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Lorentz Correction

The solution derived above is valid for continuous media. However, to know the field applied to the $ ~^{1}\mathrm{H}$ nuclei in MR, it is necessary to take account of the discrete nature of the media. The desired field is that applied externally to the nuclei, and can be calculated from the continuous field using the Lorentz Correction [9,8]. The corrected field is given by

$\displaystyle B_{LC} = B - \, \frac{2}{3} \mu_0 M = B - \, \frac{2}{3} \left( \frac{\chi}{1 + \chi} \right) B = \left( \frac{3 + \chi}{3 + 3 \chi} \right) B$ (15)

where $ M = \chi H$ is the magnetization of the material. Hence the zeroth and first order corrections can be calculated from the expansion of
$\displaystyle B_{LC}^{(0)} + \delta B_{LC}^{(1)}$ $\displaystyle =$ $\displaystyle \left( \frac{3 + \chi_0}{3 + 3 \chi_0} - \delta \frac{2 \chi_1}{3 (1 + \chi_0)^2} \right) \left( B^{(0)} + \delta B^{(1)} \right) + O(\delta^2)$ (16)
  $\displaystyle =$ $\displaystyle \frac{3 + \chi_0}{3 + 3 \chi_0} B^{(0)} + \delta \left( \frac{3 +...
...hi_0} B^{(1)} - \frac{2 \chi_1}{3 (1 + \chi_0)^2} B^{(0)} \right) + O(\delta^2)$ (17)

Consequently, the corrected field solution becomes

$\displaystyle B^{(1)}_{LC,z} = \frac{\chi_1}{3 + \chi_0} B^{(0)}_{LC,z}
- \frac...
...ial^2 G}{\partial z^2} \right) * \left( \chi_1 B^{(0)}_{LC,z} \right)
\right) .$     (18)

For the rest of this paper, only the Lorentz Corrected fields will be used although the $ {LC}$ subscript will be dropped. Note that the zeroth order fields are also Lorentz Corrected, which is appropriate if they are calibrated from NMR frequency results, since the frequency is determined by the corrected field strength.


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Next: Single Voxel Solution Up: Theory Previous: Theory