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Influence of an external magnetic field on electron scattering cross sections for oriented water molecules


In radiation therapy, the attempt is being made to apply a high radiation dose to a tumour volume to kill the tumour cells or to impede their growth, whereby the healthy tissue of the organs at risk should be protected to the greatest possible extent. To increase the quality of the radiotherapy treatment, it is helpful to be able to visualize the target volume and the surrounding tissue during the irradiation. For this purpose, instruments are currently being developed which advantageously combine a linac with a magnetic resonance tomograph. In this context, the question arises as to which influence the magnetic field of the magnetic resonance tomograph has on the interaction between the radiation and the tissue. In a first approach, the influence of a magnetic field on the electron ionization cross sections for the elastic scattering and the ionization on water molecules was investigated. This study, which was carried out within the scope of the EMRP JRP "MRI safety" [1], was aimed at finding out whether the scattering ionization cross sections would have to be modified in a conventional condensed-history Monte Carlo simulation program for dose calculation if the radiation transport were simulated in a magnetic field. To this end, the mean values of the differential electron scattering cross sections were calculated with and without a magnetic field, and then compared with each other.

In order to determine these quantities, the differential cross sections for electron scattering were, in a first step, calculated theoretically on oriented water molecules. The orientation of the water molecule was defined by means of the Euler angles (α,β,γ) which describe a sequence of rotations around the axes of the water molecule and calculate the differential electron cross sections for different values of the Euler angles from 0° to 360°. The differential cross sections for the elastic scattering were calculated by means of the Independent Atom Model (IAM) [2], whereas for the ionization, an approach was used which is based on the first Born approximation [3]. Examples of the results obtained are shown in the figure; they demonstrate the strong dependence of the scattering cross sections on the Euler orientation angles. The mean value of the ionization cross section without a magnetic field was determined by an integration over the three Euler angles.

In a second step, the presence of a magnetic field was taken into account. The magnetic field leads to an energy difference between different orientations of the water molecules with regard to the magnetic field. To determine the mean value of the differential cross section in the magnetic field, the statistic distribution of the relative number of molecules with a specific orientation was taken into account by means of a Boltzmann factor.

The results of the calculations show that there are no significant differences between the mean values of the differential cross sections for oriented water molecules with and without a magnetic field. As a consequence, a modification of the ionization cross sections in the Monte Carlo simulation program for dose calculation is not necessary if a magnetic field is to be taken into account in the simulation.


Figure: Differential cross sections for the elastic electron scattering on an oriented water molecule as a function of the Euler angles β and γ for scattering angles of 30° (left) and 90° (right) and for different energies: 50 eV (a and b), 200 eV (c and d).


  1. www.ptb.de/emrp/mri.html
  2. M. Fink, A.W. Ross and R. J. Fink, Z. Phys. D 11, 231-238 (1989)
  3. C. Champion, J. Hanssen and P. A. Hervieux, Phys. Rev. A 63, 052720