$D(z)=\widehat{D}(z)*\frac{1}{\sqrt{2\pi}\sigma}\mathrm{exp}\left(-\frac{{z}^{2}}{2{\sigma}^{2}}\right)=\frac{1}{\sqrt{2\pi}\sigma}{\displaystyle \underset{-\infty}{\overset{{R}_{\text{p}}}{\int}}\widehat{D}({z}^{\prime})}\mathrm{exp}\left(-\frac{{(z-{z}^{\prime})}^{2}}{2{\sigma}^{2}}\right)d{z}^{\prime}.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}(5.58)$

This leads to a rather complex, but closed analytic expression for the ‘real’ proton Bragg curve D(z), including some tabulated special functions. The Bragg peak now has a finite maximum and is in very good agreement with experimental data [12]. Here, we have carried out the convolution numerically (σ = 0.16 cm). The resulting Bragg curve is plotted in Figure 5.12 (solid line). A finite spectrum of initial energies further contributes to the broadening of realistic Bragg peaks. In radiotherapy with protons, the superposition of several quasi monoenergetic beams is employed to create a so called spread out Bragg peak with an almost homogeneous dose plateau in a certain ...

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