Development of a tune knob for lattice adjustment in the HLS-II storage ring

Development of a tune knob for lattice adjustment in the HLS-II storage ring 1 first-author 1 corresp Corresponding author, wxu@ustc.edu.cn Corresponding author, wxu@ustc.edu.cn wxu@ustc.edu.cn 1 1 1 1 2 National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China The betatron tune is an important parameter in a storage ring to enable stable operation. A tune adjustment tool with a small impact on the beam dynamics is useful for user operation and machine studies. Therefore, a tune knob is developed for the Hefei Light Source-II (HLS-II) storage ring. Owing to the compactness of the storage ring, a global adjustment mechanism is adopted. To reduce the impact on beam injection, only quadrupole families outside the injection section are used by the tune knob, and the β functions of the injection section remain unchanged. A code is developed based on the accelerator simulation software, MAD-X, to calculate the adjustment of the quadrupole strengths. The Accelerator Toolbox is used to double check the accuracy of the tune knob. Online measurement of the tune knob is also performed. The result shows that the tune knob works well when the tune is adjusted in a specific range. Betatron coupling measurement is also carried out, showing an application of the tune knob on machine studies. In this paper, the development of the tune knob and its experimental results in the HLS-II storage ring are reported in detail. Tune knob Storage ring Beta function Betatron coupling 1 Introduction 1 In storage rings, the betatron tune is the number of periods of the transverse oscillation in one turn. The betatron tune should avoid critical resonance lines and remain stable for user operation [ 1 1 ]. The stability of the betatron tune may be affected by several factors, such as undulator gap changes, hysteresis effects, or jitter of the magnet power supplies [ 2 2 ]. A tool for tune adjustment in a certain range is useful to compensate for tune variations and provide opportunities for machine studies. A tune knob has been developed at the Duke Free Electron Laser Laboratory (DFELL) for its storage ring [ 3 3 , 4 4 ]. The tune knob was developed with a local adjustment method: adopting only part of the quadrupoles near its south straight section (SSS), which is long enough (34 m) to produce required tune adjustment with acceptable β function variations. The Twiss parameters at both ends of the SSS are perfectly matched with those of the arc sections so that the Twiss parameters in the remainder are almost unaffected [ 5 5 , 6 6 ]. The Hefei Light Source-II (HLS-II) storage ring has a total circumference of approximately 66 m with four double bend achromat structures. Owing to its compactness, the local tune adjustment method adopted by the Duke storage ring is not applicable for the HLS-II storage ring. Therefore, a tune knob is designed based on global adjustment, in which all or selected quadrupole families are adopted by the tune knob. A preliminary result of the tune knob developed for the HLS-II storage ring was reported in an IPAC paper, in which all the quadrupole families were adopted [ 7 7 ]. In this paper, an improved scheme for tune knob development is reported in detail. This scheme maintains the Twiss parameters of the injection section, which adopts the global adjustment using selected quadrupole families outside the injection section while preserving the lattice symmetry. The simulated impact on the beam optics and the online measurement results are presented. An application of the tune knob, betatron coupling measurement, is also reported. 2 Tune adjustment for storage rings 1 Tune adjustment in a small range usually adopts a matrix method. It is known that when the variations of the quadrupole strengths are small, the induced tune change is given by [ 8 8 ] <math display="block" id="M1"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><mi>Δ</mi><mi>K</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi>β</mi><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>s</mi><mn>,</mn></mrow></math> where Δ K is the quadrupole strength change, β ( s ) refers to the beta function at location s , and Δν denotes the tune change. Adopting the hard-edge model, the above integral can be rewritten by a summation as <math display="block" id="M2"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><mi>Δ</mi><msub><mi>K</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mover accent="true"><mi>β</mi><mo>¯</mo></mover><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Δ</mi><msub><mover accent="true"><mi>β</mi><mo>¯</mo></mover><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mi>l</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>≈</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><mi>Δ</mi><msub><mi>K</mi><mi>i</mi></msub><msub><mover accent="true"><mi>β</mi><mo>¯</mo></mover><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><msub><mi>l</mi><mi>i</mi></msub><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where l i is the effective length of the i th quadrupole and <math id="M3"><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> is the average β function in this quadrupole. Assuming Δ K i and <math id="M4"><mrow><mi>Δ</mi><mover accent="true"><mi>β</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> are both small, <math id="M5"><mrow><mi>Δ</mi><msub><mi>K</mi><mi>i</mi></msub><mi>Δ</mi><msub><mover accent="true"><mi>β</mi><mo>¯</mo></mover><mrow><mi>x</mi><mn>,</mn><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> can be neglected. Thus, the relation between the tune change and quadrupole strength adjustment becomes linear and is represented in a matrix form: <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mi>Δ</mi><msub><mi>ν</mi><mi>x</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mi>Δ</mi><msub><mi>ν</mi><mi>y</mi></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>m</mi><mrow><mn>11</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>m</mi><mrow><mn>12</mn></mrow></msub></mrow></mtd><mtd><mo>…</mo></mtd><mtd><mrow><msub><mi>m</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>m</mi><mrow><mn>21</mn></mrow></msub></mrow></mtd><mtd><mrow><msub><mi>m</mi><mrow><mn>22</mn></mrow></msub></mrow></mtd><mtd><mo>…</mo></mtd><mtd><mrow><msub><mi>m</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mi>Δ</mi><msub><mi>K</mi><mn>1</mn></msub></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mrow><mi>Δ</mi><msub><mi>K</mi><mi>n</mi></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mi>Δ</mi><msub><mi>K</mi><mn>1</mn></msub></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mrow><mi>Δ</mi><msub><mi>K</mi><mi>n</mi></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where M is the tune response matrix and n is the number of quadrupoles [ 9 9 ]. When the tune is adjusted in a small range, the quadrupole adjustment can be obtained from the pseudo-inverse M - of the tune response matrix M, i.e. <math display="block" id="M7"><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mi>Δ</mi><msub><mi>K</mi><mn>1</mn></msub></mrow></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mrow><mi>Δ</mi><msub><mi>K</mi><mi>n</mi></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><msup><mstyle mathvariant="bold" mathsize="normal"><mi>M</mi></mstyle><mo>−</mo></msup><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mi>Δ</mi><msub><mi>ν</mi><mi>x</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mi>Δ</mi><msub><mi>ν</mi><mi>y</mi></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> This matrix method is approximate within a limited tune adjustment range. In addition, the control of Twiss parameters is not considered. To adjust the tune in a larger range, an alternative scheme is developed, which adopts a step-by-step method with the matching of beam optics. This scheme can adjust the tune while constraining the Twiss parameters. The step-by-step method is shown in Fig. 1 Fig. 1 , and it has the following properties: • The nominal lattice is used as the starting point for either increasing or decreasing the tune; • A proper tune change step is set to ensure the matching is accurate in each adjustment step; • The new lattice with adjusted quadrupole strengths from the previous step is used as the starting point for the next step; • After the adjustment process is completed, the adjusted quadrupole strengths as functions of the tune change are fitted by polynomials. With this scheme, the quadrupole strengths vary monotonically with the tune change. The adjustment is more accurate compared with the matrix method. 3 Tune Knob Development for the HLS-II Storage Ring 1 3.1 Tune Knob Development 2 Main parameters of the HLS-II storage ring are listed in Table 1 Table 1 . The linear lattice is shown in Fig. 1 Fig. 1 , which consists of four 4.0 m long straight sections and four 2.3 m long ones [ 10 10 ]. A total of 32 quadrupoles are grouped into eight families labeled from K1 to K8, as shown in the figure. 1 1 Parameters 1 1 Value 1 1 Circumference (m) 1 1 66.13 1 1 Energy (MeV) 1 1 800 1 1 Natural emittance (nm ⋅ rad) 1 1 38 1 1 RF frequency (MHz) 1 1 204 1 1 Harmonic number 1 1 45 1 1 Horizontal tune 1 1 4.4448 1 1 Vertical tune 1 1 2.3598 1 1 Momentum compaction factor 1 1 0.0158 The injection section of the HLS-II storage ring consists of four kicker magnets, and their corresponding locations are indicated in Fig. 2 Fig. 2 . To reduce the impact on beam injection, the Twiss parameter changes of the injection section should be small. This is achieved by the following processes. • Only the quadrupoles labeled K5 to K8 in each family are adopted by the tune knob. • Twiss parameters at the two outer kicker locations are constrained. The first condition maintains the quadrupole strengths in the injection section, whereas the second constrains the Twiss parameters of the boundary of injection section. Under these two conditions, the Twiss parameters within the injection section are maintained for each adjustment step. A code based on the accelerator simulation software MAD-X [ 11 11 ] has been developed, in which the step-by-step method is carried out using the matching module of the software. The quadrupole strengths of the four families (K5–K8) are set as the independent adjustment variables. To ensure the accuracy of t he matching, the tune adjustment step in simulation is set to Δ ν =0.001. The transverse tunes of the nominal lattice are ν x =4.4448 and ν y =2.3598. To avoid critical resonance lines, the tune adjustment range in the simulation is set to be from -0.08 to 0.04 horizontally, and from -0.10 to 0.08 vertically. To simplify the simulation process, the tune is adjusted separately in the horizontal and vertical plane. This means that when the tune in one plane is adjusted, the tune in the other plane is maintained. The quadrupole strength adjustment with respect to the tune change is shown in Fig. 3 Fig. 3 . The changes of K5 to K8 as functions of the tune change are fitted using fifth-order polynomial functions given by <math display="block" id="M8"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>K</mi><mrow><mi>i</mi><mi>x</mi></mrow></msub><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>a</mi><mrow><mi>i</mi><mn>5</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>x</mi><mn>5</mn></msubsup><mo>+</mo><msub><mi>a</mi><mrow><mi>i</mi><mn>4</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>x</mi><mn>4</mn></msubsup><mo>+</mo><msub><mi>a</mi><mrow><mi>i</mi><mn>3</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>x</mi><mn>3</mn></msubsup><mo>+</mo><msub><mi>a</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>x</mi><mn>2</mn></msubsup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mi>a</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mi>Δ</mi><msub><mi>ν</mi><mi>x</mi></msub><mo>+</mo><msub><mi>a</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub><mn>,</mn><mtext> </mtext><mi>i</mi><mo>=</mo><mn>5,</mn><mo>…</mo><mn>,8,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>K</mi><mrow><mi>i</mi><mi>y</mi></mrow></msub><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msub><mi>b</mi><mrow><mi>i</mi><mn>5</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>y</mi><mn>5</mn></msubsup><mo>+</mo><msub><mi>b</mi><mrow><mi>i</mi><mn>4</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>y</mi><mn>4</mn></msubsup><mo>+</mo><msub><mi>b</mi><mrow><mi>i</mi><mn>3</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>y</mi><mn>3</mn></msubsup><mo>+</mo><msub><mi>b</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mi>Δ</mi><msubsup><mi>ν</mi><mi>y</mi><mn>2</mn></msubsup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><msub><mi>b</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mi>Δ</mi><msub><mi>ν</mi><mi>y</mi></msub><mo>+</mo><msub><mi>b</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub><mn>,</mn><mtext> </mtext><mi>i</mi><mo>=</mo><mn>5,</mn><mo>…</mo><mn>,8.</mn></mrow></mtd></mtr></mtable></mrow></math> The fitted coefficients in the above equations are listed in Table 2 Table 2 . If ν x and ν y are both adjusted, the required quadrupole adjustment can be obtained by a combination of Eq. (5) as 1 1   1 1 a i 5 1 1 a i 4 1 1 a i 3 1 1   1 1 a i 2 1 1 a i 1 1 1 a i 0 1 1   1 1 b i 5 1 1 b i 4 1 1 b i 3 1 1   1 1 b i 2 1 1 b i 1 1 1 b i 0 1 1 K5 1 1 1.4030×10 3 1 1 1.6145×10 2 1 1 -1.5175×10 1 1 1   1 1 -5.7850 1 1 -1.0439 1 1 -3.8966×10 -6 1 1   1 1 1.0550×10 3 1 1 2.3595×10 2 1 1 3.8081×10 1 1 1   1 1 9.1661 1 1 5.3095×10 -1 1 1 5.9559×10 -5 1 1 K6 1 1 -1.3495×10 3 1 1 -1.4952×10 2 1 1 1.6001×10 1 1 1   1 1 5.6859 1 1 1.7414 1 1 3.7455×10 -6 1 1   1 1 -1.4233×10 2 1 1 -1.7375×10 1 1 1 5.0443 1 1   1 1 -8.4617×10 -1 1 1 8.0608×10 -1 1 1 -5.0335×10 -6 1 1 K7 1 1 -5.3473×10 2 1 1 -5.6627×10 1 1 1 7.0959 1 1   1 1 2.7758 1 1 -8.8913×10 -1 1 1 1.5005×10 -6 1 1   1 1 -1.3083×10 2 1 1 -2.6241×10 1 1 1 -5.6024 1 1   1 1 -2.1746 1 1 -1.3282 1 1 -6.3048×10 -6 1 1 K8 1 1 1.4314×10 3 1 1 1.5070×10 2 1 1 -1.9910×10 1 1 1   1 1 -6.7700 1 1 1.6443 1 1 -4.0451×10 -6 1 1   1 1 -1.4498×10 2 1 1 -5.1852×10 1 1 1 -4.6449 1 1   1 1 -1.7455 1 1 -7.7932×10 -1 1 1 -1.2319×10 -5 <math display="block" id="M9"><mrow><mi>Δ</mi><msub><mi>K</mi><mi>i</mi></msub><mo>=</mo><mi>Δ</mi><msub><mi>K</mi><mrow><mi>i</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>K</mi><mrow><mi>i</mi><mi>y</mi></mrow></msub><mn>,</mn><mtext> </mtext><mi>i</mi><mo>=</mo><mn>5,</mn><mo>…</mo><mn>,8.</mn></mrow></math> 3.2 β Function Variations 2 The β function variations during the tune adjustment are studied. When the tune is adjusted by the knob, the β function changes with respect to the nominal values are calculated. The results are plotted in Figs. 4 Figs. 4 and 5 5 for adjusting the tune in the horizontal and vertical plane, respectively. When ν x is adjusted in the range of -0.08 to 0.04, the largest β x change is about 0.2 m and the largest β y change is about 0.4 m. When ν y is adjusted in the range of -0.10 to 0.08, the largest β x change is about 0.8 m and the largest β y change is about 3 m. The overall β function change is larger when adjusting ν y than adjusting ν x . The results also show that the β functions change symmetrically, which means the lattice symmetry is preserved during the tune adjustment process. The β function change in the injection section is very small, which is consistent with the design requirement to keep the injection unaffected by the tune knob. The horizontal η function is also calculated and the largest η function deviation is about 0.5 m, which has little impact on normal operation. 3.3 Effect on Chromaticities 2 The tune knob affects chromaticities. In a storage ring, the corrected chromaticities are expressed by [ 8 8 ] <math display="block" id="M10"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>C</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><msub><mi>β</mi><mi>x</mi></msub><mo stretchy="false">[</mo><mi>K</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><mi>S</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi>η</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mtext>d</mtext><mi>s</mi><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>C</mi><mi>y</mi></msub><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><msub><mi>β</mi><mi>y</mi></msub><mo stretchy="false">[</mo><mi>K</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mi>S</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi>η</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mtext>d</mtext><mi>s</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where S is the sextupole strength and η x is the horizontal η function. When the tune is adjusted by the knob, the quadrupole strength K , together with β and η functions are affected, resulting in chromaticity changes. The chromaticities of the nominal HLS-II lattice are corrected to approximately one and three in horizontal and vertical planes, respectively. The chromaticity changes owing to tune adjustment are plotted in Fig. 6 Fig. 6 . The result shows that when ν x is adjusted from -0.08 to 0.04, the horizontal and vertical chromaticities vary in the range of approximately 0.7 to 1.2 and 2.6 to 3.0, respectively. When ν y is adjusted from -0.10 to 0.08, the horizontal and vertical chromaticities vary in the range of approximately 0.5 to 1.5 and 1.6 to 3.8, respectively. In practical tune adjustment, the tune is usually adjusted within a smaller range than in the simulation. The chromaticities remain in the positive range, thus avoiding the head–tail instability. 3.4 Tune Knob Survey 2 The tune knob is surveyed using the Accelerator Toolbox (AT) [ 12 12 ], in which both ν x and ν y are adjusted according to Eq. (6). The adjustment range is from -0.08 to 0.04 for Δ ν x and from -0.10 to 0.08 for Δ ν y , with the step in both planes set to Δ ν =0.005. The tunes are calculated at each point and the differences between the set value and the calculated values (Δ ν check -Δ ν set ) are plotted in Fig. 7 Fig. 7 . The tune deviations on the Δ ν x =0 and Δ ν y =0 axes are very small, showing the accuracy of tune knob fitting in Eq. (5). The deviation becomes a little larger when the tune is adjusted farther off-axis, yet the tune deviations in most regions are smaller than 10 -3 . The result shows good agreement between the set value and simulated values. 4 Online Measurement and Calibration 1 4.1 Online Measurement 2 The tune knob is measured online in the HLS-II storage ring. The quadrupole strengths are set through the machine control system based on the Experimental Physics and Industrial Control System (EPICS) [ 13 13 - 15 15 13-15 ]. To measure the tune, the beam is excited by a modulated signal with certain central frequency and bandwidth. The tune spectrum is read from the spectrometer. To obtain better measurement accuracy, the Lorentzian function is used to fit the spectrum [ 16 16 ]. In the experiment, the tune knob coefficients are set according to simulation results. The tune adjustment step is set as Δ ν =0.005, whereas the adjustment range is from -0.07 to 0.04 for Δ ν x and from -0.08 to 0.07 for Δ ν y . The measurement is carried out within a narrower range than in the simulation to avoid the half-integer resonance. Similar to the simulation procedure, the tune knob measurement in the horizontal and vertical plane is performed separately. The measurement is performed in single-bunch mode to eliminate multi-bunch effects and obtain a cleaner tune spectrum. The measured tune changes and the set values in the tune knob are plotted in Fig. 8 Fig. 8 . The residual tune variations (Δ ν meas -Δ ν set ) are also shown in the figure. When adjusting the tune in one plane, the adjusted tune changes linearly with respect to the set value, and the tune in the other plane remains unchanged. The measured and set tune changes are fitted by linear relations as <math display="block" id="M11"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mtext>meas</mtext></mrow></msub><mo>=</mo><mn>0.9723</mn><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mtext>set</mtext></mrow></msub><mo>+</mo><mn>1.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>y</mi><mn>,</mn><mtext>meas</mtext></mrow></msub><mo>=</mo><mn>0.9721</mn><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>y</mi><mn>,</mn><mtext>set</mtext></mrow></msub><mo>+</mo><mn>3.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The fitted linear coefficients of the adjusted tunes are not equal to one. One reason is the difference between a real storage ring and its model. Another factor comes from the hysteresis effect of the quadrupoles [ 17 17 ]. During tune adjustment, some of the quadrupole strengths increase and some decrease. The decreasing strengths go along a minor hysteresis loop whereas the increasing strengths go along the main hysteresis curve. The different trend of decreasing quadrupole strengths leads to a deviation of quadrupole adjustment. To make the tune knob more accurate, a calibration is needed. 4.2 Tune Knob Calibration 2 The tune knob calibration is performed by modifying the set value of tune adjustment to make the real tune change closer to the target value. This is realized by resetting the tune adjustment to <math display="block" id="M12"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mtext>calib</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mtext>set</mtext></mrow></msub></mrow><mrow><mn>0.9723</mn></mrow></mfrac><mn>,</mn><mtext> </mtext><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>y</mi><mn>,</mn><mtext>calib</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>y</mi><mn>,</mn><mtext>set</mtext></mrow></msub></mrow><mrow><mn>0.9721</mn></mrow></mfrac></mrow></math> according to the fitted linear coefficients in Eq. (8). Δ ν x ( y ),set is the designated tune adjustment and Δ ν x ( y ), calib is the calibrated value set to the machine. After calibration, the tune knob is measured under the same condition as before. The relations between the measured tune changes and set values become <math display="block" id="M13"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mtext>meas</mtext></mrow></msub><mo>=</mo><mn>0.9981</mn><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>x</mi><mn>,</mn><mtext>set</mtext></mrow></msub><mo>+</mo><mn>1.6</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>y</mi><mn>,</mn><mtext>meas</mtext></mrow></msub><mo>=</mo><mn>0.9994</mn><mi>Δ</mi><msub><mi>ν</mi><mrow><mi>y</mi><mn>,</mn><mtext>set</mtext></mrow></msub><mo>+</mo><mn>3.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The linear coefficients get closer to one than before, showing that the tune knob becomes more precise after calibration. The residual tune variation (Δ ν meas -Δ ν set ) is plotted in Fig. 9 Fig. 9 . The result shows that the maximum residual tune variation is approximately 1.5×10 -3 , which is smaller than that before calibration. The tune knob after calibration meets the requirements for tune compensation and machine studies. 5 Coupling Measurement for the HLS-II Storage Ring 1 The tune knob is a useful tool for machine studies. In the HLS-II storage ring, the betatron coupling is measured using the tune knob. The coupling coefficient C is used to evaluate the betatron coupling effects of a storage ring [ 18 18 , 19 19 ]. When measuri ng the coupling coefficient, the betatron tunes are adjusted across a difference resonance line Q x - Q y = l , where Q x and Q y are the horizontal and vertical tunes, respectively, and l is an integer. For a storage ring, the measured tunes are the eigen-tunes ν + and ν - expressed by [ 20 20 , 21 21 ] <math display="block" id="M14"><mrow><msub><mi>ν</mi><mo>±</mo></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>ν</mi><mi>x</mi></msub><mo>+</mo><msub><mi>ν</mi><mi>y</mi></msub><mo>±</mo><msqrt><mrow><msup><mi>Δ</mi><mn>2</mn></msup><mo>+</mo><mo>|</mo><mi>C</mi><msup><mo>|</mo><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></msqrt><mn>,</mn></mrow></math> where ν x and ν y are the free tunes without coupling, Δ=| ν x - ν y | is their difference, and | C | is the coupling coefficient. When the tunes are away from the difference resonance, the measured eigen-tunes are close to the uncoupled tunes. If the tune is near the difference resonance, the measured eigen-tunes become ν ± = ν x,y ±1/2| C |, and the coupling coefficient is obtained from the minimum separation between the two eigen-fractional tunes: <math display="block" id="M15"><mrow><mo>|</mo><mi>C</mi><mo>|</mo><mo>=</mo><munder><mstyle mathsize="140%" displaystyle="true"><mrow><mi>min</mi></mrow></mstyle><mrow><mo>|</mo><msub><mi>ν</mi><mi>x</mi></msub><mo>−</mo><msub><mi>ν</mi><mi>y</mi></msub><mo>|</mo><mo>→</mo><mn>0</mn></mrow></munder><mo>|</mo><msub><mi>ν</mi><mo>+</mo></msub><mo>−</mo><msub><mi>ν</mi><mo>−</mo></msub><mo>|</mo><mn>.</mn></mrow></math> For the HLS-II storage ring, the nominal tune is around Q x =4.4448 and Q y =2.3598. When measuring the betatron coupling, the tune knob is used to decrease the horizontal tune to cross the difference resonance line Q x - Q y =2. The measured fractional eigen-tunes are plotted in Fig. 10 Fig. 10 . The coupling coefficient | C | is calculated as approximately 1.26%. 6 Summary 1 The development of a tune knob for the HLS-II storage ring was presented. The tune knob adopts an improved scheme based on global adjustment of quadrupole families, preserving the lattice symmetry. To reduce the impact on beam injection, the Twiss parameters of the injection section remain unchanged. This scheme is applicable for compact storage rings with small phase advances in one cell. Online measurement and calibration was also performed. To demonstrate its application as a useful machine study tool, the betatron coupling was measured using the tune knob. The tune knob works well in the designated range and fully meets the design requirements for tune compensation. The potentials of the tune knob for user operation and machine studies can be further explored in future work. References H. Wiedemann , ( 2007 ). Particle accelerator physics (Vol. 314 ). Berlin : Springer . Particle accelerator physics B. Li , J. Li , W. Xu et al ., Beta Function Matching and Tune Compensation for HLS-II Insertion Devices . Proceedings of IPAC 2015 , Richmond, VA, USA , TUPJE015 . doi: 10.18429/JACoW-IPAC2015-TUPJE015 http://doi.org/10.18429/JACoW-IPAC2015-TUPJE015 Beta Function Matching and Tune Compensation for HLS-II Insertion Devices H. Hao , S. Mikhailov , Y. Wu et al ., Characterizing Betatron Tune Knobs at Duke Storage Ring . Proceedings of IPAC 2015 , Richmond, VA, USA , MOPMA053 . doi: 10.18429/JACoW-IPAC2015-MOPMA053 http://doi.org/10.18429/JACoW-IPAC2015-MOPMA053 Characterizing Betatron Tune Knobs at Duke Storage Ring W. Li , H. Hao , J. Li et al ., Measuring Duke Storage Ring Lattice Using Tune Based Technique . Proceedings of IPAC 2015 , Richmond, VA, USA , MOPJE003 . doi: 10.18429/JACoW-IPAC2015-MOPJE003 http://doi.org/10.18429/JACoW-IPAC2015-MOPJE003 Measuring Duke Storage Ring Lattice Using Tune Based Technique J. Li , S. Mikhailov , Y. 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Top-ac . 8 ( 1 ), 014001 ( 2005 ). doi: 10.1103/PhysRevSTAB.8.014001 http://doi.org/10.1103/PhysRevSTAB.8.014001 Measurement of global betatron coupling with skew quadrupole modulation D. Sagan , and D. Rubin , Linear analysis of coupled lattices . Phys. Rev. Spec. Top-ac . 2 ( 7 ), 074001 ( 1999 ). doi: 10.1103/PhysRevSTAB.2.074001 http://doi.org/10.1103/PhysRevSTAB.2.074001 Linear analysis of coupled lattices G. Wang , Y. Li , T. Shaftan et al . NSLS-II Storage Ring Coupling Measurement and Correction . Proceedings of IPAC 2015 , Richmond, VA, USA , TUPHA009 . doi: 10.18429/JACoW-IPAC2015-TUPHA009 http://doi.org/10.18429/JACoW-IPAC2015-TUPHA009 NSLS-II Storage Ring Coupling Measurement and Correction The authors would like to thank Prof. Y. K. Wu and Dr. H. Hao from the Duke Free Electron Laser Lab (DFELL) for their previous work on developing the tune knob. We would also like to thank the scientists and engineers at NSRL who give us valuable suggestions and helped us prepare the machine study. 10.1007/s41365-018-0513-y This work was supported by the National Natural Science Foundation of China (Nos. 11375177 and 11705200). 2018-06-08 2018-09-06 2018-10-01 2018-11-17 2018-12-01 2021-12-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0513-y http://dx.doi.org/10.1007/s41365-018-0513-y . 2018 Nuclear Science and Techniques 12 29