the liquid crystal modulator [48]. 5 DLA Undulator Similar to a conventional magnetic undulator, a DLA undulator needs to provide an oscillatory deflection force as well as transversal confinement to achieve stable beam transport and scalable radiation emission. On the long run, a DLA based radiation source would use beams provided by a DLA accelerator. However, closer perspectives to experiments favor using advanced RF accelerators which can also provide single digit femtosecond bunches at high brightness. The ARES accelerator [49] at SINBAD/DESY provides such beam parameters suitable to be injected into DLA undulators. Thus, we adapt our design study on the $1 0 7 \\mathrm { M e V }$ electron beam of ARES. 5.1 Tilted Grating Design Figure 10 shows one cell of a tilted DLA structure composed of two opposing silica $\\epsilon _ { \\mathrm { r } } = 2 . 0 6 8 1 \\$ ) diffraction gratings for the laser wavelength $\\lambda = 2 \\pi / k = 2 \\mu \\mathrm { m }$ . The laser excites a grating-periodic electromagnetic field with $k _ { \\mathrm { z } } = 2 \\pi / \\lambda _ { \\mathrm { g } }$ which imposes a deflection force [17] on the electrons. Our investigation considers two different concepts for the application of tilted DLA gratings as undulators. First, the concept introduced in refs. [17, 18] which uses a phase-synchronous DLA structure fulfilling the Wideroe condition Eq. (1.1) (see ref. [9] for an analysis of the dynamics therein). Second, a concept similar to microwave [50], terahertz [51] or laser [52] driven undulators which uses a non-synchronous DLA structure that does not fulfill Eq. (1.1).
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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The spatial harmonic focusing scheme is much less efficient than APF, since most of the damage threshold limited laser power goes into focusing rather than into acceleration gradient. However, when equipped with a focusing scheme imprinted on the laser pulse by a liquid crystal phase mask, it can operate on a generic, strictly periodic grating structure. This provides significantly improved experimental flexibility. Moreover, as the scheme intrinsically operates with different phase velocities of electromagnetic waves in the beam channel, it can be easily adapted to travelling wave structures. For a high energy collider, travelling wave structures are definitely required to meet the laser energy efficiency requirement. They can efficiently transfer energy from a co-propagating laser pulse to the electrons, until the laser pulse is depleted. Laterally driven standing wave structures cannot deplete the pulse. In the best case, on can recycle the pulse in an integrated laser cavity [57]. However, significant improvement in energy efficiency as compared to the status quo can be obtained by waveguide driven DLAs, see [44, 45]. More information about the requirements and the feasibility of DLA for a high energy collider can be found in [58]. The on-chip light source is still under theoretical development. Currently we outline a computationally optimized silica grating geometry as well as an analytical description and numerical simulations of the dynamics for electrons passing a soft X-ray radiation DLA undulator. The analytical model provides essential guidelines for the ongoing design process. The concept of a non-synchronous tilted grating structure turns out to be a promising alternative to the synchronous operation mode. The non-synchronous undulator operates without phase jumps in the structure, which relaxes the fabrication requirements and the requirements on the drive laser phase front flatness. Furthermore, variation of the laser wavelength allows direct fine tuning of the undulator period length. Preliminary results indicate that in order to achieve approximately $5 0 \\%$ beam transmission, the geometric emittance must not exceed $\\varepsilon _ { \\mathrm { y } } = 1 0 0 \\mathrm { p m }$ (at $1 0 7 \\mathrm { M e V } .$ ). Optimization of the beam focusing within the DLA undulator structures is outlined for investigations in the near future.
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augmentation
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NO
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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6 Conclusion For the study of beam dynamics in DLA, computer simulations will remain essential. With combined numerical and experimental approaches, the challenges of higher initial brightness and brightness preservation along the beamline can be tackled. The electron sources available from electron microscopy technology are feasible for experiments, however cost and size puts a major constraint on them. The upcoming immersion lens nanotip sources offer a suitable alternative. Their performance does not reach the one of the commercial microscopes yet, but one can expect significant improvements in the near future. This will enable low energy DLA experiments with high energy gain and full six-dimensional confinement soon. Full confinement is a requirement for high energy gain at low injection energies, since the low energy electrons are highly dynamical. Recent APF DLA experiments showed that (as theoretically expected) the so-called invariant dimension is in fact not invariant for the mostly used silicon pillar structures. The consequences are energy spread and emittance increase, eventually leading to beam losses. A way to overcome this is to turn towards a 3D APF scheme, which can be implemented on commercial SOI wafers. The 3D scheme has also advantages at high energy, since it avoids the focusing constants going to zero in the ultrarelativistic limit. Only the square-sum goes to zero and thus a counterphase scheme is possible with high individual focusing constants. Using a single high damage threshold material for these structures leads however to fabrication challenges.
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$$ The first term recovers the tracking equation of DLATrack6D (see ref. [9]). The correction terms contribute less than $1 \\%$ for the investigated structures. Figure 14 a) compares tracking results for the particle at the beam center of a synchronous and a non-synchronous DLA undulator. In the synchronous undulator the phase jumps by $\\Delta \\varphi _ { 0 } = \\pi$ due to a $\\lambda _ { g } / 2$ drift section after each ${ \\lambda _ { \\mathrm { u } } } / { 2 }$ such that deflection force acting on the particle always switches between its maximum and minimum value. Hence, the reference particle’s momentum $x ^ { \\prime }$ changes linearly between two segments. The subsequent triangular trajectory introduces contributions of higher harmonics into the radiation. Furthermore, the accumulation of deflections leads to a deviation from the reference trajectory for $z \\ge 1 0 \\mathrm { m m }$ and the accumulated extra distance travelled by the reference particle leads to dephasing, which damps the momentum oscillation. In the non-synchronous DLA undulator the particle trajectory follows a harmonic motion. A smooth phase shift of $\\Delta \\varphi _ { \\mathrm { 0 } } = 2 \\pi \\lambda _ { \\mathrm { z } } / \\lambda _ { \\mathrm { u } }$ per DLA cell generates a harmonically oscillating deflection which is approximately $30 \\%$ smaller compared to the synchronous DLA. A tapering of the deflection strength introduced towards $z = 0$ and $z = 1 6 . 4 \\mathrm { m m }$ ensures a smooth transition at the ends of the non-synchronous DLA undulator.
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augmentation
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NO
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$ { 0 . 4 \\mathrm { m m } }$ with a throughput of roughly $5 0 \\%$ . As shown in Fig. 4, multiple acceleration stages can be arranged on a single SOI chip. Each stage roughly doubles the energy and is characterized by the laser pulse front tilt angle, corresponding to an ’average’ beam velocity in the stage (See [15] Supplementary material for the optimal constant tilt angle within a stage). Between the stages, a vertical adjustment of the beam position can be done by electrostatic steerers, which use the substrate and another silicon on glass wafer, attached from the top as two plates of a deflecting capacitor. The contacting can be done on the device layer of the SOI wafer. Due to the small distance of the plates, voltages of only about 30V are sufficient to obtain sufficiently large deflections to counteract accumulated deflection errors over hundreds of periods. 3.2 Low Energy Applications and Experiments At low energy, acceleration gradients are not that critical, since an accelerator chip will only be of the size of a thumbnail to reach relativistic velocity, which we define as $1 \\ \\mathrm { M e V }$ electron energy. Therefore, gradient can be sacrificed to some extend for flexibility and improved beam confinement. The first sacrifice is the utilization of materials which are DC-conductive and have a high refractive index, but suffer a significantly lower laser damage threshold. The best example of such is silicon, which also allows us to use the wide range of semiconductor fabrication tools.
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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$$ \\mathbf { a } \\left( x , y , z , c t \\right) = a _ { \\mathrm { z } } \\cosh \\left( k _ { \\mathrm { y } } y \\right) \\sin \\left( k c t - k _ { \\mathrm { z } } z + k _ { \\mathrm { x } } x \\right) \\mathbf { e } _ { \\mathrm { z } } $$ with the reciprocal grating vectors of the tilted DLA cell $k _ { \\mathrm { z } }$ and $k _ { \\mathrm { x } } ~ = ~ k _ { \\mathrm { z } } \\tan \\alpha$ (see ref. [9]), ${ k _ { \\mathrm { y } } } \\equiv \\sqrt { \\left| { k ^ { 2 } - { k _ { \\mathrm { x } } } ^ { 2 } - { k _ { \\mathrm { z } } } ^ { 2 } } \\right| } ,$ , and the dimensionless amplitude defined as $$ { a } _ { \\mathrm { z } } \\equiv \\frac { q \\left| \\boldsymbol { e } _ { 1 } \\left( \\alpha \\right) \\right| / k } { m _ { 0 } c ^ { 2 } } \\mathrm { ~ . ~ }
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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Looking towards applications of dielectric laser acceleration, electron diffraction and the generation of light with particular properties are the most catching items, besides the omnipresent goal of creating a TeV collider for elementary particle physics. As such we will look into DLA-type laser driven undulators, which intrinsically are able to exploit the unique properties of DLAtype electron microbunch trains. Morever, the DLA undulators allow creating significatly shorter undulator periods than conventional magnetic undulators, thus reaching the same photon energy at lower electron energy. On the long run, this shall enable us to build table-top lightsources with unprecented parameters [16]. This paper is organized along such a light source beamline and the current efforts undertaken towards achieving it. We start with the ultralow emittance electron sources, which make use of the available technology of electron microscopes, to provide a beam suitable for injection into low energy (sub-relativisitic) DLAs. Then, news from the alternating phase focusing (APF) beam dynamics scheme is presented, especially that the scheme can be implemented in 3D fashion to confine and accelerate the highly dynamical low energy electrons. We also show a higher energy design, as APF can be applied for DLA from the lowest to the highest energies, it only requires individual structures. At high energy, also spatial harmonic focusing can be applied. This scheme gains a lot of flexibility from variable shaping of the laser pulse with a spatial light modulator (SLM) and keeping the acceleration structure fixed and strictly periodic. However, due to the large amount of laser energy required in the non-synchronous harmonics, it is rather inefficient. Last in the chain, a DLA undulator can make use both of synchronous and asynchrounous fields. We outline the ongoing development of tilted DLA grating undulators [17–19] towards first experiments in RF-accelerator facilities which provide the required high brightness ultrashort electron bunches.
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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Figure $1 4 \\mathrm { b }$ ) shows the width $\\sigma _ { \\mathrm { y } }$ for an electron beam passing the DLA undulator without particle losses. A transversal geometric emittance of $\\varepsilon _ { \\mathrm { y } } = 1 0 \\mathrm { p m }$ ensures $1 0 0 \\%$ transmission. The simulations use an electron beam with the twiss parameters $\\hat { \\alpha } = 0$ and $\\gamma = 1 / \\hat { \\beta }$ at $z = 0$ . Depending on the phase $\\varphi _ { 0 }$ the transversal momentum kick (5.8) in a DLA cell can be either focusing or defocusing in y-direction. Hence, the beam width oscillates but remains bounded for both DLA undulators. In order to achieve proper beam matching into the structure a future design study will address the focusing properties of both DLA undulator concepts in more detail. Figure 15 shows the phase space of an electron beam passing a DLA undulator in a) nonsynchronous and b) synchronous operation mode. The transversal geometric emittance $\\varepsilon _ { \\mathrm { x } } = 1 \\mathrm { n m }$ and the energy spread $\\sigma _ { \\mathrm { E } } = 0 . 0 2 \\%$ follow the design parameters of ARES [56]. The bunch length is $\\sigma _ { t } = 1$ fs. The phase space in the center of the undulator at $z \\approx 9 ~ \\mathrm { m m }$ shows that both DLA designs induce transversal electron oscillations across the whole beam. However, the transverse electron beam size is larger than one unit cell of the DLA undulator such that the particle distribution transversely ranges across several grating periods. For this reason the momentum $x ^ { \\prime }$ at $z \\approx 9 ~ \\mathrm { m m }$ varies depending on the relative phase $\\varphi _ { 0 }$ in Eq. (5.5) of the electron with respect to the laser field. The averaged momentum of the particle beam remains zero. In the non-synchronous operation mode the particles experience an averaged deflection and focusing force. Thus, all electron trajectories are similar, but differ by a constant drift motion along the $\\mathbf { \\boldsymbol { x } }$ -coordinate. The drift depends on the initial phase $\\varphi _ { 0 }$ at which the particle enters the undulator. In the synchronous mode each particle experiences a different deflection and focusing force which accumulates additive along one undulator period $\\lambda _ { \\mathrm { u } }$ . The oscillation of each particle depends on its phase $\\varphi _ { 0 }$ . Thus, a substructure in the phase space slightly visible at $z \\approx 9 \\mathrm { { m m } }$ and more prominent towards the exit at $z \\approx 1 6 . 4 \\ : \\mathrm { m m }$ develops as the beam passes the undulator.
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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Moving forward, there are possibilities of increasing the acceleration gradients in DC biased electron sources through the use of novel electrode materials and preparation [25, 27] to 20 to 70 $\\mathbf { k V / m m }$ or greater gradients. These higher gradients, combined with the local field enhancement at a nanotip emitter to just short of the field emission threshold, will result in higher brightness and lower emittance beams to be focused by downstream optics. Triggering the cathode with laser pulses matched to its work function also helps minimize the beam energy spread and reduce chromatic aberrations [26]. Beam energy spread is determined by the net effects of space-charge and excess cathode trigger photon energy, and can range from $0 . 6 ~ \\mathrm { e V }$ FWHM for low charge single-photon excitation to over 5 eV FWHM for 100 electrons per shot [21]. This excess energy spread increased the electron bunch duration from a minimum of 200 fs to over 1 ps FWHM at high charge in a TEM [21]. The majority of the space charge induced energy spread occurs within a few microns of the emitter, emphasizing the importance of having a maximum acceleration field at the emitter [28].
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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Usually the laser pulses are impinging laterally on the structures, with the polarization in the direction of electron beam propagation. Short pulses thus allow interaction with the electron beam only over a short distance. This lack of length scalability can be overcome by pulse front tilt (PFT), which can be obtained from dispersive optical elements, such as diffraction gratings or prisms [10]. At ultrarelativistic energy, a 45 degree tilted laser pulse can thus interact arbitrary long with an electron, while the interaction with each DLA structure cell can be arbitrary short. The current record gradient of $8 5 0 ~ \\mathrm { M e V / m }$ [11] and record energy gain of $3 1 5 { \\mathrm { ~ k e V } }$ [12] in DLA could be obtained in this way. Generally, the pulse front tilt angle $\\alpha$ must fulfill $$ \\tan \\alpha = { \\frac { 1 } { \\beta } } $$ in order to remain synchronous with the electron [11, 13, 14]. This requires a curved pulse front shape, especially for electron acceleration at low energy, where the speed increment is nonnegligible. A general derivation of the pulse front shape required for a given acceleration ramp design is given in [15], where also pulse length minima are discussed when the curved shape is approximated by linear pieces.
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augmentation
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NO
| 0
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expert
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What is the Wideroe condition?
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the synchronization of optical near fields to relativistic electrons
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definition
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Beam_Dynamics_in_Dielectric_Laser_Acceleration.pdf
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There have been two major approaches to producing injectors of sufficiently high brightness. The first approach uses a nanotip cold field or Schottky emitter in an electron microscope column that has been modified for laser access to the cathode [14, 21, 22]. One can then leverage the decades of development that have been put into transmission electron microscopes with aberration corrected lenses. Such systems are capable of producing beams down to $2 \\mathrm { n m }$ rms with 6 mrad divergence while preserving about $3 \\ \\%$ of the electrons produced at the cathode [23], which meets the ${ \\sim } 1 0$ pm-rad DLA injector requirement for one electron per pulse. Transmission electron microscopes are also available at beam energies up to $3 0 0 \\mathrm { k e V }$ or $\\beta = 0 . 7 8$ , which would reduce the complexity of the sub-relativistic DLA stage compared to a $3 0 \\mathrm { k e V }$ or $\\beta = 0 . 3 3$ injection energy. Moreover, at higher injection energy, the aperture can be chosen wider, which respectively eases the emittance requirement. Laser triggered cold-field emission style TEMs tend to require more frequent flashing of the tip to remove contaminants and have larger photoemission fluctuations than comparable laser triggered Schottky emitter TEMs [21, 23].
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augmentation
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NO
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IPAC
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ \\begin{array} { r l } & { \\quad \\displaystyle \\int _ { - \\infty } ^ { t } \\mathrm { d } t ^ { \\prime } \\beta ( t ^ { \\prime } ) \\frac { \\partial ^ { 2 } } { \\partial \\phi ^ { 2 } } \\mathcal { S } ( \\phi ) } \\\\ & { = \\beta ( t ) \\frac { \\partial t ^ { \\prime } } { \\partial \\phi } | _ { t ^ { \\prime } = t } \\delta ^ { \\prime } ( \\phi ( t ) ) - \\frac { \\partial } { \\partial t ^ { \\prime } } ( \\beta \\frac { \\partial t ^ { \\prime } } { \\partial \\phi } ) | _ { t ^ { \\prime } = t } \\delta ( \\phi ( t ) ) } \\\\ & { \\quad + \\displaystyle \\frac { 1 } { c } [ \\frac { \\partial } { \\partial t ^ { \\prime } } ( \\frac { \\partial } { \\partial t ^ { \\prime } } ( \\beta \\frac { \\partial t ^ { \\prime } } { \\partial \\phi } ) \\frac { \\partial t ^ { \\prime } } { \\partial \\phi } ) ] _ { \\mathrm { r e t } } } \\end{array}
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augmentation
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NO
| 0
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IPAC
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ From this equation we may either (i) find the elements $F _ { i , j }$ of column $j$ of matrix $\\mathbf { F }$ by performing the inverse $\\textbf { \\em z }$ -transform; or (ii) investigate the recursion as a function of $j$ and $p$ ; we do the latter. For example, when $M = 0$ (i.e. no lift) then $d _ { n + 1 } = F _ { 0 } ( z ) d _ { n } [ j ] / z ^ { j }$ for all $j$ ; in which case every sequence converges if $| F _ { 0 } ( z ) | \\le 1$ . For example, when $M = 1$ then $d _ { n + 1 } = F _ { 0 } ( z ) d _ { n } [ 0 ]$ if $j = 0$ , and $d _ { n + 1 } = F _ { 1 } ( z ) d _ { n } [ j ] / z ^ { j }$ if $j > 0$ . Hence there are two simultaneous conditions for MC: $| F _ { 0 } ( z ) | \\le 1$ and $| F _ { 1 } ( z ) | \\le$ 1 for all $z = e ^ { i \\theta }$ . And the equipment operating point must satisfy them both!
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augmentation
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NO
| 0
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where we have used the Twiss parameter identity $\\alpha _ { y } ^ { 2 } = \\beta _ { y } \\gamma _ { y }$ . The change in emittance is thus proportional to the emittance, with a proportionality factor $- d p / P _ { 0 }$ . We thus have an exponentially decreasing emittance (the factor 2 is by convention) $$ \\varepsilon _ { y } ( t ) = \\varepsilon _ { y } ( 0 ) \\cdot \\exp \\left( - 2 \\frac { t } { \\tau _ { y } } \\right) . $$ This result underscores the value of the chosen variable transformation. By using action and angle variables, we can get an understanding of a key characteristic of the electron bunch: its emittance. This variable transformation is not just a mathematical maneuver; it serves as a powerful tool, offering clarity and depth to our exploration. Note that we assume the momentum of the photon to be much smaller than the reference momentum. As a result, we see a slow (i.e. an adiabatic) damping of the emittance. To proceed our determination of the vertical damping time, i.e. the decay constant of the emittance, we need to quantify the energy lost by a particle due to synchrotron radiation for each turn in the storage
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1
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NO
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Fourtihogneneexrtartaiocnti:oFnr.oAmccthoerd2i0n1g0lsyonewiatrhdes.r tAhlseoskonuorwcne assizdeiffnraocrtitohneliamnitgeudlsatrorage rings (DLSsRpsr),etahde sewfearciel itoiepstifematiuzreds iagsn ifaicra natsl ytrhed urcaedihaotiriozno nitsa lceomnicttearnnce,di.ncreasing the coherent flux siVgenirfyi casnotloy.n Wdeewdillclaotoekd atftahceislietinesd, etawilhinerSe ctihoen I.u1n0i.4q.uEe xapmuprlepsoisnec luodfe MA iXnItVh in Lutnhd,e Sewledcetnr,oan datcheceulpecroamtionr g SwLaSs 2t.0o inseVrilvlie geans, Sa wiltizgehrlta nsdo. Synchrotronesleacretrtohne dset-ofraactgoestraindgasr,dwfohrerreseetarhceh eulseincgtrconhserecinrtcXu-lratyebaetamas. Tnhsetyanarte operated by nationaleonreErguryo paenadn trhese raracdh alatbiorna tlorises s,is wrheoplmeankieshthed mbay vaRiFl apbloewtoera. cBadEeSmSiYc I aind indusftrioalm researchers.BeSrylnicnh, roGtreornms arney naonwds utphpel eNmaetnitoednabl ySfryenecehlercotrtornonl asLeirgsh(t FSEoLus)r,cwe h(icNhSLmSa)ke usetorfoan linear acceler ator to generate u ltrabr ight electron be ams that radi ate c oherent ly in lo ng undulators. FaEnLds are treated in Chapte r II I.7. The key properties of synchrotron radiation are: – Broad spectrum available, – High flux, – High spectral brightness, – High degree of transverse coherence, – Polarization can be controlled, – Pulsed time structure, – Stability, – Power can be computed from first principles. We will now navigate through the electromagnetic theory to understand how synchrotron radiation is generated when relativistic electrons are subjected to magnetic fields, noting in particular undulators, insertion devices present in every synchrotron radiation source. We will then look at the effect of the emission of synchrotron radiation on the particle bunches in a storage ring, and come to the surprising conclusion that this actually improves the emittance of the beam. We will then explore recent technological advancements in accelerator physics, which allow improving the transverse coherence of the $\\mathrm { \\Delta } X$ -ray beams significantly. Finally, we will look at the interaction of $\\mathrm { \\Delta } X$ -rays with matter, and give an overview of scientific uses of synchrotron radiation.
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1
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NO
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ Shifting our view to a broader perspective, we now consider the properties of the entire electron bunch. By definition, the emittance is given as the ensemble average of the action. The change in emittance follows thus from the change in action $$ \\begin{array} { r c l } { d \\varepsilon _ { y } } & { = } & { \\langle d J _ { y } \\rangle } \\\\ & { = } & { - \\langle \\alpha _ { y } \\underbrace { \\langle g p _ { y } \\rangle } _ { = - \\alpha _ { y } \\varepsilon _ { y } } + \\beta _ { y } \\underbrace { \\langle p _ { y } ^ { 2 } \\rangle } _ { = \\gamma _ { \\mathrm { t r } } \\varepsilon _ { y } } \\frac { \\partial p _ { \\mathrm { t } } } { \\partial P _ { 0 } } } \\\\ & & { - \\langle - \\alpha _ { y } ^ { 2 } \\varepsilon _ { y } + \\beta _ { y } \\gamma _ { y } \\varepsilon _ { y } \\rangle \\frac { \\partial p _ { \\mathrm { t } } } { \\partial P _ { 0 } } } \\\\ & { = } & { - \\varepsilon _ { y } \\langle \\underbrace { \\delta _ { y } \\gamma _ { y } } _ { \\mathrm { 1 } } - \\alpha _ { y } ^ { 2 } \\rangle \\frac { d p _ { \\mathrm { f } } } { P _ { 0 } } } \\\\ & & { - \\varepsilon _ { y } \\frac { d p _ { \\mathrm { f } } } { D _ { 0 } } , } \\end{array}
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1
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NO
| 0
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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The natural energy spread $\\Delta E / E$ is given by $$ \\frac { \\Delta E } { E } = \\sqrt { \\frac { C _ { q } \\gamma ^ { 2 } } { 2 j _ { z } \\langle \\rho \\rangle } } , $$ where $\\langle \\rho \\rangle$ is the average radius of curvature in the storage ring. Finally, in the vertical phase space of accelerators, the dynamics are somewhat different than in the horizontal or longitudinal phase spaces. This is primarily because of typically negligible dispersion in the vertical plane, and because this phase space is typically not coupled to the other dimensions. This means that, under normal conditions, variations in the energy of a particle do not significantly affect its vertical position. However, that does not exempt the vertical phase space from the effects of quantum excitation. Three effects remain that counterbalance radiation damping even in the vertical plane: – A (small) vertical component of the emitted photon, – Intra-beam scattering, – A remnant coupling between the horizontal and vertical plane. In most accelerators, the last point usually dominates, despite a careful set-up of the accelerator lattice that avoids coupling terms. It is worth noting that quantum effects determine macroscopic effects such as the beam size in a synchrotron. In fact, the value of Planck’s constant $\\hbar$ has just the right magnitude to make practical the construction of large electron synchrotrons [3].
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1
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NO
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ Figure I.10.2 shows the development of the peak brilliance of $\\mathrm { \\Delta X }$ -ray sources during the last century. Scientists working in synchrotron radiation facilities have gotten accustomed to an extremely high flux, as well as an excellent stability of their X-ray source. The flux is controlled on the permille level, and the position stability is measured in micrometers. Synchrotron radiation was first observed on April 24, 1947 by Herb Pollock, Robert Langmuir, Frank Elder, and Anatole Gurewitsch, when they saw a gleam of bluish-white light emerging from the transparent vacuum tube of their new $7 0 \\mathrm { M e V }$ electron synchrotron at General Electric’s Research Laboratory in Schenectady, New York1. It was first considered a nuisance because it caused the particles to lose energy, but it was recognised in the 1960s as radiation with exceptional properties that overcame the shortcomings of X-ray tubes. Furthermore, it was discovered that the emission of radiation improved the emittance of the beams in electron storage rings, and additional series of dipole magnets were installed at the Cambridge Electron Accelerator (CEA) at Harvard University to provide additional damping of betatron and synchrotron oscillations. The evolution of synchrotron sources has proceeded in four generations, where each new generation made use of unique new features in science and engineering to increase the coherent flux available to experiments:
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1
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NO
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IPAC
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ However, there is one remaining term that neither groups with any factor of $z$ or $\\delta$ on the right-hand side nor does it appear to fit into the form of the derivative on the left. In order to allow for this additional constraint on the system, we define a new function $\\mathcal { F }$ , such that $$ \\frac { \\mathrm { d } \\mathcal { F } } { \\mathrm { d } \\sigma } = - c _ { 4 } \\omega \\tan ( \\omega t + \\theta ) , $$ and we may then move this resulting function inside the derivative on the left-hand side of Eq. (12). Since we have already solved for $c _ { 4 }$ in Eq. (13), we may substitute the result into the expression above and integrate over $t$ after another change of the integration variable, and we may write the resultant invariant only in terms of the constant $c _ { 2 }$ , $$ \\begin{array} { c } { { \\displaystyle I _ { \\mathrm { R } } = \\frac { c _ { 2 } \\delta ^ { 2 } } { 2 } - c _ { 2 } \\delta \\big [ \\ln ( | \\cos ( \\omega t + \\theta ) | ) - \\ln ( | \\cos ( \\theta ) | ) \\big ] } } \\\\ { { + \\displaystyle \\frac { c _ { 2 } } { 2 } \\big [ \\ln ^ { 2 } \\left( | \\cos ( \\omega t + \\theta ) | \\right) + \\ln ^ { 2 } ( | \\cos ( \\theta ) | ) \\big ] } } \\\\ { { - c _ { 2 } \\ln ( | \\cos ( \\omega t + \\theta ) | ) \\ln ( | \\cos ( \\theta ) | ) . } } \\end{array}
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1
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NO
| 0
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Photoelectric absorption: absorption by electrons bound to atoms; – Thomson scattering: elastic scattering, i.e. scattering without energy transfer between the X-ray and a free electron; – Compton scattering: inelastic scattering, where energy is transferred from the X-ray photon to an electron. I.10.5.1 Interaction of X-rays with free electrons Thomson scattering occurs when photons with an energy that is much lower than the binding energy of electrons in atoms interact with free or loosely bound electrons. The incident photons are then scattered elastically, i.e. there is no energy transfer between the X-ray photon and the electron. The photon wavelength is inversely proportional to its energy, thus it remains constant in an elastic process. A collision, however, implies in general a change in direction, thus the momentum $\\vec { k }$ of the photon will change its direction. We can describe Thomson scattering by classical electromagnetism, considering a free electron that encounters an electromagnetic wave. The electron will start oscillating and radiate in all directions except the direction of the oscillation, with an intensity given by $I = I _ { 0 } \\cos ^ { 2 } \\vartheta$ . This re-radiated light has the same frequency as the incident light because the electron’s oscillation frequency is driven by the frequency of the electromagnetic wave, and there’s no energy loss in the system.
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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $J$ is the Bessel function of the first kind. As $K$ increases, the higher harmonics play a more signicificant role, but the fundamental harmonic always has the highest flux. I.10.3 Effects of the emission of radiation on beam dynamics In this section, we will delve deeper into the interplay between the radiation emission and the ensuing dynamics of the beam. The treatment closely follows the book by Wolski [4]. First, we will explore the energy transfer that occurs when an electron emits a photon. Following this, we will make a coordinate transformation to the more beneficial action and angle variables, providing a clearer perspective on the underlying mechanisms. We will then proceed to compute the ensemble average to calculate the implications on the emittance of the beam. A noteworthy observation will emerge from our analysis: the emittance decreases exponentially, plateauing at a limit dictated by the fundamental principles of quantum mechanics. This revelation underscores the intricate ties between quantum mechanics and relativistic beam dynamics, shedding light on the broader consequences of radiation emission in storage rings. In the following sections, we will make use of Hamiltonian mechanics. Those not familiar with this matter are invited to watch two introductory videos: "Hamiltonian formalism $1 ^ { \\dag 6 }$ and "Hamiltonian formalism 2"7.
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– The Lorentz factor $\\gamma$ , – The magnetic field to bend the beam (assume for simplicity that the ring consists of a uniform dipole field), – The critical energy of the synchrotron radiation, – The energy emitted by each electron through synchrotron radiation in one turn. I.10.7.7 Future Circular Collider: FCC-hh Particle physicists are evaluating the potential of building a future circular collider, which aims at colliding two proton beams with $5 0 0 \\mathrm { m A }$ current each and $1 0 0 \\mathrm { T e V }$ particle energy (FCC-hh). The protons would be circulating in a storage ring with $1 0 0 \\mathrm { k m }$ circumference, guided by superconducting magnets. The dipoles aim at a field of $1 6 \\mathrm { T }$ Calculate: – The Lorentz factor $\\gamma$ , – The critical energy of the synchrotron radiation, – The total power emitted by both beams through synchrotron radiation. I.10.7.8 Simple storage ring Let’s build a very simple synchrotron! Consider a storage ring that is located at the (magnetic) North Pole of the Earth. Assume that the Earth’s magnetic field of $5 0 \\mu \\mathrm { T }$ is used to confine electrons to a circular orbit, and ignore the need for focusing magnets. As a particle source, we will use the electron gun of an old TV, which accelerates the particles with a DC voltage of $2 5 \\mathrm { k V }$ (we will ignore the requirement of an injection system).
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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File Name:Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation.pdf Chapter I.10 Synchrotron radiation Rasmus Ischebeck Paul Scherrer Institut, Villigen, Switzerland Electrons circulating in a storage ring emit synchrotron radiation. The spectrum of this powerful radiation spans from the far infrared to the $\\boldsymbol { \\mathrm { \\Sigma } } _ { \\mathrm { X } }$ -rays. Synchrotron radiation has evolved from being a mere byproduct of particle acceleration to a powerful tool leveraged in diverse scientific and engineering fields. Indeed, synchrotrons are the most brilliant X-ray sources on Earth, and they find use in a wide range of fields in research. In this chapter, we will look at the generation of radiation of charged particles in an accelerator, at the influence of this on the beam dynamics, and on the physics behind applications of synchrotron radiation for research. I.10.1 Introduction It is difficult to overstate the importance of X-rays for medicine, research and industry. Already a few years after their discovery by Wilhelm Conrad Röntgen, their ability to penetrate matter established Xrays as an important diagnostics tool in medicine. Experiments with $\\mathrm { \\Delta X }$ -rays have come a long way since the inception of the first $\\mathrm { \\Delta } X$ -ray tubes. The short wavelength of $\\mathrm { \\Delta X }$ -rays allowed Rosalynd Franklin and Raymond Gosling to take diffraction images that would lead to the discovery of the structure of DNA. Today, X-ray diffraction is an indispensable tool in structural biology and in pharmaceutical research. Industrial applications of X-rays range from cargo inspection to sterilization and crack detection.
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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ring. We start with the radiation power $$ P _ { \\gamma } = \\frac { C _ { \\gamma } } { 2 \\pi } c \\frac { E ^ { 4 } } { \\rho ^ { 2 } } . $$ As the energy spread and the change in energy around one turn will be small, we can replace the particle energy $E$ by the nominal energy $E _ { \\mathrm { n o m } }$ . The emission of synchrotron radiation leads to a decrease in the energy of the particles within a storage ring. In order to maintain these particles within the beam pipe, it is imperative to counterbalance this energy loss. This compensation is achieved using radio frequency (RF) cavities. These cavities are specifically designed to accelerate particles in the forward direction, ensuring their continued trajectory within the ring. The $y$ component of the momentum is thus unchanged. In Fig. I.10.6, the momentum of a particle, after undergoing energy diminution due to radiation emission and subsequent re-acceleration by the RF cavities, is denoted as $p ^ { \\prime \\prime }$ . Let us get back to the change of emittance in one turn
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Free electrons, – Electrons bound to an atom, – Crystals. The interaction of X-rays with matter is determined by the cross-section, which is itself proportional to the square of the so-called Thomson radius. The Thomson radius, in turn, is inversely proportional to the mass of the charged particle. Consequently, considering the substantial mass difference between protons and electrons, the interaction with protons can be ignored. Furthermore, neutrons, which have the same mass as protons but lack electric charge, so do not interact with electromagnetic radiation, such as $\\mathrm { \\Delta } X$ -rays. They can thus be entirely ignored. The attenuation of $\\mathrm { \\Delta X }$ -rays in matter can be described by Beer’s Law $$ I ( z ) = I _ { 0 } \\exp ( - \\mu z ) , $$ where $\\mu$ is the attenuation coefficient. One commonly normalizes to the density $\\rho$ , and defines the mass attenuation coefficient as $\\mu / \\rho$ . Values for attenuation coefficient can be found in the $\\mathrm { \\Delta } X$ -ray data booklet [6] or at https://henke.lbl.gov/optical_constants/atten2.html. The relevant processes that contribute to the X-ray cross section are shown in Fig. I.10.9. Nuclear processes are only relevant for gamma rays, i.e. at photon energies far higher than what can be achieved by presently available synchrotrons. Pair production can occur only for photon energies above twice the electron rest energy, $2 \\times 5 1 1 \\mathrm { k e V } .$ The only processes relevant in synchrotrons are:
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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– The required magnetic field in the dipoles, – The losses through synchrotron radiation per particle per turn. How would the situation change for electrons of the same energy? Calculate the required magnetic field in the dipoles, and the synchrotron losses! I.10.7.43 A particle accelerator on the Moon Imagine a particle accelerator around the circumference of a great circle of the Moon (Fig. I.10.19) [8]. Assuming a circumference of $1 1 0 0 0 { \\mathrm { k m } }$ , and a dipole field of $2 0 \\mathrm { T }$ , what center-of-mass energy could be achieved – For electrons? – For protons? For simplicity, assume the same dipole filling factor as in LHC, i.e. $67 \\%$ of the circumference is occupied by dipole magnets. What is the fundamental problem with the electron accelerator? (hint: calculate the synchrotron radiation power loss per turn, and compare to the space available for acceleration. Which accelerating gradient would be required?) Calculate the horizontal damping time for proton beams circulating in this ring. What are the implications for the operation? I.10.7.44 Diffraction Why is diffraction often used in place of imaging when using X-rays? What is the phase problem in X-ray diffraction?
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $E _ { \\mathrm { n o m } }$ is the nominal beam energy and $$ C _ { \\gamma } = \\frac { e ^ { 2 } } { 3 \\varepsilon _ { 0 } ( m _ { e } c ^ { 2 } ) ^ { 4 } } . $$ We define the following integral as the second synchrotron radiation integral $$ I _ { 2 } : = \\oint \\frac { 1 } { \\rho ^ { 2 } } d s . $$ From the energy lost per turn $U _ { 0 }$ and the critical photon energy $E _ { c }$ , we can calculate an average number of photons to be approximately $$ \\langle n _ { \\gamma } \\rangle \\approx \\frac { 1 6 \\pi } { 9 } \\alpha _ { \\mathrm { f i n e } } \\gamma , $$ where $\\alpha _ { \\mathrm { f i n e } } \\approx 1 / 1 3 7$ is the fine structure constant. This is a relatively small number; we will therefore have to consider the quantum nature of the radiation, and we will see later how this quantum nature ultimately defines the beam emittance in an electron storage ring.
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Elastic scattering: in contrast to Thomson scattering, which occurs for free electrons, Rayleigh scattering describes the scattering on bound electrons, incorporating the quantum mechanical nature of the atoms; – Inelastic scattering: while we looked at free electrons previously, Compton scattering can still occur with weakly bound electrons in heavier atoms where the binding energy is much lower than the energy of the incident X-ray photon; – Photoelectric effect: when the energy of the incoming photon is greater than the binding energy of the electron in the atom, it can be completely absorbed, ejecting the bound electron (now referred to as a photoelectron) from the atom. The energy of the photoelectron is equal to the energy of the incident X-ray photon minus the binding energy of the electron in its original orbital; – Absorption edges: the requirement that X-rays have a minimum energy to ionize an electron in a given orbital leads to the formation of absorption edges. These edges are specific to each element, and are widely used to characterize samples; – Fluorescence: when an inner-shell electron is ejected (as in the photoelectric effect), an electron from a higher energy level falls into the lower energy vacancy, emitting an $\\mathrm { \\Delta } X$ -ray photon with a characteristic energy specific to the atom;
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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The term diffraction-limited refers to a system, typically in optics or imaging, where the resolution or image detail is primarily restricted by the fundamental diffraction of light rather than by imperfections or aberrations in the source, or in imaging components. In such a system, the performance reaches the theoretical physical limit dictated by the wave nature of the radiation. Achieving diffraction-limited performance means maximizing image sharpness and detail by minimizing all other sources of distortion or blurring to the extent that diffraction becomes the overriding factor in limiting resolution. The horizontal emittance, conversely, is typically an order of magnitude larger. The X-ray beams are thus not diffraction-limited in this dimension. Diffraction-limited storage rings (DLSRs) overcome these constraints by minimizing the horizontal emittance to a level such that horizontal and vertical beam sizes in the undulators are similar. The diffraction limit of the X-ray beam thus becomes the defining factor for the source size, which leads to beams that are transversely fully coherent. This increased coherence translates into improved resolution and contrast in experimental techniques like X-ray imaging and scattering. The implications of achieving diffraction-limited performance are profound. The significantly improved coherence of the X-ray beams allow scientists to use the full beam for diffraction experiments, opening doors to previously intractable scientific questions. We will now see how this is achieved, and discuss briefly the challenges for design, construction and operation.
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.7.55 Practical applications of synchrotron radiation The Italian Light Source Elettra is a 3rd generation synchrotron source with $2 5 9 \\mathrm { m }$ circumference, and can operate at beam energies of either $2 . 0 \\mathrm { G e V }$ or $2 . 4 \\mathrm { G e V } ,$ with beam currents of $3 1 0 \\mathrm { m A }$ and $1 6 0 \\mathrm { m A }$ , respectively. The Machine Director is feeling thirsty, and would like to use Elettra to make a splendid espresso. By assuming that all radiation emitted as SR from the dipole magnets can be converted into heat, calculate how much time is needed for the $2 . 0 \\mathrm { G e V }$ beam to heat up the espresso water from $2 0 ^ { \\circ } \\mathrm { C }$ to $8 8 ^ { \\circ } \\mathrm { C }$ . One espresso is $3 0 \\mathrm { m L }$ . The radius of curvature in the dipoles is $5 . 5 \\mathrm { m }$ . Neglect potential insertion devices! Hint: the specific heat capacity of water is $\\begin{array} { r } { c _ { w } = 4 . 1 8 6 \\frac { \\mathrm { ~ J ~ } } { \\mathrm { g K } } } \\end{array}$
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