3.1 Beam spectrum

The amplitude and frequency of the main lines of the beam current spectrum directly affect the probability and intensity of HOMs in SRF accelerating structures.

Two cases of the beam in the SHINE are considered in this paper: Case 1 and Case 2. In Case 1, the bunch repetition frequency is assumed to be constant and equal to 1 MHz, and total change Qb = 300 pC. In Case 2, the bunch repetition frequency is assumed to be constant and equal to 0.6 MHz, and total change Q_b = 100 pC. The calculated beam spectra of Case 1 and Case 2 are shown in Fig. 2.

Fig. 2Full-screen View

Idealized beam spectrum of Beam Cases 1 and 2

3.2 Cavity HOM spectrum

The spectra and impedance of the higher-order modes in the SHINE are evaluated in this section. The main contribution of HOM loss comes from the frequency mode with the highest impedance and a frequency approaching the main line of the beam spectrum.

The HOM spectrum of a 1.3-GHz Tesla-type cavity is evaluated by using a CST simulation [10]. For longitudinal monopole modes, the cutoff frequency in a cylindrical pipe is defined as $f_{\text{cutoff}}=2\pi c\frac{X_{01}}{r}\approx 2\pi c\frac{2.4048}{r},$ where $X_{01}$ is the first root of J₀(r), the Bessel function of the first kind of order 0. According to the above formula, the cutoff frequency of a 1.3-GHz Tesla-type cavity is equal to 2.94 GHz. The spectrum and R/Q of a 1.3-GHz Tesla-type cavity are shown in Fig. 3a and Fig. 3b, respectively.

Fig. 3Full-screen View

(a) Spectrum and (b) impedance of monopole HOMs in 1.3-GHz Tesla-type cavity

3.3 Power loss calculation

The beam traversing a cavity excites various modes of the cavity. The main accelerating mode will be compensated, but there are also high-order modes in the cavity. The excitation of HOMs leads to losses of beam power. In addition, it is necessary to know which modes are used to evaluate these losses. Given the shapes of the cavities, all needed information can be obtained by using a CST simulation. After that, power losses are calculated as described below.

According to Ref. [11], the surface resistance of superconducting Nb is

where R_res=10 nΩ, and the BCS part is parameterized as

Trapped modes may have a higher value of Q_h [12], where Q_h is the loaded quality factor of the HOMs. The following Q_h values are used in our analysis: Q_h = 2×10⁵, 1×10⁶, and 1×10⁷. The HOM frequency spread owing to the manufacturing mechanical tolerances will be considered in our simulation. The total power loss in the cavity walls is calculated as the sum of losses by the individual beam harmonics.

The power losses can be calculated as [13]

where

3.4 Results

To accurately estimate the probability of resonance HOM excitation in a SHINE linear accelerator by the beam component, a statistical analysis must be carried out. This analysis requires the propagation of data for the HOM parameters (frequency, impedance, and quality factor). Thus, the manufacturing mechanical tolerances will be taken into account in order to obtain this information. The acceleration mode should be perfectly adjusted so a frequency of exactly 1.3 GHz is used in calculations. About 3000 random runs are made for each cavity in order to estimate the probability of the RF losses per cryomodule [14].

The distributions of power loss for the two beam cases in a 1.3-GHz Tesla-type cavity are shown in Fig. 4a and Fig. 4b, respectively.

Fig. 4Full-screen View

Calculations of power losses in 1.3-GHz Tesla-type cavity for (a) Beam Case 1 and (b) Beam Case 2

4.1 HOM power generated in SHINE linac

When an electron bunch traverses the 1.3-GHz cryomodule shown in Fig. 2, its wake energy is radiated into modes (untrapped longitudinal HOMs) that are much higher than the cutoff frequency. The primary source of excitation of untrapped longitudinal HOMs in the SHINE is the irises of nine-cell cavities (called a geometric wake). In addition, another source is generated by resistive wall wakefields in the 1.3-GHz cryomodule beam pipe of the SHINE.

The HOM power generated by the beam is

In the SHINE, the maximum HOM power is generated for Case Q_b = 300 pC and f_rep = 1 MHz, and the minimum HOM power is generated for Case Q_b = 100 pC and f_rep = 0.6 MHz. Thus, the above two cases will be considered in this section.

According to Eq. (5), we need a K_loss parameter, which is defined by the following equation:

where q is the bunch charge, $W_{\parallel }(s)$ is the longitudinal wake potential, and λ(s) is the bunch distribution, which usually has a Gaussian distribution [18].

The longitudinal wake potential is a convolution of a longitudinal wake function and bunch distribution is described in the following equation:

where $w_{\parallel }(s)$ is the wake function of a pointlike charge.

In a periodic structure, a short-range wake can be approximated by [8, 19]

The 1.3-GHz SHINE linac can be considered a multiperiodic structure: the first elementary period is the cavity cell, the second is the nine-cell cavity with a bellows and beam tubes, and the third is the cryomodule, which houses eight cavities with nine bellows. Thus, Eq. (8) can be used in our calculation.

4.2 Steady-state wake losses

When the beam enters the first cryomodule in a string, it will first encounter transient wakefields that gradually change to steady-state wakes. The change occurs over a distance on the order of the catchup distance, $Z_{\text{cu}}=a^2/2{\sigma }_{\text{z}}$. For the SHINE, the catchup distance Z_cu=0.2, 1.5, and 29 m and Z_cu= 0.6, 4.4, and 87.5 m in the L1, L2, and L3 sections for Beam Case 1 and Beam Case 2, respectively. According to Fig. 2, the length of a 1.3-GHz cryomodule is about 12 m. To reach a steady-state solution, 1, 1, and 3 cryomodules need to be considered in the L1, L2, and L3 sections for Beam Case 1. Similarly, 1, 1, and 8 cryomodules need to be considered in the L1, L2, and L3 sections for Beam Case 2. Therefore, the wake function is fitted by calculating the wake potentials of different σ_z bunches in the chain of eight cryomodules with ECHO [20], which is shown in Fig. 5 [21].

Fig. 5Full-screen View

Longitudinal wake potential and fitted wake function in eighth cryomodule

From the fit of the numerical data to Eq. (8), we obtain the following equation:

In this section, Beam Case 1 and Beam Case 2 in the L3 part are primarily considered in our calculations. The wake potential is calculated by the convolution of the longitudinal wake function and Gaussian bunch distribution shown in Fig. 6a and Fig. 6b for Beam Case 1 and Beam Case 2, respectively [22].

Fig. 6Full-screen View

Wake potential and bunch distribution for a bunch length of ${\sigma }_{\text{z}}=21\text{ μm}$ for (a) Beam Case 1 and (b) ${\sigma }_{\text{z}}=7\text{ μm}$ for Beam Case 2 owing to 1.3-GHz cryomodule

With the above wake potential, K_loss can be calculated according to Eq. (6). For an RMS bunch length of 1 mm, 0.14 mm, and 7μm in L1, L2, and L3, the loss factor is 89, 137, and 169 V/pC per cryomodule, respectively. The steady-state HOM power generated for Q_b = 100 pC and f_rep = 0.6 MHz (Beam Case 2) is 0.54, 0.82, and 1.01 W/Cryomodule, respectively.

For an RMS bunch length of 3 mm, 0.42 mm, and 21 μm in L1, L2, and L3 for Beam Case 1, the loss factor is 57, 113, and 162 V/pC per cryomodule, respectively. The steady-state HOM power generated for Q_b = 300 pC and f_rep = 1 MHz for Beam Case 1 is 5.12, 10.17, and 14.55 W/Cryomodule, respectively.

4.3 Transient wake losses

As mentioned in section 4.2, the beam wake reaches a steady-state wake in sections L1 and L2 as long as the beam traverses one cryomodule owing to the catchup distance. To reach a steady-state solution, however, a structure of eight cryomodules with a total length of 96 m needs to be considered when the beam traverses the L3 section. Thus, transient wake losses generated by a 7-μm beam in L3 are the focus of our calculations.

When the 7-μm beam enters the first cell of the first cavity of the first cryomodule in L3, the wake induced is well approximated by a diffraction model. In subsequent cells and cavities, the wake gradually reaches its steady state. According to the diffraction model, the loss factor for a Gaussian bunch traversing the first cell of a cavity is given by [5]

where g is the cell gap. For a 1.3-GHz cryomodule in L3, a = 35 mm (the cell iris radius), g = 89 mm, and ${\sigma }_{\text{z}}=7\text{ μm}$. Using Eq. (10), we find that k_cell=18.9 V/pC is the contribution of the first cell. For the first cavity, the diffraction model gives this value multiplied by the number of cells in a cavity: k_cavity=170.1 V/pC.

To estimate the loss factor, k_n, we consider the model [5]

where m is the cavity number, and p is the cell number. ${\overline{k}}_{\text{tr}}$ and ${\overline{k}}_{\text{ss}}$ are, respectively, the transient and steady-state cavity loss factors, and ${\alpha }_{nmp}=[72(n-1)+9(m-1)+p-1]/9d_{\text{c}}$, where $d_{\text{c}}$ is the declination per cavity of the transient component. The transient loss factor ${\overline{k}}_{\text{tr}}$ is given by the diffraction Eq. (10), where ${\sigma }_{\text{z}}=7\text{ μm}$, and is then multiplied by 9 (the number of cells in a cavity) so that the value of ${\overline{k}}_{\text{tr}}$ is 170.1 V/pC per cavity.

${\overline{k}}_{\text{ss}}$ is taken from section 4.2 (Steady State Wake Losses). $K_{\text{loss}}$ = 169V/pC per cryomodule. Divided by 8 (the number of cavities in a cryomodule), the value of ${\overline{k}}_{\text{ss}}$ is 21.13 V/pC per cavity. Note that the declination d_c is equivalent to a distance of 8.93 m, which is much shorter than the catchup distance Z_cu=87.5 m, the distance after which the wake experienced by the beam is within a few percent of the steady-state wake.

Taking ${\overline{k}}_{\text{tr}}=170.1\text{ V/pc}$ (from the diffraction model), ${\overline{k}}_{\text{ss}}=21.13\text{ V/pc}$ (from the steady-state section above), and d_c=8.93 m, we obtain the result that for the first 10 cryomodules, the loss factor and power losses by the 100-pC beam at 1 MHz are shown in Fig. 7a and Table 2.

Fig. 7Full-screen View

K_loss and transient wake power of nth cryomodule for (a) ${\sigma }_{\text{z}}=7\mu \text{m}$ and (b) ${\sigma }_{\text{z}}=21\mu \text{m}$ bunches

It is known from Fig. 7a and Table 2 that K_loss is unchanged from the eighth cryomodule of L3, and K_loss is decremented from the first cryomodule to the eighth cryomodule. That is to say, after the beam traverses the eighth cryomodule, it reaches the steady state.

For completeness, the calculations for the beam passing through the initial cryomodules of L1 and L2 in Beam Case 2 are repeated. We find that in L1, the loss in the first cryomodule is 0.54 W, and the result for all the others is 0.54 W. In L2, the loss in the first cryomodule is 0.86 W, and the result for all the others is 0.82 W.

Similarly, the loss factor and power losses in L3 of Beam Case 1 are calculated again in the same way as shown in Fig. 7b and Table 3. For completeness, the calculations for the beam passing through the initial cryomodules of L1 and L2 in Beam Case 1 are repeated. We find that in L1, the loss in the first cryomodule is 5.13 W, and the result for all the others is 5.12 W. In L2, the loss in the first cryomodule is 10.32 W, and the result for all the others is 10.17 W.

4.4 HOM power spectrum

In order to understand the distribution of power among the modes, the HOM power spectrum can be evaluated using the following Equation [9]:

where $Z_{\parallel }$ is the longitudinal wake impedance. $Z_{\parallel }$ can be obtained by performing a Fourier transform on the wake function. Fig. 8 shows the differential HOM power spectra for Beam Cases 1 and 2 in the 1.3-GHz cryomodule of the SHINE linac.

Fig. 8Full-screen View

Differential power spectrum about Beam Case 1 (blue) and Beam Case 2 (red) in 1.3-GHz cryomodule of SHINE linac

The frequency range of the excited modes can be approximated as

In the SHINE linac, the bunch length is as short as 7 μm, and the spectrum of the HOM frequency extends up to Terahertz. In order to estimate the fraction of HOM power dissipating at 2 K in the cryomodule, the total HOM power can be determined as the sum of two parts, i.e.,

where the first term is the power below the cutoff frequency (${\omega }_{\text{c}}$). This power is effectively extracted from the operating environment using HOM couplers. The second term in Eq. (14) corresponds to HOMs that have frequencies above the cutoff frequency and is called the untrapped HOM power. According to Eq. (12) and (14), the untrapped HOM power of Beam Case 1 for L1, L2, and L3 is 3.47, 8.34, and 12.95 W/Cryomodule, respectively. Similarly, the untrapped HOM power of Beam Case 2 for L1, L2, and L3 is 0.43, 0.70, and 0.938 W/Cryomodule, respectively.

4.5 Power loss generated by resistive wall wakefields in cryomodule beam pipe

Another source of beam power loss is the resistive wall wakefields in the 1.3-GHz cryomodule beam pipe of SHINE. Because the beam pipe is maintained at cryogenic temperatures, resistive wall effects are typically weak or, more commonly, they enter the anomalous skin effect regime (ASE) where the AC conductivity of metals is substantially different from the normal skin effect (NSE). The beam pipe of the 1.3-GHz cryomodule in SHINE is round. The impedance in this case is in the form of Ref. [17]:

where a is the inner radius of the beam pipe, and $\zeta (k)$ can be known from Ref. [17]. From Eq. (15), we obtain the real and imaginary parts of the impedance as a function of the frequency for copper and for a pipe of radius a = 39 mm, which is shown in Fig. 9. Then, according to Eq. (12), the power density spectra for the resistive wall wakefields in Beam Cases 1 and 2 in the 1.3-GHz cryomodule beam pipe of the SHINE linac are calculated and shown in Fig. 10.

Fig. 9Full-screen View

Real (red lines) and imaginary (blue lines) parts of impedance for copper (the residual resistivity ratio (RRR) = 30 [17])

Fig. 10Full-screen View

Differential power spectrum about Beam Case 1 (blue) and Beam Case 2 (red) in 1.3-GHz cryomodule beam pipe of SHINE linac

4.6 Diffusion model

One way to estimate the distribution of the HOM power absorption is to use a diffusion-like model [9] in which the radiation fills the available volume like a gas. This is a good approximation that is well above the cutoff frequency and where the surface reflection coefficient is close to unity.

The power absorption is proportional to the impedance of the element:

where S_i is the surface area of the ith element, i is the type of element (such as a cavity, bellows, or beam pipe), n is the number of elements, ω is the angular frequency, $\frac{\text{d}p(\omega )}{\text{d}\omega }$ is the HOM spectral density, and $\mathrm{Re}(Z_i(\omega ))$ is the real part of the surface impedance.

The power absorbed by the ith-type element is

In the end, we have to count the total power loss that contains the power loss P_{geom_wake} of the untrapped longitudinal HOMs generated by the geometric wakefield and the heat load P_{resis_wake} of the resistive wake generated by the resistive wall of the beam pipe.

4.7 Surface impedance of elements

The surface impedance of a superconducting Nb cavity can be found in Eq. (1) in section 3.3. The surface impedance of steel is computed using

where k is the electrical conductivity. We used k=10⁶Ω^-1m^-1for stainless steel.

Copper exhibits anomalous effects at 4 K at higher frequencies:

where $A=3.3\times {10}^{-10}\Omega {\text{s}}^{2/3}$.

The surface impedance of the absorber, R_absorber, can be calculated in the following way [23]:

where α is the attenuation coefficient in a lossy medium; ${\xi }_{\text{r}}^{\text{'}}$ and ${\xi }_{\text{r}}^{\text{'}\text{'}}$ are the real and imaginary parts of the relative electrical permittivity, respectively; and ${\mu }_{\text{r}}^{\text{'}}$ is the relative permeability. Aluminum nitride ceramic is chosen as the absorber material. The loss tangent for this material is larger than 0.4, and ${\xi }_{\text{r}}^{\text{'}}$ < 30.

Calculation formulas for the equivalent impedance of the radiation elements are derived from [24]:

The equivalent surface impedances for all of the beam line components can be obtained by solving Eq. (21) above. The result is shown in Fig. 11 with a comparison of the impedances of the stainless steel, copper, and superconducting cavity.

Fig. 11Full-screen View

Equivalent surface impedances of beamline components and impedance of cavity and copper and steel materials

4.8 Distribution of untrapped radiation in 1.3-GHz cryomodule

Using Eq. (16) and (17), the dependence of the surface impedance on the frequency and HOM differential power spectrum, and the distribution of the total power loss (including the power loss P_{geom_wake} of the untrapped longitudinal HOMs generated by the geometric wakefield and the heat load P_{resis_wake} of the resistive wake generated by the resistive wall of the beam pipe), in every element of the 1.3-GHz cryomodule are shown in Fig. 12a and Fig. 12b for Beam Cases 1 and Case 2, respectively.

Fig. 12Full-screen View

Distribution of total power losses in 1.3-GHz SHINE cryomodule L3 part for (a) Beam Case 1 and (b) Beam Case 2 with radiation loss and no radiation loss, respectively

From Fig. 12, it can be observed that most of the power is deposited to the beam line absorber placed outside the operating temperature of 2 K. Only a small fraction is dissipated at 2 K, and this is distributed among different elements (such as the cavity, flanges, and bellows) when there is no radiation loss to the coupler ports. If the radiation loss is considered, then the power losses are uniformly distributed between the fundamental power coupler (FPC) and the beam line absorber.