1. Overall consideration for ESS

The basic element of beam optics is the achromatic beam transport, which is independent of the beam energy spread. Wherever possible, the beam transport system consists of symmetrical optical components which reduce the occurrence of optical aberrations and phase space dependence, and minimize the effects of small errors in the settings of beamline components. The layout of ESS, shown in Fig. 1, is designed based on the calculation of beam optics, which consists of a fixed energy beamline (FEB), an energy degrader, two collimators, and a double bend achromatic beam transport system with an energy selection slit. The FEB is situated next to the cyclotron, and composed of four quadrupoles (QC1-QC4) with a total length of 5.2 m. The energy degrader consists of two multi-wedge graphite blocks aimed at the energy modulation through adjusting its thickness. Two collimators (COL1 and COL2) are used to control the beam emittance. A double bend achromatic beam transport system consists of two dipoles (Dipole1 and Dipole2, each bending the beam over typically 45º–65º), eight quadrupoles (QE1-QB2) and a slit, which is located behind the collimator system. And the main optics considerations about ESS are summarized as follow:

Fig. 1.Full-screen View

Schematic layout of SC200 ESS (Steering magnets and BPMs are hidden)

· A passive degrader with two multi-wedge blocks placed oppositely with each other, which is located just behind FEB.

· An energy analysis system is placed behind the degrader, which includes a symmetrical double bend achromatic section with a size-changeable energy slit

· At the matching point (MP), mirror and symmetrical round beam (x=y, x’=y’) is designed to match with beam transfer line.

· For verification of the optics and correcting orbit distortion along the ESS, a series of beam monitors for both directions (in x and y) and steering magnets are required.

2. Beam optics design

Figure 2 shows the first order transport envelopes between the cyclotron reference point and matching point (MP). The optical design for ESS is divided into two different sections.

Fig. 2.Full-screen View

(Color online) Beam optics in the ESS. The half beam size is plotted as a function of the position along the beamline. Above the x-axis, the vertical beam size is plotted as a solid line and the 1%-dispersion trajectory as a dashed line. Below the x-axis, the horizontal beam size is plotted. And at the dispersive focus both envelopes for dp/p=0%, 0.5% and 1% are drawn with blue, red and black line

· Cyclotron reference point to degrader (fixed energy beamline)

· Degrader to matching point (EAS)

These are discussed in more detail in the following section.

2.2. Energy analysis system

Behind the fixed energy beamline, a wedge graphite energy degrader is used to decrease the fixed energy proton beam to any value in the range of 70−200 MeV. Due to multiple scattering in the degrader material, beam divergence and beam size increase toward the exit of the degrader when the exit energy decreases. A collimation system for emittance matching is designed to limit the emittance of the beam leaving the degrader whose emittance can be calculated with Eq. (1).

$\epsilon =2R_1{R}_2/L$ (1)

where, R₁ is the radius of the entrance collimator, R₂ is the radius of the exiting collimator, L is the total length between two collimators, and the emittance definition of Eq. (1) is valid only when the alpha parameters are zero. The first collimator immediately following the degrader is used to control the beam size and the second collimator at a distance of approximately 1–1.5 m behind the degrader is used to control the beam divergence. Phase space is shown in Fig. 3.

Fig. 3.Full-screen View

(Color online) Phase space: (a) Before the entrance collimator (b) After the exiting collimator

After the collimator, the proton beam is focused using one focusing and one defocusing quadrupole, guided into the double bend achromatic section. It is better to design a small vertical beam envelope or a vertical beam waist lactating in the bending magnet as much as possible, so that a gap size of 60 mm is sufficient. The first bending magnet will create a large horizontal dispersion to the beam which will be enlarged using a subsequent horizontal defocusing quadrupole. The dispersion will then reach a maximum value using a focusing horizontal quadrupole, which will remain unchanged in the drift space because the derivative of dispersion is zero ($R_{26}=0$).

For a beam with a different momentum spread, the beam will be spread out across the horizontal plane. A correlation exists between the momentum of the protons and their distance to the centre trajectory, as shown in Fig. 4.

Fig. 4.Full-screen View

(Color online) The correlation between the momentum spread and the horizontal distance (x) to beam axis at slit of the energy selection system. The slope of the ellipse indicates the dispersion R₁₆ with −27.54 mm/%. The calculations have been performed by the computer code TURTLE [7] and TRANSPORT.

Therefore, a horizontal slit is set in the middle of the two horizontal focusing quadrupoles to select the required energy for clinical therapy by adjusting its aperture. The half-aperture of the slit for a different momentum spread can be calculated with equation (2).

${\sigma }_x={\sigma }_{x0}+{\delta }^2{R}_{\text{16}}{}^2$ (2)

where, $\sqrt{{\sigma }_x}$ is the half envelope of beam, $\sqrt{{\sigma }_{x0}}$ is the half envelope of beam located at the slit depend on the beam itself regardless of dispersion which can be given by the Eq.(3).

${\sigma }_{x0}=R_{11}^2{\sigma }_{x011}+2R_{11}{R}_{12}{\sigma }_{x012}+R_{12}^2{\sigma }_{X022}$ (3)

where, ${\sigma }_{x012}$=0, and $R_{12}$ should be set as zero to ensure the beam envelope is independent of the initial beam divergence; $R_{11}$, $R_{12}$ are the elements of the transfer matrix; ${\sigma }_{x011}$, ${\sigma }_{x012}$, ${\sigma }_{x022}$ are the elements of initial beam matrix. However, for beams with momentum $P_0$ and $P_0+\delta P$, the centre distance of such beams can be calculated with equation (4).

$x_{\eta }=R_{16}\delta$ (4)

It is obvious that $x_{\eta }$ must be more than 2$\sqrt{{\sigma }_{x0}}$ to separate them completely. Therefore, the minimum momentum spread can be obtained by equation (5).

${\frac{\Delta P}{P}}_{\min }={\delta }_{\min }=\frac{2\sqrt{{\sigma }_{x0}}}{R_{16}}=\left|\frac{R_{11}}{R_{16}}\right|*2\sqrt{{\sigma }_{x011}}$ (5)

According to the final design, the momentum spread can reach a range about 0.1% to 1% for SC200 ESS, which is equal to the energy spread of about 0.193%-1.93% for a proton beam calculated with equation (6).

$\frac{\Delta P}{P}=\frac{E_{\text{K}}+E_0}{E_{\text{K}}+2E_0}\frac{\Delta E_{\text{K}}}{E_{\text{K}}}$ (6)

where $E_{\text{K}}$ is the kinetic energy, $E_0$ is the rest energy of proton, $\Delta E_{\text{K}}/E_{\text{K}}$ is the energy spread, and $\Delta P/P$ is the momentum spread. Eventually, the maximum half-aperture of the slit is set as 28 mm for a maximum of 1% relative beam momentum spread. Meanwhile, all the above elements are aimed at forming a double achromatic system (R₁₆=R₂₆=0) to make the beam transport almost independent of the beam energy.

3. Beam optics optimization

Fig. 5a shows the beam optics of different initial beams with the same emittance. The result shows that the beam envelopes will decrease when the aperture of the first collimator increases. Table 2 shows the detailed parameters at the slit for different collimation systems, which represent the different initial beam, are calculated based on Eq. (1). It can be seen that the aperture of the second collimator will decrease when the aperture of the first collimator increases. The result also indicates that a smaller aperture will have a higher momentum spread precision. Therefore, it is better to set a smaller aperture after the wedge degrader, which has higher transmission efficiency as shown in Fig. 5b. An important aspect to consider is the location of the maximum beam dimensions, in other words, the location for possible beam losses. In the final design of ESS for SC200, the configuration of R₁=1.2, R₂=8 is used for an emittance of 16 πmmmrad. The others can be calculated by equation (1).

Fig. 5.Full-screen View

(Color online) (a) Beam envelope for a different initial beam with the same emittance (b) Transmission efficiency

Fig. 6 (a) shows the beam optics of different emittances from 8 to 24 π mm mrad with the same Twiss parameter. It is obvious that the beam envelope increases with beam emittance regularity, and can be calculated with equation (7).

Fig. 6.Full-screen View

(Color online) (a) Beam envelope for different emittance (b) Transmission efficiency

This also shows that the dispersion remains unchanged, but the beam size at the slit which increases with the emittance will lead to an enlarging of the minimum momentum spread according to Eq.(3). It also can be seen that the transmission efficiency will increase with the emittance or energy as shown in Fig. 6 (b). Therefore, the commission model will be determined by the requirement of beam size, transmission efficiency and minimum momentum spread during patient treatment. A smaller beam size and higher beam transmission efficiency can be realized with the proton beam of higher energy and lower emittance. That is to say, it will be difficult to have a small beam size and high beam intensity at low energy level.

4. Analysis of transmission efficiency

The ESS transmission efficiency has been analysed using a Monte-Carlo simulation computer code LISE++ ^[8], and characteristics of the particle beam should be determined once the initial design is completed and the number of protons tracked is 10⁶. In the ESS beamline, there are large beam losses at such points as the degrader, emittance matching, and energy selection as shown in Fig. 7 (a) and (b). Therefore, the degrader to the energy slit section will be the most activated region of the ESS beamline. Some local radiation protection calculation (e.g. shielding wall thicknesses, labyrinths, activations, etc) must be taken into consideration. Fig. 7 (c) shows the contribution of the individual components degrader, collimator, and slit system to the loss percentage. The transmission efficiency after the energy slit is strongly dependent on the chosen kinetic energy of the protons, which decreases when the beam energy degrades, and also shows there is almost no beam loss after the slit. As described above, the dose rate of all sources will be determined at relevant locations by assuming certain operating parameters. For example, the most conservative assumption that can be made is as follows: Irradiation shall be given at 70 MeV. Since the cyclotron produces a beam with E=200 MeV, the beam must be decelerated in the degrader to 70 MeV. Due to the low transmission at 70 MeV, it needs to be further assumed that the beam current from the cyclotron is at its maximum, that is, up to 500 nA.

Fig. 7.Full-screen View

(Color online) (a) Beam trajectory for all particles. (b) Beam trajectory only transport through ESS. (c) Transmission efficiency at the degrader, collimator, slit system and MP. (d) Transmission efficiency for different momentum spread after the slit.

An investigation into the transmission efficiency at the slit is introduced in detail. After the degrader, the beam momentum spread distribution for all used energy will obviously be enlarged. A horizontal slit is used to select the required proton and stop others. The relationship between dp/p and the transmission efficiency at the slit is shown in Fig. 7 (d). Transmission efficiency increases with the momentum spread for a 70-185MeV proton beam. Consequently, only 0.204% of the beam intensity from the SC200 cyclotron reaches the isocentre of the beam delivery system of the nozzle when proton beams degrade from 200 to 70 MeV, with an acceptance of 24π mm.mrad in each transverse plane and a momentum acceptance of ±0.6%.

5. Orbit Distortion Correction

Due to the deviations introduced by the magnetic field errors and the misalignments, a series of beam profile monitors for both directions (in x and y) with single steering magnets are used to verify the beam optics and correct the centre trajectory of the proton beam along ESS. In this paper, orbit distortion correction has been calculated with a response matrix and SVD algorithm based on MADX code ^[9-11]. A statistical analysis simulating 1000 different trajectories with random errors have been carried out before and after the correction. The layout of ESS with six beam position monitors and ten steering magnets is shown in Fig. 2. The position of the beam profile monitors are marked with vertical lines pointing upwards and having labels like BPMn. The positions of the steering magnets required for centring the beam are marked similarly with vertical lines pointing downwards and having labels like Stxn/Styn. The distributions of the RMS orbit distortion (OD) with respect to ESS are evaluated before and after correction as shown in Fig. 8. It is obvious that the beam position at the slit has an extremely high precision after correction as shown in Fig. 9 at about 0.15 mm. Fig. 10 shows the distributions of the maximum absolute corrector strength for both horizontal and vertical planes at top energy (E=200 MeV). On the basis of the error analysis for ESS, the main specifications for steering magnets are summarized in Table 3.

Fig. 8.Full-screen View

(Color online) Excursion distributions of ESS (a) XCORMS before correction (b) XCORMS after correction (c) YCORMS before correction (d) YCORMS after correction

Fig. 9.Full-screen View

(Color online)(a) Excursion distributions at the slit before correction (b) Excursion distributions at the slit after correction

Fig. 10.Full-screen View

(Color online) Maximum absolute corrector strength (a) Horizontal corrector strength (b) Vertical corrector strength