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Energy degrader optimization for medical beam lines

This paper describes the optimization of a variable energy degrader design for the Midwest Proton Radiotherapy Institute (MPRI) [1]. To optimize the energy degrader design we investigate the choice of an optimal material for the degrader, the beam emittance growth in the degrader, and the matching of the degraded beam with the acceptance of a medical beam line.


Introduction
Many practical applications of particle accelerators require varying the beam energy in an experimental beam line without changing the settings of the accelerator. Installing a variable thickness energy degrader provides a fast, reliable and reproducible way of setting the beam energy, which is especially important for medical applications. Several proton radiation treatment centers including MPRI [1] and NPTC [2] plan on using variable energy degraders to set the beam energy for radiation treatments. A typical setup includes a variable thickness degrader followed by a momentum selection beam line that transfers the beam to a treatment room. Multiple scattering in the energy degrader causes beam emittance growth and a rotation in phase space, which results in a mismatch of the degraded beam with the acceptance of the momentum selection beam line. In order to reduce beam losses in the energy degrading process it is important to minimize both effects. In this paper we investigate an optimal configuration for the degrader including the choice of material and geometry. An excellent review of various energy degrader configurations in medical beam lines is given in [3]. We consider three generic options for the variable thickness degrader geometry (shown in Figure 1): a wedge with a flat front face, a wedge with a flat exit face, and a wedge with a fixed center of gravity (or double wedge system). This study expands the analysis of the degrader to beam line matching performed at NPTC [4].

Emittance growth in energy degrader
The beam emittance is determined by the rms size and angular spread of the beam. Both the beam spot size and the angular spread grow due to multiple Coulomb scattering in the degrader material. The lateral displacement of a scattered particle accumulates as an integral of the multiple scattering angle over the degrader thickness. Empirically, it is proportional to the rms multiple scattering angle and to the material thickness L [5]: With a typical thickness of an energy degrader being of the order of a few centimeters, the angular scattering is always an order of magnitude larger than the lateral scattering, and therefore, its contribution dominates in the emittance growth. The effect of the multiple scattering to the beam emittance is illustrated in Figure 2. The rms multiple scattering angle in one plane is given by [5]: where βc, z and p are velocity, charge and momentum of the incident particle and L0 is the radiation length in the degrader material of thickness L. Multiple scattering is statistically independent from the beam initial conditions. Therefore, multiple scattering angle should be added in quadrature to the rms beam angular spread To express the beam emittance growth due to multiple scattering we will use the following emittance parameterization: (4) where α, β, and γ are the optical functions describing beam envelope evolution along the beam line. Using equations (3) and (4) and neglecting the contribution of the lateral scattering, one can conclude that the beam emittance increase after a thin degrader is proportional to the beta-function and the mean square multiple scattering angle: The beta-function describes the beam spot size. Therefore, the emittance growth is minimal when the beam is focused to a waist in the middle of the degrader. This is illustrated in Figure 3.

Energy degrader material
As we saw in the previous section, the main contribution to the beam emittance growth comes from the angular spread induced by the multiple scattering in the degrader material. The amount of angular scattering and, thus, the increase in the beam emittance depends on the degrader material. Here we will study how the multiple scattering angle varies for different materials when their thickness is set to provide the same amount of energy degradation. For proton beam energies below 250 MeV the Bethe-Bloch equation expressing material stopping power can be written ignoring the density correction term, which is important only above 1 GeV: where K = 0.307 MeV⋅cm 2 /g. Expressed in the units of mass thickness, stopping power varies very little over a wide range of materials. The dependence on the material properties only comes in the atomic number to atomic weight ratio (Z/A ≈ 0.5) and in a weak logarithmic dependence on the material's mean excitation energy I(Z) which is tabulated for different materials in [7]. Thus, different energy degraders with the same mass thickness (ρdx) will cause similar amount of energy degradation (dE).
Next, we estimate how the multiple scattering angle varies in materials of the same mass thickness. Equation (2) indicates that the multiple scattering angle is determined by the ratio of the material thickness to the material radiation length. The radiation length for different materials can be parameterized in the same units as mass thickness [5]: Equation (7) clearly indicates that the low Z elements have larger radiation length and, therefore, smaller multiple scattering angle. In the following table we quote radiation length for different materials that could be used for degrading the beam energy. Note, that while lithium appears to have the largest radiation length, its usage is limited by very low melting temperature. Therefore, the best practical degrader material is beryllium.

Energy degrader matching to the momentum selection line
Next, we discuss how the degrader geometry effects the beam emittance matching to the subsequent beam line. If orientation of the degraded beam emittance is tilted with respect to the beam line acceptance, then only a fraction of the beam will be transmitted through the beam line even if the areas are similar.
To investigate the problem we consider the beam emittance evolution through three different degrader configurations shown in Figure 1. The initial conditions are the same for all degrader configurations. We assume that the beam is focused to a waist at z=0.
We then calculate the beam emittance as a function of the degrader thickness at z=L, which corresponds to the maximum degrader thickness. There are two effects that should be taken into account: 1) evolution of the beam emittance through drift space and 2) emittance growth in the degrader. Figure 4 illustrates evolution of the beam emittance through a drift space, while Figure 2 shows the effect of multiple scattering in a degrader to the beam emittance. The beam emittance starts with the upright orientation at the focal point (z=0). The drift space "tilts" the beam emittance in the phase space, while multiple scattering in the degrader "straitens" the beam emittance orientation. This qualitative analysis leads to a conclusion that drift space followed by energy degrader is preferable configuration to keep the beam emittance orientation constant at the entrance to the momentum selection beam line. This corresponds to the degrader geometry with a flat exit.
Next, we perform a more rigorous quantitative analysis. We will use the sigma transport matrix notation [8] to describe the beam emittance evolution.
Then, the emittance evolution in a drift space of length L can be written as,

Conclusions
We can draw several conclusions from this study of performance optimization for a variable energy degrader. First, beryllium appears to be an optimal material for the energy degrader. For the same amount of energy degradation, beryllium causes the least amount of multiple scattering. Second, it appears to be very important to focus the incoming beam to a double waist in the middle of the degrader. Focusing the beam both vertically and horizontally minimizes the amount of emittance growth in any given energy degrader. Therefore, an optimal degrader performance would be achieved in a geometry with fixed mid-point (such as double wedge system). Finally, the emittance matching to the subsequent momentum selection beam line appears to be less critical, although a degrader with flat exit face offers somewhat better beam matching.
The results of this study have been integrated into the energy degrader design for the MPRI project [1]. As a part of the MPRI project, this study has been supported by a construction grant from the State of Indiana. The author is grateful to B. Gottschalk and N. Schreuder for raising questions that inspired this study.  Beam emittance after degrader Beam emittance before degrader