Improving Touschek lifetime and synchrotron frequency spread by passive harmonic cavity in the storage ring of SSRF

Improving Touschek lifetime and synchrotron frequency spread by passive harmonic cavity in the storage ring of SSRF 1 2 first-author 1 1 1 1 1 1 2 corresp Corresponding author. E-mail address: zhaozhentang@sinap.ac.cn Corresponding author. E-mail address: zhaozhentang@sinap.ac.cn zhaozhentang@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Zhangjiang campus, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Zhangjiang campus, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Beam lifetime of a synchrotron is dominated by Touschek scattering. In the beamline Phase II project of Shanghai synchrotron radiation facility (SSRF), a passive third harmonic cavity is to be installed for bunch-lengthening and instability-suppressing. In this paper, the beam dynamics of the cavity is investigated. The parameters of passive operation are optimized to cancel the slope of RF voltage and lengthen the bunches. The Touschek lifetime increases are estimated for optimum and non-optimum voltage flattening. A tolerance of the operation is studied in case that there is a shift on detuning angle. The effect caused by reduction of harmonic voltage generated by lengthened bunch distribution is also estimated using iteration method. An increase of synchrotron frequency spread due to nonlinearity of the voltage giving to the bunch is found by using tracking simulation. This spread can help in damping coupled bunch instability through Landau damping. Landou damping passive harmonic cavity SSRF 1 Introduction 1 Shanghai synchrotron radiation facility (SSRF) has a low emittance of 3.9 nm-rad [ 1 1 ] . Its beam lifetime is dominated by the Touschek scattering effect caused by the momentum deviation transferred from elastic collisions of electrons in the bunch at higher bunch charge [ 2 2 ] . Higher harmonic cavity is an effective method to increase the beam lifetime without affecting the brightness [ 3 3 - 5 5 3-5 ] . It can suppress coupled bunch instabilities [ 6 6 - 8 8 6-8 ] and single bunch instabilities through increased synchrotron frequency spread [ 9 9 ] . One can operate a higher harmonic cavity in active or passive mode. An active mode needs external RF power supplies. In passive operation, a harmonic cavity voltage is induced by the beam itself, hence no need of an external RF power source, being compact and economic. The passive cavities are widely used in third generation light sources [ 10 10 - 12 12 10-12 ] . Therefore, in Phase-II project of SSRF, we have planned to install a passive third harmonic cavity to the storage ring. In this paper, longitudinal beam dynamics with passive cavity is reviewed including with investigation on the lifetime improvement, factor and tolerance of bunch lengthening. Due to passive operation, the fields excited in the harmonic cavity depend on the bunch distribution. The influence of lengthening bunch on the lifetime improvement is also discussed. In the last session, an increase in synchrotron frequency spread is studied using tracking codes for synchrotron motion. 2 Longitudinal beam dynamics with passive cavity 1 The total voltage seen by the beam can be defined by[ 13 13 ] <math display="block" id="M1"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mrow><mtext>rf</mtext></mrow></msub><mi>sin</mi><mo stretchy="false">(</mo><msub><mi>φ</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>ω</mi><mrow><mtext>rf</mtext></mrow></msub><mi>τ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><msub><mi>I</mi><mtext>b</mtext></msub><mi>F</mi><msub><mi>R</mi><mtext>s</mtext></msub><mtext> cos</mtext><msub><mi>ψ</mi><mtext>h</mtext></msub><mi>cos</mi><mtext>(</mtext><msub><mi>ψ</mi><mtext>h</mtext></msub><mtext>-n</mtext><msub><mi>ω</mi><mrow><mtext>rf</mtext></mrow></msub><mi>τ</mi><mtext>),</mtext></mrow></math> where, τ is the relative time deviation, V rf is the main RF voltage, φ s is the synchronous phase of the main RF cavity, ω rf is the angular frequency, I b is beam current, R s is the shunt impedance; ψ h is detuning angle of the harmonic cavity; n is the harmonic number, and F is the bunch form factor given by <math display="block" id="M2"><mrow><mi>F</mi><mo>=</mo><mtext>exp</mtext><mrow><mo>[</mo><mrow><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>n</mi><msub><mi>ω</mi><mrow><mtext>rf</mtext></mrow></msub><msub><mi>σ</mi><mtext>τ</mtext></msub></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> where σ τ is the RMS bunch length; According to Eq. (1), the phase of harmonic voltage and the amplitude are related to detuning angle. To optimize bunch lengthening, the following conditions should be satisfied: <math display="block" id="M3"><mrow><mi>sin</mi><msub><mi>φ</mi><mtext>s</mtext></msub><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup></mrow><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mfrac><mrow><msub><mi>U</mi><mn>0</mn></msub></mrow><mrow><msub><mi>V</mi><mrow><mtext>rf</mtext></mrow></msub></mrow></mfrac><mo>,</mo></mrow></math> <math display="block" id="M4"><mrow><mi>tan</mi><msub><mi>ψ</mi><mtext>h</mtext></msub><mo>=</mo><mo>−</mo><mi>n</mi><mi>cot</mi><msub><mi>φ</mi><mtext>s</mtext></msub><mo>,</mo></mrow></math> <math display="block" id="M5"><mrow><msub><mi>R</mi><mtext>s</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mrow><mtext>rf</mtext></mrow></msub><mi>sin</mi><msub><mi>φ</mi><mtext>s</mtext></msub></mrow><mrow><mn>2</mn><msub><mi>I</mi><mtext>b</mtext></msub><mi>F</mi><msup><mi>n</mi><mn>2</mn></msup><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><msub><mi>ψ</mi><mtext>h</mtext></msub></mrow></mfrac><mo>.</mo></mrow></math> Parameters of the SSRF storage ring are given in Table 1 Table 1 . From Eqs.(3)–(5), passive operation parameters for optimum bunch lengthening can be calculated as R s =21.32 MΩ and ψ h = 96.86°. 1 1 Parameters 1 1 Value 1 1 Energy (GeV) 1 1 3.5 1 1 Circumference (m) 1 1 432 1 1 Beam current (mA) 1 1 300 1 1 RF frequency (MHz) 1 1 500 1 1 Harmonic number 1 1 720 1 1 Momentum compaction 1 1 0.00042 1 1 Energy spread 1 1 0.00098 1 1 Radiation loss (MeV) 1 1 1.44 1 1 Main RF voltage (MV) 1 1 4.8 1 1 Nominal bunch length (mm) 1 1 3.8 In passive operation, the resonant frequency of harmonic cavity should be detuned as Eq. (6): <math display="block" id="M6"><mrow><mi>tan</mi><msub><mi>ψ</mi><mtext>h</mtext></msub><mo>=</mo><mn>2</mn><mi>Q</mi><mo stretchy="false">(</mo><mi>δ</mi><mi>f</mi><mo>/</mo><msub><mi>f</mi><mtext>r</mtext></msub><mo stretchy="false">)</mo><mo>,</mo></mrow></math> where, δf = f r −3 f rf is detuning frequency, with f r being the resonant frequency of the cavity and f rf being the main RF frequency, and Q is the quality factor, which is very high for a superconducting cavity. When δf ≈ fr / Q, the induced voltage can be given by [ 14 14 ] <math display="block" id="M7"><mrow><msub><mi>V</mi><mtext>h</mtext></msub><mo>≈</mo><msub><mi>I</mi><mtext>b</mtext></msub><mo stretchy="false">(</mo><msub><mi>R</mi><mtext>s</mtext></msub><mo>/</mo><mi>Q</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>f</mi><mtext>r</mtext></msub><mo>/</mo><mi>δ</mi><mi>f</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> so that the flattened RF potential at the synchrotron phase is established and thus the bunch is lengthened. For Rs/Q = 180 Ω, δf =53.43 kHz is needed to gain the voltage of 1.52 MV. 3 Bunch lengthening and the lifetime improvement 1 To investigate how much the bunch can be lengthened, the longitudinal charge density distribution of the bunch is determined from Gaussian energy distribution in the potential well. The longitudinal density distribution is given by [ 13 13 ] <math display="block" id="M8"><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub><mtext>Exp</mtext><mrow><mo>[</mo><mrow><mi>Φ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mo stretchy="false">(</mo><mi>α</mi><msub><mi>σ</mi><mtext>e</mtext></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mo>,</mo></mrow></math> where, ρ 0 is a normalization constant, Φ ( z ) is the potential defined by line integral of the voltage in Eq.(1) [ 13 13 ] , α is momentum compaction, and σ e is the energy spread. The RF potential is calculated from integral of the RF voltage. Touschek loss rate is proportional to probability for scattering beyond the RF acceptance and the square of the volume charge density in the bunch. The change in volume density can be simply calculated from the longitudinal density distribution in Eq. (8). The scattering probability is inversely proportional to the square of the RF acceptance, δ rf . Therefore, an improvement factor defined by a ratio of the lifetimes with and without harmonic voltage can be calculated by Eq. (9) [ 13 13 ] . <math display="block" id="M9"><mrow><mi>R</mi><mo>=</mo><mfrac><mrow><msubsup><mi>δ</mi><mrow><mtext>hc</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>δ</mi><mrow><mtext>rf</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mtext>d</mtext><mi>z</mi><msubsup><mi>ρ</mi><mrow><mtext>rf</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><mtext>d</mtext><mi>z</mi><msubsup><mi>ρ</mi><mrow><mtext>hc</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mrow></mfrac></mrow></math> The lifetime improvement factor in Eq.(9) depends significantly on the distribution of the particles. According to Eq. (1), particle distribution in passive operation is mainly corresponding to beam current, shunt impedance, and detuning angle. When shunt impedance and detuning angle satisfy the flat voltage condition, the potential well becomes flatten and bunch distribution becomes broader. The Touschek lifetime consequently improves by a factor of 3.77, when the shunt impedance is fixed by the beam intensity of 300 mA, as shown in Fig.1 Fig.1 . A change on beam current can greatly perturb the bunch lengthening since amplitude of harmonic voltage changes. Below 300 mA of beam intensity case, the lifetime improvement decreases rapidly; while over 300 mA of beam intensity, it rises until a peak valve of 4.47, where it begins to drop sharply. In Eqs. (2)–(4), the only parameter that depends on the beam current is the shunt impedance. The easiest way to achieve the desired bunch lengthening effect with fluctuated beam intensity is to install a variable resistor on the harmonic cavity and adjust the proper shunt impedance by observing the intensity of the electron bunches. When the shunt impedance is not adjustable, one can only adjust the detuning angle instead. From Eq.(1), the detuning angle is related to both amplitude and phase of harmonic RF voltage. As shown in Fig. 2(a) Fig. 2(a) , the detuning angle shift affects the particle distribution. When the detuning angle is slightly tuned from 96.86° of the flattened potential condition, the distribution becomes broader and transforms to two bunch-lets. This condition, known as overstretch condition, can improve the Touschek lifetime until reaching the maximum factor of 4.16 at detuning angle of 97.22° as shows in Fig.2(b) Fig.2(b) . When detuning angle is tuned further, there is a significant distortion on the particle distribution, leading to a drop on the improvement factor. When the detuning angle is below the flattened potential condition, there will be much less shifting effect on distribution than that in overstretch condition. The distribution is narrower than that with flattened potential condition and even turns to be shortened when the detuning is less than 90°. As a result, the lifetime improvement factor decreases. From Fig. 2(b) Fig. 2(b) , the decrease of improvement factor is sensitivity to the detuning angle. The distortion of particle distribution caused by detuning shift can greatly disturb the lengthening effect of the harmonic cavity. From Eq. (6), it is estimated that to maintain the lifetime improvement to be more than 80% of that in flattened potential case, the detuning frequency should be 47.21–57.44 kHz. Regarding the stable beam intensity in top-up mode operation of SSRF, small fluctuation of beam current can be handled by varying detuning frequency. Thus this issue is far from our concern. On the other hand, the effect of lengthened bunch distribution on passive operation is the one we are interested and will be discussed in next section. 4 Investigation on lengthened bunch effects 1 The influence of the bunch shape on the excitation of the fields in the harmonic cavity is taken into account by the bunch form factor in Eq.(2). From previous calculation, the effect of lengthened bunch was neglected. The bunch form factor was estimated as an original short bunch even when the bunch was much lengthened. In this section, the influence of lengthened bunch distribution on the lengthening effect is investigated. To acquire the efficient bunch distribution after the bunch was lengthened, the self-consistent method was used. We started the calculation to obtain an initial bunch distribution as we did in Sections 2 and 3 using the form factor in Eq. (2). Assuming that the distribution is in Gaussian form, the RMS bunch length could be calculated from full width half maximum of the distribution divided by the factor of 2.3548. By substituting the calculated RMS bunch length to Eq. (2), the new harmonic voltage and the new potential could be found. Then we achieved the new longitudinal distribution. A new form factor from this longitudinal distribution was used to re-calculate the parameters again. Within 10 times of iteration, the parameters converged and saturated. The form factor and Touschek lifetime improvement factor resulted from the iteration method in flattened potential case are shown in Fig.3 Fig.3 . The high value of form factor which is close to 1 indicates the shorter bunch and the lower value refers to broader bunch. Figure 4(a) Figure 4(a) shows the distribution of a lengthened bunch, together with the bunch distribution calculated using the original bunch in flattened potential condition. Because the harmonic voltage corresponds to the form factor related to the bunch shape, an increase in bunch length can limit the bunch lengthening effect through a decrease of voltage of harmonic RF. The peak harmonic voltage generated by lengthened bunch distribution drops to 1.41 MV, hence the degraded lengthening efficiency and the reduced lifetime improvement factor from 3.77 to 3.05. To reach the flattened potential distribution, the harmonic voltage amplitude of 1.74 MV is needed to compensate the reduced harmonic voltage due to the lengthened bunch. As discussed in previous section, increasing shunt impedance is the simplest method to provide desired harmonic voltage without affecting the detuning angle. By increasing shunt impedance to 24.51 MΩ, the overstretch distribution can be obtained from original short bunch calculation. After the bunch-lengthening reaches saturation, the broad distribution in flattened potential condition is shown in Fig.4(b) Fig.4(b) . At a fixed shunt impedance, the harmonic voltage can be controlled only by adjusting detuning angle through resonant frequency of the cavity. With shunt impedance of 21.32 MΩ from previous flattened potential condition, the desired harmonic voltage can be reached by increasing detuning angle to 97.89°. In this case, the bunch distribution is distorted and shifted due to detuning angle deviation. Therefore, the broad distribution from flattened potential cannot be achieved in this situation. At fixed shunt impedance, the more bunch distribution is lengthened, the more distortion of bunch distribution appears. This effect causes a limitation on the lifetime improvement especially when it reaches high value. Figure 5 Figure 5 shows the Touschek lifetime improvement factors of the original and lengthened bunches. At detuning angle of 97.66°, the improvement factor from lengthened bunch iterative method is maximized at 3.81,which is still 8.5% less than 4.16 from the original bunch. 5 Synchrotron frequency spread investigation 1 A nonlinear acceleration field from third harmonic cavity gives a large spread in synchrotron frequency. To investigate an increase of synchrotron frequency spread , the distribution in the ( φ , ε ) phase space was generated by tracking. The bunch was considered as a macro-particle and difference equations of longitudinal motion were developed corresponding to the voltage and the phase in the main and harmonic cavities [ 15 15 ] . Difference equations for the synchrotron oscillations of each bunch can be expressed as <math display="block" id="M10"><mrow><msub><mi>ε</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>ε</mi><mi>i</mi></msub><mo>+</mo><msub><mi>V</mi><mrow><mtext>rf</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>φ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>U</mi><mn>0</mn></msub><mo>−</mo><mn>2</mn><mfrac><mrow><msub><mi>T</mi><mn>0</mn></msub></mrow><mrow><msub><mi>τ</mi><mi>ε</mi></msub></mrow></mfrac><msub><mi>ε</mi><mi>i</mi></msub><mo>,</mo></mrow></math> <math display="block" id="M11"><mrow><msub><mi>φ</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>φ</mi><mi>i</mi></msub><mo>+</mo><mfrac><mrow><mi>α</mi><msub><mi>T</mi><mn>0</mn></msub></mrow><mrow><msub><mi>E</mi><mn>0</mn></msub></mrow></mfrac><msub><mi>ε</mi><mi>i</mi></msub><mo>,</mo></mrow></math> where, ε i is the relative beam energy deviation for the i th turn of calculation, φ i is the bunch phase with respect to the nominal synchronous phase, T 0 is the revolution period, and E 0 is the beam energy. Synchrotron frequency of each particle can be estimated from Fourier transform of the phase oscillation of all particles in the bunch using tracking method by neglecting radiation damping in the last term of Eq.(10). In this part, the specific effect of lengthen bunch is neglected but it can be considered as an imperfection of harmonic voltage. As shown in Fig. 6 Fig. 6 , for the SSRF storage ring, the synchrotron frequency shifts in downwards direction, and the synchrotron frequency spread increases with the harmonic voltage. The largest spread occurs at flat voltage condition. These results agree with analytical method reported earlier [ 16 16 ] . Since the harmonic cavity is tuned above the RF beam harmonic, Robinson instability can be excited. The growth rate has been studied concerning on this instability. For small amplitude oscillation, the synchrotron frequency is [ 17 17 ] <math display="block" id="M12"><mrow><mfrac><mrow><msub><mi>ω</mi><mtext>s</mtext></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>ω</mi><mrow><mtext>s0</mtext></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><msqrt><mrow><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mn>6</mn></mfrac></mrow></msqrt><mfrac><mrow><msub><mi>ω</mi><mrow><mtext>rf</mtext></mrow></msub><mi>τ</mi></mrow><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msqrt><mn>2</mn></msqrt><mo stretchy="false">)</mo></mrow></mfrac><msqrt><mrow><mfrac><mrow><mi>cos</mi><msub><mi>φ</mi><mtext>s</mtext></msub></mrow><mrow><mi>cos</mi><msub><mi>φ</mi><mrow><mtext>s0</mtext></mrow></msub></mrow></mfrac></mrow></msqrt><mo>,</mo></mrow></math> where ω s0 is the synchrotron frequency at zero amplitude when there is no higher harmonic voltage, and K (2 −1/2 )=1.854 is the complete elliptic integral for the first kind which is defined as <math display="block" id="M13"><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mrow><mfrac><mrow><mtext>d</mtext><mi>θ</mi></mrow><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>t</mi><mn>2</mn></msup><mi>sin</mi><mi>θ</mi></mrow></msqrt></mrow></mfrac><mo>.</mo></mrow></mrow></mstyle></mrow></math> RMS frequency spread can be estimated from [ 17 17 ] <math display="block" id="M14"><mrow><msub><mi>σ</mi><mrow><mi>ω</mi><mo>−</mo><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>π</mi><msub><mi>ω</mi><mrow><mtext>s0</mtext></mrow></msub></mrow><mn>2</mn></mfrac><msqrt><mrow><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mn>6</mn></mfrac></mrow></msqrt><mfrac><mrow><msub><mi>ω</mi><mrow><mtext>rf</mtext></mrow></msub><msub><mi>σ</mi><mi>τ</mi></msub></mrow><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msqrt><mn>2</mn></msqrt><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></mrow></math> The growth rate without damping can be approximately given by [ 17 17 ] <math display="block" id="M15"><mrow><mfrac><mn>1</mn><mi>τ</mi></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><mi>η</mi><mi>e</mi><msub><mi>I</mi><mtext>b</mtext></msub></mrow><mrow><msub><mi>E</mi><mn>0</mn></msub><msub><mi>T</mi><mn>0</mn></msub></mrow></mfrac><mrow><mo>[</mo><mrow><mfrac><mrow><mn>2</mn><mi>Δ</mi></mrow><mrow><msub><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mtext>s</mtext></msub></mrow></mfrac><msub><mi>Q</mi><mrow><mtext>s0</mtext></mrow></msub><msub><mi>R</mi><mrow><mtext>s0</mtext></mrow></msub><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><msub><mi>ψ</mi><mrow><mtext>ω-s</mtext></mrow></msub><mi>sin</mi><msub><mi>ψ</mi><mrow><mtext>ω-s</mtext></mrow></msub><mo>+</mo><msub><mi>Q</mi><mtext>h</mtext></msub><msub><mi>R</mi><mtext>h</mtext></msub><msup><mrow><mi>cos</mi></mrow><mn>2</mn></msup><msub><mi>ψ</mi><mtext>h</mtext></msub><mi>sin</mi><msub><mi>ψ</mi><mtext>h</mtext></msub></mrow><mo>]</mo></mrow><mo>,</mo></mrow></math> where, the first and second terms correspond to fundamental and the third harmonic frequencies, respectively; η is the slip factor; Δ is a main RF cavity detuning frequency; Q s0 and R s0 are the quality factor and shunt impedance of main RF cavity, respectively; ψ ω-s is the detuning angle of the main RF cavity; and <math id="M16"><mrow><msub><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mi>s</mi></msub></mrow></math> is the mean angular synchrotron frequency defined as [ 17 17 ] <math display="block" id="M17"><mrow><msub><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mtext>s</mtext></msub><mo>=</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mrow><mi>ω</mi><mtext>-s</mtext></mrow></msub><mo>.</mo></mrow></math> For SSRF storage mode, the detuning is about −2.5 to −3.8 kHz. The main RF cavity shunt impedance is 15.84 MΩ. The damping time calculated from Eq.(15) ranges 0.012 ms to 0.016 ms. It can be seen that even though the harmonic resonant frequency is tuned to the Robinson unstable side, compared to SSRF longitudinal damping time of 3.5 ms, the stability from fundamental cavity is still large enough to keep the beam stable. From the stability criterion, the m th azimuthal mode will be stable if 1/ τ is close to or small than m 1/2 σ ω -s /2. [ 17 17 , 18 18 ] From Eq.(14), the synchrotron frequency spread in flat voltage condition is about 1.40 kHz for SSRF which is able to damp an instability in the dipole azimuthal mode with the growth time, τ , longer than 1.4 ms. 6 Conclusion 1 Bunch lengthening induced by a third harmonic RF cavity is studied in order to increase beam lifetime of the SSRF storage ring. The detuning angle is simulated numerically to optimize the bunch lengthening. The broad potential and distribution occurs in flattened potential case lead to an increase in Touschek lifetime with the improvement factor of about 3.77. In passive operation, the beam current is an important parameter for passive operation. The change in beam current can alter the bunch lengthening effect. The shunt impedance can be adjusted to compensate the change of beam current without affecting the phase of harmonic RF, but the detuning angle becomes alternative choice for invariable shunt impedance case,. The effect of detuning angle variation is investigated. The highest improvement factor reaches 4.16 when the detuning angle is tuned slightly above flattened potential condition. However, the improvement factor rapidly drops when detuning angle is tuned further. The limitation of the detuning angle and detuning frequency variation are estimated. Because the harmonic voltage is generated by the beam current corresponding to the bunch shape through the form factor, an increase of the bunch length can cause a decrease on improvement factor. The detuning angle can be adjusted to achieve the highest improvement factor, but it cannot give flattened potential distribution due to the distortion caused by detuning angle deviation. An effect on synchrotron frequency spread induced by third harmonic cavity in the SSRF storage ring is investigated, showing the shift and an increase in synchrotron frequency spread. This spread can help in damping longitudinal coupled bunch instability and increases stability of the beam. The instability of the beam due to Robinson unstable of the harmonic cavity is investigated, too. Although the higher harmonic cavity resonant frequency is tuned to be above the third harmonic frequency, the beam is still stable. References G.M. Liu , Z.M. Dai , H.H. Li , et al . Lattice Design for SSRF Storage Ring . High Engry Phys & Nucl Phys , 2006 , 30 , Supp. I , 144 - 146 . DOI: 10.3321/j.issn:0254-3052.2006.z1.047 http://doi.org/10.3321/j.issn:0254-3052.2006.z1.047 Lattice Design for SSRF Storage Ring B.C. Jiang , Z.T. Zhao , G.M. Liu . Study of Touschek lifetime in SSRF storage ring , High Engry Phys & Nucl Phys , 2006 , 30 , 693 - 698 . Study of Touschek lifetime in SSRF storage ring M. Boscolo . Beam Lifetime in Low Emittance Rings . Proceedings of IPAC2013 , Shanghai, China , 2013 , 1574 - 1576 . Beam Lifetime in Low Emittance Rings L.H. Chang , C. Wang , M.C. Lin , et al .,, Effects of the passive harmonic cavity on the beam bunch . Proceedings of 2005 Particle Accelerator Conference , Knoxville, Tennessee , 2005 , 3904 - 3906 . Effects of the passive harmonic cavity on the beam bunch M. Migliorati , L. Palumbo , M. Zobov . Bunch length control in DAQNE by a higher harmonic cavity . Nucl Instrum Meth A , 1995 , 354 , 215 - 223 . Bunch length control in DAQNE by a higher harmonic cavity R.A. Bosch , C.S. Hsue . Suppression of longitudinal coupled-bunch instabilities by a passive higher harmonic cavity , Proceedings of 1993 Particle Accelerator Conference . Washington, DC , 1993 , 5 , 3369 - 3371 . DOI: 10.1109/PAC.1993.309653 http://doi.org/10.1109/PAC.1993.309653 Suppression of longitudinal coupled-bunch instabilities by a passive higher harmonic cavity R.A. Bosch , K.J. Kleman , J.J. Bisognano . Robinson instabilities with a higher-harmonic cavity . Phys Rev Spec Top-AC , 2001 , 4 , 074401 . DOI: 10.1103/PhysRevSTAB.4.074401 http://doi.org/10.1103/PhysRevSTAB.4.074401 Robinson instabilities with a higher-harmonic cavity P.F. Tavares , S.C. Leemann , M. Sjostrom , et al . The MAX IV storage ring project . J Synchrotron Radiat , 2014 , 21 , 862 - 877 . DOI: 10.1107/S1600577514011503 http://doi.org/10.1107/S1600577514011503 The MAX IV storage ring project G. Skripka , P.F. Tavares , M. Klein , et al . Transverse instabilities in the MAX IV 3 GeV ring . Proceedings of IPAC2014 , Dresden, Germany , 2014 , 1689 - 1691 . Transverse instabilities in the MAX IV 3 GeV ring H. Fan , C.F. Wu , G.C. Liu , Calculations of operating schemes for a passive harmonic cavity in HLS-II , Chinese Phys C , 2012 , 36 , 895 - 899 . doi: 10.1088/1674-1137/36/9/018 http://doi.org/10.1088/1674-1137/36/9/018 Calculations of operating schemes for a passive harmonic cavity in HLS-II M. Georgsson . Landau cavities in third generation synchrotron light sources . Proceedings of the 2001 Particle Accelerator Conference , Chicago, IL , 2001 , 4 , 2689 - 2691 . DOI: 10.1109/PAC.2001.987874 http://doi.org/10.1109/PAC.2001.987874 Landau cavities in third generation synchrotron light sources M. Georgsson , A. Andersson , M. Eriksson . Landau cavities at MAX II . Nucl Instrum Meth A , 1998 , 416 , 465 - 474 . DOI: 10.1016/S0168-9002(98)00667-6 http://doi.org/10.1016/S0168-9002(98)00667-6 Landau cavities at MAX II J.M. Byrd , M. Georgsson . Lifetime increase using passive harmonic cavities in synchrotron light sources . Phys Rev Spec Top-AC , 2001 , 4 , 030701 . DOI: 10.1103/PhysRevSTAB.4.030701 http://doi.org/10.1103/PhysRevSTAB.4.030701 Lifetime increase using passive harmonic cavities in synchrotron light sources G.M. Ma , Z.T. Zhao , J.F. Liu . Design of a higher harmonic cavity for the SSRF storage ring . Chinese Phys C , 2008 , 32 , 275 . doi: 10.1088/1674-1137/32/4/007 http://doi.org/10.1088/1674-1137/32/4/007 Design of a higher harmonic cavity for the SSRF storage ring H. Fan , C.F. Wu , L. Wang . Tracking simulations for the HLS- II with a passive harmonic cavity in the symmetric and asymmetric fill patterns . Chinese Phys C , 2012 , 36 , 1111 - 1115 . doi: 10.1088/1674-1137/36/11/013 http://doi.org/10.1088/1674-1137/36/11/013 Tracking simulations for the HLS- II with a passive harmonic cavity in the symmetric and asymmetric fill patterns A. Hofmann , S. Myers . Beam dynamics in a double RF system . Proc. of the 11th Int. Conf. on High Energy Acc , CERN , Geneva, Switzerland , 1980 , 610 - 14 . DOI: 10.1007/978-3-0348-5540-2_83 http://doi.org/10.1007/978-3-0348-5540-2_83 Beam dynamics in a double RF system Ng. King-yuen . Passive landau cavity for the LNLS light source electron ring, FERMILAB-FN-0696 , 2000 . Passive landau cavity for the LNLS light source electron ring, FERMILAB-FN-0696 E. Métral . Stability of the longitudinal bunched-beam coherent dipole mode . Proceedings of APAC 2004 , Gyeongju, Korea , 2004 , 137 - 139 . Stability of the longitudinal bunched-beam coherent dipole mode 10.1007/s41365-017-0259-y **This work was supported by the National Key Research and Development Program of China (No.2016YFA0402001) 2016-07-21 2017-01-20 2017-01-24 2017-06-29 2017-08-01 2021-04-11 © 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. 2017 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/s41365-017-0259-y http://dx.doi.org/10.1007/s41365-017-0259-y . 2017 Nuclear Science and Techniques 08 28