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Calculation of the diamagnetic tensor of the water molecule


The quality of radiotherapy treatment can be increased by visualizing the tumour volume during the irradiation (so-called "image-guided radiotherapy", IGRT). Facilities for image-guided radiotherapy are currently being developed at different places where MRI is used as an imaging method. Within the scope of the EMRP JRP "MRI safety" [1], the influence of the magnetic field on the interaction between radiation and matter is, among other things, being investigated – as is dosimetry.

The current study thus aims to determine the polarization of the water molecules by the external magnetic field. Since the water molecule is not rotationally symmetric, the (dia)magnetic moment induced by the magnetic field depends on the orientation of the water molecules and, thus, on the Boltzmann factor for this orientation. Hereby, the inertia tensor of the electron distribution in the molecule appears in the energy term.

The components of the inertia tensor were calculated as the moments of the coordinates by means of the integral below:

E(tm) = ∫∫∫ tm p(x,y,z)dxdydz

Hereby t∈{x, y, z} is m∈{0, 1, 2} and the function p(x,y,z) corresponds to the three-dimensional probability density distribution of the electrons in the water molecule. It takes the five molecular orbitals of water into account, each of which is expressed as a linear combination of atomic orbitals. The quantum-chemical program "Gaussian09" was used to determine the coefficients of the linear combinations by means of the self-consistent field procedure. Hereby, each atomic orbital itself was described as a linear combination of Gaussian functions which are centred onto the respective atom. To this end, we used the so-called split-valence, triple-zeta Gaussian basis set 6-311G. Based on its mathematical representation, analytical expressions for E(tm) were derived and implemented in MatLab for the numerical evaluation.

The results were used to determine the energy shift of the oriented water molecules in the magnetic field and the corresponding Boltzmann distribution which can then be used to calculate the expectation and the standard deviation of the differential electron scattering cross-sections.


  1. Opens external link in new windowhttp://www.ptb.de/emrp/mri.html